# Global solvability and convergence to stationary solutions
in singular quasilinear stochastic PDEs

Tadahisa Funaki∗ and Bin Xie⋆

###### Abstract

We consider singular quasilinear stochastic partial differential equations
(SPDEs) studied in [7], which are defined in paracontrolled sense. The main
aim of the present article is to establish the global-in-time solvability for
a particular class of SPDEs with origin in particle systems and, under a
certain additional condition on the noise, prove the convergence of the
solutions to stationary solutions as $t\to\infty$. We apply the method of
energy inequality and Poincaré inequality. It is essential that the Poincaré
constant can be taken uniformly in an approximating sequence of the noise. We
also use the continuity of the solutions in the enhanced noise, initial values
and coefficients of the equation, which we prove in this article for general
SPDEs discussed in [7] except that in the enhanced noise. Moreover, we apply
the initial layer property of improving regularity of the solutions in a short
time.

†† ∗Department of Mathematics, School of Fundamental Science and Engineering,
Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan. e-mail:
<EMAIL_ADDRESS>
⋆Department of Mathematical Sciences, Faculty of Science, Shinshu University,
3-1-1 Asahi, Matsumoto, Nagano 390-8621, Japan. e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
##  1 Introduction

We studied in [7] the following quasilinear stochastic partial differential
equation (SPDE) defined in paracontrolled sense

(1.1) $\partial_{t}u=a(\nabla u)\Delta u+g(\nabla u)\cdot\xi,$

on one dimensional torus ${\mathbb{T}}\simeq[0,1)$ having the spatial noise
$\xi\in C^{\alpha-2},\alpha\in(\frac{4}{3},\frac{3}{2})$, where
$\nabla=\partial_{x}$, $\Delta=\partial_{x}^{2}$ and $C^{\alpha}\equiv
C^{\alpha}({\mathbb{T}})=B_{\infty,\infty}^{\alpha}({\mathbb{T}})$ denotes the
Hölder-Besov space on ${\mathbb{T}}$ with regularity exponent
$\alpha\in{\mathbb{R}}$ equipped with the norm $\|\cdot\|_{C^{\alpha}}$. We
showed the local-in-time solvability and the continuity of the solution in the
enhanced noise $\hat{\xi}$. More precisely, assuming that the coefficients
satisfy $a,g\in C_{b}^{3}({\mathbb{R}})$ and

(1.2) $c_{-}\leq a(v)\leq c_{+},$

for some $c_{-},c_{+}>0$, it was shown that (1.1) with the initial value
$u_{0}\in C^{\alpha}$ has a solution $u$ up to some $T_{*}>0$ (see (2.23)) and
if the enhanced noise $\hat{\xi}=(\xi,\Pi(\nabla X,\xi))$ converges in
$C^{\alpha-2}\times C^{2\alpha-3}$, then the corresponding solution
$u=u^{\hat{\xi}}$ converges in $\mathcal{L}_{T}^{\alpha}$, where
$\mathcal{L}_{T}^{\alpha}=C([0,T],C^{\alpha})\cap
C^{\alpha/2}([0,T],L^{\infty})$ taking $T>0$ uniformly in a neighborhood of
some $\hat{\xi}$ and $L^{\infty}=L^{\infty}({\mathbb{T}})$. Here
$X=(-\Delta)^{-1}(\xi-\xi({\mathbb{T}})),\xi({\mathbb{T}})\equiv\langle\xi,1\rangle$,
and $\Pi(\nabla X,\xi)$ denotes the resonant term in the paraproduct of
$\nabla X$ and $\xi$. In particular, we see
$u(t)\in\cap_{\delta>0}C^{\frac{3}{2}-\delta}$ for $t<T_{*}$, see [7] for
details. We remark that the same result still holds under a weaker assumption:
$a,g\in C^{3}({\mathbb{R}})$ satisfying (1.2), see Lemma 1.4 below.

###  1.1 The aim of the article

The present article is a continuation of [7]. Assuming that two coefficients
$a$ and $g$ satisfy the relation $a=g^{\prime}$, we establish the global-in-
time solvability and convergence of the solution to a stationary solution as
$t\to\infty$, see Theorems 1.1 and 1.2. For the convergence to the stationary
solution, we assume $|\mu_{\xi}|$ is small enough for the noise $\xi$, where
$\mu_{\xi}$ is the constant defined from $\xi$ by (1.7) below. A typical
example of the noise $\xi$ is the derivative of a periodic Brownian motion
$w=w(x),x\in{\mathbb{T}}$: $\xi=\dot{w}$ and, for this $\xi$, $\mu_{\xi}=0$
holds. Then, in general without assuming such conditions for the coefficients
$a,g$ and noise $\xi$, we show the continuity of the local-in-time solution in
initial values, see Theorem 1.3. This is used for the proofs of Theorems 1.1
and 1.2.

Let $\varphi\in C^{4}({\mathbb{R}})$ satisfying

(1.3) $\displaystyle c_{-}\leq\varphi^{\prime}(v)\leq c_{+}$

for some $c_{-},c_{+}>0$ and $\chi\in C^{3}({\mathbb{R}})$ be given, and
consider the SPDE

(1.4) $\partial_{t}v=\Delta\\{\varphi(v)\\}+\nabla\\{\chi(v)\xi\\},\quad
x\in{\mathbb{T}}.$

Note that the linear case $\varphi(v)=\chi(v)=v$ is included. Then, for every
$m\in{\mathbb{R}}$, if $u\equiv u_{m}$ is a solution of (1.1) with
$a(v)=\varphi^{\prime}(v+m)$, $g(v)=\chi(v+m)$, $v:=\nabla u_{m}+m$ solves the
equation (1.4) under the assumption $a,g\in C_{b}^{3}({\mathbb{R}})$, see
Section 1.2 of [7] and an indirect Definition 1.1 of the solution of (1.4)
below. Note that, if $\xi$ is smooth, this is true in classical sense. In
particular, (1.4) has a local-in-time solution $v$ in
$\mathcal{L}_{T}^{\alpha-1}$. See Subsection 1.4 for more explanation on the
relation between $u$ and $v$ under the weak assumption $a,g\in
C^{3}({\mathbb{R}})$. We note that the equation (1.4) has a mass conservation
law:

(1.5) $\int_{\mathbb{T}}v(t,x)dx=m$

for all $t\geq 0$ with a constant $m\in{\mathbb{R}}$, which is determined from
its initial value $v_{0}$.

As we mentioned above, the main aim of the present article is to show the
global-in-time solvability and establish the convergence of the solution to
the stationary solution as $t\to\infty$ for the SPDE (1.1) when the
coefficients satisfy $a=g^{\prime}$. For this purpose, it turns out to be more
convenient to study the SPDE in the form of (1.4). Due to the discussions in
Subsection 1.4, especially by Definition 1.1, the result for (1.4) implies
that for $\nabla u$ for the solution $u$ of (1.1). Our assumption
$a=g^{\prime}$ for (1.1) corresponds to $\chi=\varphi$ for (1.4) so that we
consider the equation of the special form

(1.6) $\partial_{t}v=\Delta\\{\varphi(v)\\}+\nabla\\{\varphi(v)\xi\\},$

on ${\mathbb{T}}$. The equation (1.6) has a physical meaning in the sense that
it can be derived from a microscopic particle system in random environment,
see [14].

Our another aim is to establish the continuity of the solution of (1.1) in
initial values in general without assuming $a=g^{\prime}$. We use such
property in the study of the SPDE (1.6), but we show it for the general case.
The solution determines a continuous flow on the state space
$C^{\alpha}\cup\\{\Delta\\}$ with a death point $\Delta$ added to cover the
explosion of the solutions, see Remark 2.4 in Subsection 2.2. The equation
(1.1) can be considered as if a deterministic PDE once $\xi$ is fixed, so that
it is different from the SPDEs driven by the space-time white noise, but the
continuity in initial value $u_{0}$ in our case corresponds to the strong
Feller property for such SPDEs, cf. [11].

###  1.2 Global solution in time and convergence to stationary solutions

We consider the SPDE (1.6) with $\xi\in
C^{\alpha-2},\alpha\in(\frac{4}{3},\frac{3}{2})$. To describe its stationary
solutions, for a given $\xi$, we define its integral
$\eta(x):=\langle\xi,1_{[0,x]}\rangle\equiv\int_{0}^{x}\xi(y)dy,x\in{\mathbb{T}}$.
Note that $\eta$ is not periodic, but
$\tilde{\eta}(x):=\langle\xi-\sigma,1_{[0,x]}\rangle=\eta(x)-\sigma x$ is
periodic, where

$\sigma\equiv\sigma_{\xi}:=\xi({\mathbb{T}})=\langle\xi,1\rangle=\eta(1)\in{\mathbb{R}}.$

It is known that $\eta\in C^{\alpha-1}({\mathbb{R}})$ and especially $\eta\in
C([0,1])$, see Lemma A.10 of [9]. Typically, we can take
$\xi=\dot{w}(x)+\sigma$ with a periodic Brownian motion $w(x)$,
$x\in{\mathbb{T}}$ and $\sigma\in{\mathbb{R}}$. The most interesting noise is
$\xi=\dot{w}$ and, in this case, $\sigma_{\xi}=0$ holds.

Then, from $\eta$ or equivalently from $\xi$, we define a function
$\theta(x)=\theta_{\xi}(x)$ on ${\mathbb{T}}$ and a constant
$\mu=\mu_{\xi}\in{\mathbb{R}}$, respectively, by

(1.7)
$\displaystyle\theta(x):=e^{-\eta(x)}\Big{\\{}\mu\int_{0}^{x}e^{\eta(y)}dy+1\Big{\\}},\quad
x\in{\mathbb{T}},$
$\displaystyle\mu:=\frac{e^{\eta(1)}-1}{\int_{0}^{1}e^{\eta(y)}dy}.$

Note that $\theta(x)>0$ so it is uniformly positive by its continuity. Indeed,
this is obvious if $\mu\geq 0$, while $\theta(x)\geq
e^{-\eta(x)}\\{\mu\int_{0}^{1}e^{\eta(y)}dy+1\\}=e^{-\eta(x)+\eta(1)}>0$ if
$\mu<0$. Moreover, $\theta$ is periodic: $\theta(0)=\theta(1)=1$. In case
$\sigma=0$, we have $\mu=0$ (and vice versa) and $\theta(x)=e^{-\eta(x)}$.
Note that $\theta=\theta(x)$ satisfies

(1.8) $\displaystyle\nabla\theta+\xi\theta=\mu$

at least if $\xi\in C({\mathbb{T}})$, that is, $\eta\in C^{1}([0,1])$, and
therefore $\Delta(z\theta)+\nabla(z\theta\cdot\xi)=0$ holds for every
$z\in{\mathbb{R}}$. In particular, $v=\varphi^{-1}(z\theta)$ are stationary
solutions of (1.6), where $\varphi^{-1}$ is the inverse function of $\varphi$.

For each conserved mass $m\in{\mathbb{R}}$ given as in (1.5), determine
$z=z_{m}\in{\mathbb{R}}$ uniquely by the relation

(1.9) $m=\int_{\mathbb{T}}\varphi^{-1}(z\theta(x))dx.$

Note that, since $\varphi$ satisfies (1.3) and in particular, it is strictly
increasing, this determines a one to one relation between $z$ and $m$. Then,

(1.10) $\bar{v}(x)\equiv\bar{v}_{m}(x):=\varphi^{-1}(z_{m}\theta(x))$

is a stationary solution of (1.6) satisfying $\int_{\mathbb{T}}vdx=m$ in
distributional sense, or at least if $\xi\in C({\mathbb{T}})$. Indeed, the
mass conservation law is clear and, as we noted above,

$\displaystyle\Delta\\{\varphi(\bar{v}_{m})\\}+\nabla\\{\varphi(\bar{v}_{m})\xi\\}$
$\displaystyle=z_{m}\nabla\\{\nabla\theta+\theta\xi\\}=z_{m}\nabla\mu=0,$

so that the right hand side of (1.6) vanishes for $v=\bar{v}_{m}$.

As we will see, at least if $|\mu_{\xi}|$ is small enough, $\bar{v}_{m}$ is
the unique stationary solution of the SPDE (1.6) for each fixed $m$, where
$\mu_{\xi}$ is the constant determined in (1.7) from the noise $\xi$. In fact,
we will show in Section 2 that, for initial value $v_{0}(x)$ satisfying (1.5)
(with $v(t)$ replaced by $v_{0}$), the solution $v(t,x)$ of the SPDE (1.6)
converges to the stationary solution $\bar{v}_{m}$ as $t\to\infty$ at least if
$|\mu_{\xi}|$ is small. Moreover, without assuming the smallness of
$|\mu_{\xi}|$, the SPDE (1.6) has a global solution in time.

######  Theorem 1.1.

Let $\varphi\in C^{4}({\mathbb{R}})$ satisfy (1.3) and
$\alpha\in(\frac{13}{9},\frac{3}{2})$. Then, for every initial value $v_{0}\in
C^{\alpha-1}$, the SPDE (1.6) has a global-in-time solution $v(t)\in
C^{\alpha-1}$ for all $t\geq 0$. Moreover, if $|\mu_{\xi}|$ is sufficiently
small, $v(t)$ converges exponentially fast to $\bar{v}_{m}$ in $C^{\alpha-1}$
as $t\to\infty$:

(1.11) $\displaystyle\|v(t)-\bar{v}_{m}\|_{C^{\alpha-1}}\leq Ce^{-ct},$

for some $c,C>0$, where $m$ is determined from $v_{0}$ as
$m=\int_{\mathbb{T}}v_{0}(x)dx$.

We apply the energy inequality, Poincaré inequality, the continuity of the
solutions in enhanced noise and initial values, and also the initial layer
property to show Theorem 1.1. See Remark 2.2 below for the SPDE (1.4) with
general $\chi$ instead of (1.6).

This theorem for the slope $v(t)=\nabla u(t)$ of $u(t)$ implies the following
result for $u(t)$ itself.

######  Theorem 1.2.

Assume $a=g^{\prime}\in C^{3}({\mathbb{R}})$ (so that $g\in
C^{4}({\mathbb{R}})$), the condition (1.2) and
$\alpha\in(\frac{13}{9},\frac{3}{2})$. Then, the SPDE (1.1) has a global-in-
time solution $u(t)\in C^{\alpha}$ for all $t\geq 0$. Moreover, if
$|\mu_{\xi}|$ is sufficiently small as in Theorem 1.1, $u(t)$ has the
following uniform bound in $t$:

(1.12) $\sup_{t\geq 0}\|u(t)-z_{0}\mu_{\xi}t\|_{C^{\alpha}}<\infty,$

where $z_{0}$ is defined by (1.9) with $m=0$. In particular, we have

$\lim_{t\to\infty}\frac{1}{t}u(t,x)=z_{0}\mu_{\xi}$

uniformly in $x\in{\mathbb{T}}$.

######  Remark 1.1.

(i) In Theorems 1.1 and 1.2, we assume $\alpha\in(\frac{13}{9},\frac{3}{2})$,
which is slightly more restrictive than the original assumption
$\alpha\in(\frac{4}{3},\frac{3}{2})$ in [7]. This is because of Theorem 1.3,
which has been used for the proofs of Theorems 1.1 and 1.2, see Proposition
2.6 in Section 2. For the reason for changing the range of $\alpha$, see
Remark 3.2. Except this point, the other statements in Section 2 hold for
$\alpha\in(\frac{4}{3},\frac{3}{2})$. So, unless otherwise noted, we still
assume $\alpha\in(\frac{4}{3},\frac{3}{2})$ throughout this article.
(ii) For the space-time white noise case as in [8], the average (i.e. integral
on ${\mathbb{T}}$) of $u(t)$ behaves as a Brownian motion and it never
converges as $t\to\infty$. But, in our case, noise is only spatially dependent
and the situation is different. Removing the constant drift, $u(t)$ stays
bounded in $t$.

We remark that the existence of global solutions of singular semilinear SPDEs
are known, for example, for the following models. The linear equation (1.1)
with $a=1,g=v$ on ${\mathbb{R}}^{d}$ (i.e., the equation (1.6) with
$\varphi(v)=v$) is studied in [4] and respectively, the generalized parabolic
Anderson model (PAM) (i.e., the equation (1.6) with $\varphi(v)=v$ and without
$\nabla$) in [9] (Remark 5.4) and [2] by different approaches. For the
nonlinear case, the dynamic $\phi_{3}^{4}$-model on ${\mathbb{T}}^{3}$ is
studied in [16] by establishing a priori estimate, and the complex Ginzburg-
Landau equation on ${\mathbb{T}}^{3}$ in [12]. The global existence for multi-
component coupled Kardar-Parisi-Zhang (KPZ) equation is shown in [6] under the
trilinear condition by studying its stationary measure. In addition, there are
few works on the exponential decay in time of the solutions of singular
semilinear SPDEs. For instance, [17] showed the exponential decay for the
dynamic $P(\phi)_{2}$-model on ${\mathbb{T}}^{2}$ to its unique invariant
measure with respect to the total variation norm as $t\to\infty$, and recently
[10] showed the exponential $L^{2}$-ergodicity of conservative stochastic
Burgers equation on ${\mathbb{T}}$ based on the approach of the martingale
problem. Among these, to the best of our knowledge, our result is the first
one in quasilinear case.

###  1.3 Continuity of the local solution in initial values and parameter $m$

Let $u(t,\hat{\xi},u_{0}),t\leq T$ denote the local-in-time paracontrolled
solution of (1.1) for each fixed enhanced noise $\hat{\xi}:=(\xi,\Pi(\nabla
X,\xi))$ and initial value $u_{0}$. In [7], it is shown that the
paracontrolled solution $u(t,\hat{\xi},u_{0})$ is continuous in the enhanced
noise $\hat{\xi}$ for each fixed initial value $u_{0}$, see Theorem 3.1-(ii)
of [7]. In Section 3, we show the joint continuity of $u(t,\hat{\xi},u_{0})$
in $(\hat{\xi},u_{0})$ without the restriction $a=g^{\prime}$. More precisely,
the main result of Section 3 is the next theorem.

######  Theorem 1.3.

Let $\alpha\in(\frac{13}{9},\frac{3}{2})$ and $u(t,\hat{\xi},u_{0})\in
C^{\alpha}$ be the unique local paracontrolled solution of (1.1) with initial
value $u_{0}\in C^{\alpha}$ at least up to time
$T=T(\|u_{0}\|_{C^{\alpha}},\|\hat{\xi}\|_{C^{\alpha-2}\times
C^{2\alpha-2}})$. Then, $u(t,\hat{\xi},u_{0})$ is continuous in
$(t,\hat{\xi},u_{0})$ in the region
$\\{(t,\hat{\xi},u_{0})\in[0,\infty)\times(C^{\alpha-2}\times
C^{2\alpha-3})\times C^{\alpha}\,;\,t\leq T\\}$. In particular, for each
$0<t\leq T$, the solution $u(t)$ of (1.1) is continuous in its initial values.

This theorem can be easily proved by Theorem 3.1 in Section 3, see also Remark
3.2. Furthermore, the continuity in $m$ of the solution
$u=u(t,m,\hat{\xi},u_{0})$ of (1.1) with the coefficients $a$ and $g$ replaced
by $a(\cdot+m)$ and $g(\cdot+m)$, respectively, can be shown by a
straightforward extension of our estimates, see Remark 3.3.

######  Remark 1.2.

(i) For the special case $a=g^{\prime}$, we have that $u(t,\hat{\xi},u_{0})$
of (1.1) is continuous in $(t,\hat{\xi},u_{0})$ in the region
$\\{(t,\hat{\xi},u_{0})\in[0,\infty)\times(C^{\alpha-2}\times
C^{2\alpha-3})\times C^{\alpha}\\}$ by Theorems 1.2 and 1.3.
(ii) From the explanation in Subsection 1.4, we see that Theorem 1.3 implies
that the solution $v(t,\hat{\xi},v_{0})$ of (1.4) is continuous in the region
$\\{(t,\hat{\xi},v_{0})\in[0,\infty)\times(C^{\alpha-2}\times
C^{2\alpha-3})\times C^{\alpha-1};t\leq T\\}$ and in particular,
$v(t,\hat{\xi},v_{0})$ is continuous in its initial values $v_{0}$ up to time
$T$. Moreover, by Theorem 1.1, for the special type of SPDE (1.6), we know
that its solution $v(t,v_{0})$ is continuous in $(t,v_{0})$ in the region
$\\{(t,v_{0})\in[0,\infty)\times C^{\alpha-1}\\}$.

######  Remark 1.3.

Hairer and Mattingly [11] discussed as follows: Let $U$ be the state space of
solutions of certain SPDE and $\bar{U}:=U\cup\\{\Delta\\}$ by adding the death
point $\Delta$ to cover blow-up of solutions. They consider the solution of
SPDE as a random dynamical system $\Phi_{s,t}:\bar{U}\times M\to\bar{U}$,
where $M$ denotes the space of admissible models for a given regularity
structure. In our case, we may fix the noise (or more precisely, the enhanced
noise), and consider the solution of SPDE as a deterministic dynamical system
($\Phi_{t}:\bar{U}\to\bar{U}$ would be enough). Their Assumption 1 is for the
continuity of the map $\Phi_{s,t}$. They first prove the strong Feller
property under general setting, and then show the assumptions formulated in
general setting for several singular SPDEs. In our case, once the noise is
fixed, the solution is deterministic so that what we need is the continuity of
the solution in the initial value.

###  1.4 Definition of the solution of (1.4) and relation to (1.1)

Recall that the solution $u(t)$ of (1.1) was defined in paracontrolled sense
by solving the fixed point problem for the map $\Phi$ defined by (2.16) in the
class $\mathcal{B}_{T}(\lambda)$, see [7].

Let us first remark the following lemma which generalizes Theorem 1.1 of [7]
and is shown by the cut-off argument.

######  Lemma 1.4.

Let $\alpha\in(\frac{4}{3},\frac{3}{2})$, $a,g\in C^{3}({\mathbb{R}})$ and the
condition (1.2) be satisfied. Then, the SPDE (1.1) has a unique local-in-time
solution $u(t)$ defined in paracontrolled sense.

###### Proof.

For a given $a\in C^{3}({\mathbb{R}})$, we can take a sequence of functions
$a^{n}\in C_{b}^{3}({\mathbb{R}})$ such that $0<c_{-}\leq a^{n}\leq c_{+}$,
$a^{n}(v)=a(v),|v|\leq n$ and $a^{n}$ converges uniformly to $a$ on each
compact set. Similarly, for a given $g\in C^{3}({\mathbb{R}})$, let us take a
sequence of functions $g^{n}\in C_{0}^{3}({\mathbb{R}})$ such that
$g^{n}(v)=g(v),|v|\leq n$ and $g^{n}$ converges uniformly to $g$ on each
compact set. Let us consider the equation

(1.13) $\partial_{t}u^{n}=a^{n}(\nabla u^{n})\Delta u^{n}+g^{n}(\nabla
u^{n})\cdot\xi$

starting from $u_{0}\in C^{\alpha}$. Then, the assumptions of Theorem 1.1 of
[7] are satisfied and we know that (1.13) has a unique paracontrolled solution
$u^{n}=u^{n}(t)$ up to a time $T_{*}^{n}>0\ a.s.$, which is similarly defined
by (2.23). Without loss of generality, we may assume $T_{*}^{n}<\infty\ a.s.$
Let us now define $\tau^{n}$ by
$\tau^{n}=\inf\\{t>0:\|u^{n}(t)\|_{C^{\alpha}}\geq n\\}$. Then, it is clear
that $\tau^{n}>0\ a.s.,$ whenever $\|u_{0}\|_{C^{\alpha}}<n$ and Lemma 2.5
below gives that $\tau^{n}\leq T_{*}^{n}$. Moreover, we have
$u^{m}(t)=u^{n}(t),t\leq\tau^{m}\wedge\tau^{n}$. Therefore, one knows that
(1.1) has a unique solution $u(t)$ in the paracontrolled sense at least up to
the time $\lim_{n\to\infty}\tau^{n}>0\ a.s.$ ∎

As we mentioned, (1.4) is obtained at least for a smooth noise from (1.1) with
proper modification in $m$ by differentiation. Motivated by this, we give the
meaning to the equation (1.4) indirectly via the equation (1.1). Let an
initial value $v_{0}\in C^{\alpha-1},\alpha\in(\frac{4}{3},\frac{3}{2})$ of
(1.4) be given. Then, set $m:=\int_{\mathbb{T}}v_{0}(x)dx$ and define
$u_{0}\in C^{\alpha}$ by integrating $v_{0}-m$ as

(1.14) $u_{0}(x)=\int_{0}^{x}(v_{0}(y)-m)dy+C,\quad x\in{\mathbb{R}},$

for any constant $C\in{\mathbb{R}}$. We solve (1.1) with
$a(v)=\varphi^{\prime}(v+m),g(v)=\chi(v+m)$ and this initial value $u_{0}$ in
paracontrolled sense. The solution is denoted by $u(t)=u(t;v_{0},C)$. Recall
$u(t)\in\mathcal{L}_{T}^{\alpha}$ for some $T>0$.

######  Definition 1.1.

We call $v(t):=\nabla u(t;v_{0},C)+m\in\mathcal{L}_{T}^{\alpha-1}$ the
solution of the SPDE (1.4) with initial value $v_{0}$.

Note that, if the noise $\xi$ (and $v_{0}$) is smooth, $v(t)$ is a smooth
classical solution of (1.4). Indeed, in such case, $u(t)$ is a smooth solution
of (1.1) so that this follows by differentiation. Note also that
$v(t;v_{0},C)$ does not depend on the choice of $C$. Indeed, again for smooth
smeared noise $\xi^{\varepsilon}$, we easily see
$u^{\varepsilon}(t;v_{0},C)=u^{\varepsilon}(t;v_{0},0)+C$ for the
corresponding solutions $u^{\varepsilon}$ of (1.1). Thus, by applying Theorem
1.1 of [7] and taking the limit $\varepsilon\downarrow 0$, we see that
$u(t;v_{0},C)=u(t;v_{0},0)+C$ holds for general noise $\xi$. This implies
$\nabla u(t;v_{0},C)=\nabla u(t;v_{0},0)$. In particular, $v(t)$ is well-
defined.

Conversely, $u(t)$ can be recovered from $v(t)=\nabla u(t)$ (with $m=0$) as
follows. Assume $\xi\in C^{\infty}({\mathbb{T}})$ and let the initial value
$u_{0}\in C^{\alpha},\alpha\in(\frac{4}{3},\frac{3}{2})$ of (1.1) be given.
Then, we determine $v(t)$ by solving (1.4) with initial value $v_{0}:=\nabla
u_{0}$, and set

$u(t,x):=\int_{0}^{x}v(t,y)dy+\int_{\mathbb{T}}u_{0}(y)dy-\int_{\mathbb{T}}(1-y)v(t,y)dy+\int_{0}^{t}ds\int_{\mathbb{T}}\chi(v(s,y))\cdot\xi(y)dy.$

(In the right hand side, especially in the first and third terms, we regard
${\mathbb{T}}=[0,1)$.) Then, one can show that $u(t)$ solves the equation
(1.1) with $a=\varphi^{\prime},g=\chi$, see Lemma 2.10 below. At least if
$\xi$ is smooth, the equivalence between (1.1) and (1.4) is established.

Moreover, concerning the renormalizations, the equation (1.1) in integrated
form does not require them, since the resonant term $\Pi(\nabla X,\xi)$
involves the derivative of $X$ as we discussed in [7] (though (1.1) is an
analog of KPZ equation). In particular, the equation (1.4) in differentiated
form does not require them too.

The remainder of this article is organized as follows. Section 2 is for the
proofs of Theorems 1.1 and 1.2. We first formulate the energy estimate for
(1.6) driven by a smooth noise in Subsection 2.1, see Proposition 2.2. Then,
the proof of Theorem 1.1 is given in Subsection 2.2. We note the continuity of
the solution in the enhanced noise $\hat{\xi}$, the initial values and the
parameter $m$ in the coefficients. We derive Poincaré inequality and show that
the Poincaré constant can be taken uniformly in the approximating sequence of
the noise. We also rely on the initial layer type property of the solution of
the SPDE (1.6), that is, the regularity of the solution is improved in an
arbitrary short time. Subsection 2.3 is devoted to the proof of Theorem 1.2
based on the relation between (1.1) and (1.4). In Section 3, we show Theorem
1.3 by establishing Theorem 3.1.

##  2 Global solvability and convergence to stationary solution

We show the global solvability of (1.6) based on the energy inequality and
Poincaré inequality. This gives the exponentially fast convergence in
$C^{\alpha-1}$ of the solution $v(t)$ to the stationary one, first for the
initial value $v(0)\in\mathcal{D}$, at least if $|\mu_{\xi}|$ is sufficiently
small. Here $\mathcal{D}$ is the class of all functions $v\in C^{\alpha-1}$
satisfying $\varphi(v)\theta^{-1}\in H^{1}$, where $H^{1}=H^{1}({\mathbb{T}})$
is the Sobolev space on ${\mathbb{T}}$. Then, this result will be extended to
general initial values $v(0)\in C^{\alpha-1}$ by the initial layer type
property of the solution. Note that $v(0)\in
C^{\beta},\beta\in(\frac{1}{3},\alpha-1)$, is equivalent to writing $v(0)\in
C^{\alpha-1}$, $\alpha\in(\frac{4}{3},\frac{3}{2})$, by tuning in $\alpha$. In
addition, we will use $C$ to denote a positive generic constant that may
change from line to line in this section.

###  2.1 Method of energy inequality

In this subsection, we assume $\xi\in C^{\infty}({\mathbb{T}})$ (or at least
$\xi\in C^{2}({\mathbb{T}})$) and consider a differentiable solution $v(t,x)$
of (1.6). More precisely, if $\xi\in C^{2}({\mathbb{T}})$, the equation (1.4)
is a classical PDE of divergence form:

(2.1) $\displaystyle\partial_{t}v=\nabla\\{a_{1}(v,\nabla
v)\\}-a_{2}(x,v,\nabla v),\quad x\in{\mathbb{T}},$ or, we can further rewrite
it as (2.2) $\displaystyle\partial_{t}v=a_{3}(v)\Delta v-A(x,v,\nabla v),\quad
x\in{\mathbb{T}},$

where $a_{1}(v,p)=\varphi^{\prime}(v)p$,
$a_{2}(x,v,p)=-\big{(}\dot{\xi}(x)\chi(v)+\xi(x)\chi^{\prime}(v)p\big{)}$,
$a_{3}(v)=\varphi^{\prime}(v)$ and
$A(x,v,p)=-\varphi^{\prime\prime}(v)p^{2}+a_{2}(x,v,p)$. Note that (2.1) and
(2.2) are written in the forms of (6.1) and (6.4) in [13] (p.449, p.450),
respectively. We actually consider (1.6) so that $\chi=\varphi$. Recall that
$\varphi\in C^{4}({\mathbb{R}})$ satisfies (1.3) and this, in particular,
implies the linear growth property of $\varphi$: $|\varphi(v)|\leq C(|v|+1)$.
Thus, we see that the conditions a)–d) of Theorem 6.1 ([13], p.452),
especially,

$\displaystyle\partial_{p}a_{1}(v,p)\,(=\varphi^{\prime}(v))\geq 0,\quad
A(x,v,0)v\geq-b_{1}v^{2}-b_{2},$ $\displaystyle
0<c_{-}\leq\partial_{p}a_{1}(v,p)\leq
c_{+},\quad(|a_{1}|+|\partial_{v}a_{1}|)(1+|p|)+|a_{2}|\leq C_{M}(p^{2}+1)$

hold for $x\in{\mathbb{T}},|v|\leq M,p\in{\mathbb{R}}$. Indeed, the second
bound follows from

$\displaystyle|A(x,v,0)v|=|a_{2}(x,v,0)v|=|\dot{\xi}(x)|\,|\varphi(v)v|\leq
C_{2}(v^{2}+1).$

Note that $M>0$ given in (6.8) of [13] can be taken in our situation by
applying the maximum principle, see Remark 2.1-(i) below. The condition
$\xi\in C^{2}({\mathbb{T}})$ is required for the condition c). The condition
d) is shown by the boundedness of $\partial_{v}a_{1}$, $\partial_{p}a_{2}$,
$\partial_{v}a_{2}$ for $|v|\leq M$ and $|p|\leq M_{1}$ for each $M,M_{1}>0$.
Therefore, by Theorem 6.1 of [13], (1.4) has a unique global-in-time classical
solution $v(t,x)\,(\in H^{1+\beta/2,2+\beta}([0,T]\times{\mathbb{T}}))$ if
$\xi\in C^{2}({\mathbb{T}})$ and the initial value $v(0)\in
C^{2+\beta}({\mathbb{T}})$ for some $\beta\in(0,1)$. Furthermore, noting that
$|a_{1}(v,p)|+|\partial_{v}a_{1}(v,p)|\leq C_{M}|p|,|v|\leq
M,p\in{\mathbb{R}}$, by Theorem 6.4 ([13], p.460), the existence of global-in-
time classical solution is known if $\xi\in C^{2}({\mathbb{T}})$ and $v(0)\in
C^{\beta}({\mathbb{T}})$, $\beta\in(0,1)$. Note that, from examples in p.99 of
[1] or p.62 -L. 8 of [9], we know that $C^{\alpha}=H^{\alpha}$ (Hölder space
used in [13]) for all $\alpha\in{\mathbb{R}}^{+}\setminus{\mathbb{N}}$. One
can easily check that the classical solution of (1.4) is a solution in
paracontrolled sense. This shows that the life time of the solution $v(t)\in
C^{\alpha-1}$ of (1.6) equals to infinity, i.e.,

$\displaystyle T_{*}\equiv T_{*}(\hat{\xi},v(0)):=\sup\\{t\geq 0;\
\text{solution of }\eqref{eq:0-v}\ \text{with initial value}\ v(0)\
\text{exists}\\}=\infty,$

if $\xi\in C^{2}({\mathbb{T}})$ and $v(0)\in C^{\alpha-1}$, $\alpha>1$. We
expect that $T_{*}$ is lower semicontinuous in $\hat{\xi}$ as in [6], but this
combined with $T_{*}=\infty$ for $\xi\in C^{2}({\mathbb{T}})$ does not imply
the same for general $\xi\in C^{\alpha-2}$.

As we mentioned, in this subsection, we assume $\xi\in
C^{\infty}({\mathbb{T}})$ and $v(t,x)$ is a smooth global-in-time solution of
(1.6), i.e., $v\in C^{1,2}((0,\infty)\times{\mathbb{T}})\cap
C([0,\infty)\times{\mathbb{T}})$. We define $f(t,x)$ as

(2.3) $f(t,x):=\frac{\varphi(v(t,x))}{\theta(x)},$

where $\theta=\theta_{\xi}(x)$ is defined in (1.7). Note that $f(t,x)$ is
periodic in $x\in{\mathbb{T}}$. Then, we have

$\displaystyle\nabla\varphi(v)$ $\displaystyle=\nabla(f\theta)=\nabla
f\cdot\theta+f\nabla\theta$ $\displaystyle=\nabla
f\cdot\theta+f(-\xi\theta+\mu)$ $\displaystyle=\nabla f\cdot\theta+\mu
f-\xi\varphi(v).$

Therefore, the equation (1.6) can be rewritten as

(2.4) $\partial_{t}v=\nabla(\theta\nabla f+\mu f).$

Let $L^{2}_{\theta}:=L^{2}({\mathbb{T}},\theta dx)$ and
$H^{1}_{\theta}:=H^{1}({\mathbb{T}},\theta dx)$ be the spaces equipped with
the norms $\|f\|_{L_{\theta}^{2}}:=\big{(}\int_{\mathbb{T}}f^{2}\theta
dx\big{)}^{1/2}$ and
$\|f\|_{H_{\theta}^{1}}:=\big{(}\int_{\mathbb{T}}\\{f^{2}+(\nabla
f)^{2}\\}\theta dx\big{)}^{1/2}$, respectively. We define the functional
$\Phi(f)\equiv\Phi_{\theta}(f)$ of $f\in H_{\theta}^{1}$ as

$\Phi(f)\equiv\Phi_{\theta}(f):=\frac{1}{2}\int_{\mathbb{T}}(\nabla
f)^{2}\theta dx.$

######  Lemma 2.1.

The functional $\Phi$ is Fréchet differentiable in $H_{\theta}^{1}$ and its
Fréchet derivative $D\Phi(f)\in(H_{\theta}^{1})^{*}$ is given by

(2.5) $D\Phi(f)(\psi)\equiv{}_{(H_{\theta}^{1})^{*}}\langle
D\Phi(f),\psi\rangle_{H_{\theta}^{1}}=\int_{\mathbb{T}}\nabla
f\nabla\psi\,\theta dx,$

for $\psi\in H_{\theta}^{1}$. If $f\in C^{2}({\mathbb{T}})$, this is further
rewritten as

$\int_{\mathbb{T}}D\Phi(x,f)\psi(x)\,\theta dx,$

with

(2.6) $D\Phi(x,f)=-\theta^{-1}\nabla(\theta\nabla f),$

note that $\theta^{-1}$ means $\frac{1}{\theta}$.

###### Proof.

Take $\psi\in H_{\theta}^{1}$ and define $D\Phi(f)(\psi)$ as (2.5). Then,

$\Phi(f+\psi)-\Phi(f)-D\Phi(f)(\psi)=\Phi(\psi)=o(\|\psi\|_{H_{\theta}^{1}})$

as $\|\psi\|_{H_{\theta}^{1}}\to 0$. This shows the Fréchet differentiability
of $\Phi$ and the formula (2.5). The formula (2.6) for $D\Phi(x,f)$ is shown
by a simple integration by parts:

$\displaystyle\int_{\mathbb{T}}\nabla f\nabla\psi\,\theta
dx=-\int_{\mathbb{T}}\theta^{-1}\nabla(\theta\nabla f)\psi\,\theta dx.$

∎

Noting $v=\varphi^{-1}(f\theta)$ from (2.3) and using (2.6), (2.4) is
rewritten as

(2.7) $\partial_{t}(\varphi^{-1}(f\theta))=-\theta D\Phi(x,f)+\mu\nabla f.$

Set $G(x,f)=(\varphi^{-1})^{\prime}(f\theta(x))>0$ and

$\displaystyle
K(x,f)=\frac{1}{G(x,f)}=\varphi^{\prime}(\varphi^{-1}(f\theta))\,(=\varphi^{\prime}(v))\geq
c_{-}>0,$

recall the assumption (1.3). Then, since
$\partial_{t}\varphi^{-1}(f\theta)=G(x,f)\partial_{t}f\cdot\theta$, (2.7) can
be further rewritten as

(2.8) $\partial_{t}f=K(x,f)(-D\Phi(x,f)+\mu\theta^{-1}\nabla f),$

which is sometimes called Onsager equation at least when $\mu=0$, see [15],
p.193. See also Remark 2.1 below for this equation.

######  Proposition 2.2.

Assume $\xi\in C^{\infty}({\mathbb{T}})$ and $v(t,x)$ is a smooth global-in-
time solution of (1.6). Then, for $f(t)$ defined by (2.3), if $f(0)\in
H_{\theta}^{1}$, we have the bound

(2.9) $\Phi(f(t))\leq\Phi(f(0))e^{C(\theta)t},$

where $\theta=\theta_{\xi}$,

$C(\theta)=-\frac{c_{-}}{2c_{2}(\theta)}+\frac{1}{2c_{-}}\mu^{2}c_{1}(\theta)^{2},$

$c_{1}(\theta)=c_{1}(\min\theta)$ defined by (2.12) and $c_{2}(\theta)>0$ is
the constant given in (2.15) in Poincaré inequality. (Note that
$c_{2}(\theta)$ stays finite for every $\eta\in C([0,1])$.) In particular, if
$|\mu|=|\mu_{\xi}|$ is small enough, $C(\theta)<0$ and this shows the
exponential decay of $\Phi(f(t))$ as $t\to\infty$:

(2.10) $\Phi(f(t))\leq\Phi(f(0))e^{-c_{*}t},$

for some $c_{*}>0$. When $\mu_{\xi}=0$, in particular, when
$\sigma=\langle\xi,1\rangle=0$, one can take
$c_{*}=\frac{c_{-}}{c_{2}(\theta)}$ (better than $C(\theta)$ with
$\mu_{\xi}=0$).

###### Proof.

Recalling $\theta(x)>0$ and $K(x,f)=\varphi^{\prime}(v)\geq c_{-}>0$, we
obtain from (2.8)

(2.11) $\displaystyle\partial_{t}\Phi(f)$
$\displaystyle=\langle\partial_{t}f,D\Phi(\cdot,f)\rangle_{L_{\theta}^{2}}$
$\displaystyle=-\int_{\mathbb{T}}K(x,f)D\Phi(x,f)^{2}\theta
dx+\mu\int_{\mathbb{T}}K(x,f)D\Phi(x,f)\nabla fdx$ $\displaystyle\leq-
c_{-}\|D\Phi(\cdot,f)\|_{L_{\theta}^{2}}^{2}+\mu\int_{\mathbb{T}}K(x,f)D\Phi(x,f)\nabla
fdx.$

For the second term, since $\varphi$ satisfies (1.3), we have $c_{-}\leq
K(x,f)\leq c_{+}$ and this shows

$\leq|\mu|\,c_{+}\|D\Phi(\cdot,f)\|_{L^{2}_{\theta}}\|\nabla
f\|_{L_{\theta^{-1}}^{2}}.$

However, since $\theta$ is uniformly positive, $\theta^{-1}(x)\leq
c(\theta)\theta(x)$ for
$c(\theta):=(\min_{x\in{\mathbb{T}}}\theta^{2}(x))^{-1}>0$ and therefore
$\|\nabla f\|_{L_{\theta^{-1}}^{2}}\ \leq\sqrt{c(\theta)}\|\nabla
f\|_{L_{\theta}^{2}}=\sqrt{2c(\theta)}\Phi(f)^{\frac{1}{2}}$. Thus, the second
term is bounded by

$\displaystyle\leq|\mu|\,c_{1}(\theta)\|D\Phi(\cdot,f)\|_{L^{2}_{\theta}}\Phi(f)^{\frac{1}{2}}$
$\displaystyle\leq\tfrac{c_{-}}{2}\|D\Phi(\cdot,f)\|_{L^{2}_{\theta}}^{2}+\tfrac{1}{2c_{-}}\mu^{2}c_{1}(\theta)^{2}\Phi(f),$

where

(2.12) $\displaystyle c_{1}(\theta)\equiv
c_{1}(\min\theta):=c_{+}\sqrt{2c(\theta)}>0.$

Therefore, we obtain

(2.13)
$\displaystyle\partial_{t}\Phi(f)\leq-\tfrac{c_{-}}{2}\|D\Phi(\cdot,f)\|_{L^{2}_{\theta}}^{2}+\tfrac{1}{2c_{-}}\mu^{2}c_{1}(\theta)^{2}\Phi(f).$

We now apply Poincaré inequality $\Phi(f)\leq
c_{2}(\theta)\|D\Phi(\cdot,f)\|_{L^{2}_{\theta}}^{2}$ given in Lemma 2.3
below, and then (2.13) shows that

$\displaystyle\partial_{t}\Phi(f)\leq-\tfrac{c_{-}}{2c_{2}(\theta)}\Phi(f)+\tfrac{1}{2c_{-}}\mu^{2}c_{1}(\theta)^{2}\Phi(f)=C(\theta)\Phi(f).$

This implies $\partial_{t}\big{(}e^{-C(\theta)t}\Phi(f)\big{)}\leq 0$ and
leads to the bound (2.9). (2.10) is immediate from (2.9). When $\mu_{\xi}=0$,
$\partial_{t}\Phi(f)\leq-c_{-}\|D\Phi(\cdot,f)\|_{L^{2}_{\theta}}^{2}$ holds
by (2.11), which is simpler than (2.13). Therefore, one can take
$c_{*}=\frac{c_{-}}{c_{2}(\theta)}$ in this case by Lemma 2.3. ∎

The following is the Poincaré inequality used in the proof of Proposition 2.2.

######  Lemma 2.3.

For every $f\in C^{2}({\mathbb{T}})$, we have

(2.14) $\Phi(f)\leq c_{2}(\theta)\|D\Phi(\cdot,f)\|_{L^{2}_{\theta}}^{2},$

where

(2.15)
$c_{2}(\theta):=\frac{1}{2}\int_{\mathbb{T}}\theta^{-1}(x)dx\int_{\mathbb{T}}\theta(y)dy.$

###### Proof.

Set $g:=\theta\nabla f$ and note that

$\int_{\mathbb{T}}g\theta^{-1}dx=\int_{\mathbb{T}}\nabla fdx=0$

holds by the periodicity of $f$. Then, noting that

$\displaystyle\Phi(f)=\frac{1}{2}\int_{\mathbb{T}}(g\theta^{-1})^{2}\theta
dx=\frac{1}{2}\int_{\mathbb{T}}g^{2}\theta^{-1}dx,$
$\displaystyle\|D\Phi(\cdot,f)\|_{L^{2}_{\theta}}^{2}=\int_{\mathbb{T}}(\nabla
g)^{2}\theta^{-1}dx,$

and setting

$Z:=\int_{\mathbb{T}}\theta^{-1}dx,$

we have

$\displaystyle\Phi(f)$
$\displaystyle=\frac{1}{2}\int_{\mathbb{T}}\Big{(}g(x)-\frac{1}{Z}\int_{\mathbb{T}}g(y)\theta^{-1}(y)dy\Big{)}^{2}\theta^{-1}(x)dx$
$\displaystyle=\frac{1}{2}\int_{\mathbb{T}}\Big{(}\int_{\mathbb{T}}\big{(}g(x)-g(y)\big{)}\frac{1}{Z}\theta^{-1}(y)dy\Big{)}^{2}\theta^{-1}(x)dx$
$\displaystyle\leq\frac{1}{2}\int_{\mathbb{T}}\theta^{-1}(x)dx\int_{\mathbb{T}}\big{(}g(x)-g(y)\big{)}^{2}\frac{1}{Z}\theta^{-1}(y)dy$
$\displaystyle=\frac{1}{2Z}\int_{\mathbb{T}}\theta^{-1}(x)dx\int_{\mathbb{T}}\theta^{-1}(y)dy\Big{(}\int_{x}^{y}\nabla
g(z)dz\Big{)}^{2}$
$\displaystyle\leq\frac{1}{2Z}\int_{\mathbb{T}}\theta^{-1}(x)dx\int_{\mathbb{T}}\theta^{-1}(y)dy\int_{\mathbb{T}}(\nabla
g(z))^{2}\theta^{-1}(z)dz\int_{\mathbb{T}}\theta(z)dz$
$\displaystyle=c_{2}(\theta)\|D\Phi(\cdot,f)\|_{L^{2}_{\theta}}^{2},$

where we have used Schwarz’s inequality twice. This shows the conclusion. ∎

######  Remark 2.1.

(i) The equation (2.8) is rewritten as

(2.16) $\partial_{t}f=K(x,f)\theta^{-1}\nabla(\theta\nabla f)+\mu
K(x,f)\theta^{-1}\nabla f.$

As we saw above, under the assumption $\xi\in C^{\infty}({\mathbb{T}})$, $f$
exists globally in time and (2.16) can be considered as a linear PDE for $f$
regarding the coefficient $K(x,f)$ is given. Then, since the right hand side
of (2.16) has no zeroth-order term in $f$, it satisfies the maximum principle
and we have

$\min_{x\in{\mathbb{T}}}f(0,x)\leq f(t,x)\leq\max_{x\in{\mathbb{T}}}f(0,x),$

see, for example, [5] p.368. Based on this observation and taking the limit in
$\xi$, one can cover the case $\theta\in C({\mathbb{T}})$ and show the global-
in-time solvability of (1.6). This provides another proof of the first part of
Theorem 1.1, though the present article relies on the method of energy
inequality.
(ii) (Linear case) When $\varphi(v)=v$, we have $G(x,f)=K(x,f)=1$. In
addition, if $\mu_{\xi}=0$, or equivalently, if $\sigma_{\xi}=0$ holds, (2.8)
defines a simple gradient flow:

(2.17) $\partial_{t}f=-D\Phi(x,f),$

and this implies
$\partial_{t}\Phi(f)=-\|D\Phi(\cdot,f)\|_{L^{2}_{\theta}}^{2}$.

######  Remark 2.2.

For the SPDE (1.4) with general $\chi$ and smooth $\xi$, the stationary
solution is a periodic solution $v=v(x)$ of the ordinary differential equation

$\Delta\\{\varphi(v)\\}+\nabla\\{\chi(v)\xi\\}=0.$

As before, setting $\theta=\varphi(v)$, this equation is rewritten as

(2.18) $\nabla\theta+\psi(\theta)\xi=\mu,$

where $\psi(\theta):=\chi(\varphi^{-1}(\theta))$ and $\mu$ is any constant. If
$\mu=0$, the equation (2.18) is of separable type and solved as

$\Psi(\theta)\equiv\int_{0}^{\theta}\frac{d\theta^{\prime}}{\psi(\theta^{\prime})}=-\eta(x)+C,\quad
x\in{\mathbb{T}}.$

For simplicity, if $\chi>0$, then $\Psi$ is increasing and (2.18) is solved as

$\theta_{C}(x)=\Psi^{-1}(-\eta(x)+C).$

For $\theta_{C}$ to be periodic, $\eta$ should satisfy $\eta(0)=\eta(1)$, that
is, $\sigma=0$. In other words, the condition $\mu=0$ implies $\sigma=0$. On
the other hand, the constant $C=C_{m}$ is determined from the conservation law
(1.5).

Once stationary solutions are found, to link them to the SPDE (1.4), we need
to find a proper transformation like (2.3) from $v$ to $f$, which extracts the
factor $z$, that is $C$ in the present setting for general $\chi$, and also a
proper functional $\Phi(f)$ of $f$. However, this looks nontrivial.

Note that, in case $\chi=\varphi$, $\psi(\theta)=\theta$,
$\Psi(\theta)=\log|\theta|$ and $\theta_{C}(x)=\pm
e^{-\eta(x)+C}=ze^{-\eta(x)}$, assuming $\mu=0$.

###  2.2 Proof of Theorem 1.1

Now we consider general $\xi\in C^{\alpha-2}$,
$\alpha\in(\frac{4}{3},\frac{3}{2})$. We are discussing $v(t)$, but here start
with $u(t)$, i.e., the unique local-in-time paracontrolled solution of (1.1)
with the initial value $u_{0}\in C^{\alpha}$.

Let us recall Theorem 3.1 (i) in [7]. In that theorem, it is declared that the
map $\Phi$ defined by (2.16) in [7] (or see (3.1) below, which is defined in a
little different setting from the original one) is contractive from
$\mathcal{B}_{T}(\lambda)$ (a variant of (3.4) below) into itself for some
large enough $\lambda$ and small enough $T>0$. But, the explicit choices of
$\lambda$ and $T$ were not given. To show Lemma 2.5 below, let us explicitly
choose $\lambda$ and $T$. They can be constructed easily by the estimates
(3.48) and (3.50) obtained in the proof of Theorem 3.1 (i) in [7]. In fact, by
these estimates, we know that there exists a large enough constant $M>0$ such
that for all ${\bf u}:=(u,u^{\prime})\in\mathcal{B}_{T}(\lambda)$,

(2.19) $\displaystyle\|\Phi({\bf u})\|_{\alpha,\beta,\gamma}\leq
M\Big{(}T^{\frac{\alpha+\beta-\gamma}{2}}K(\|{\bf
u}\|_{\alpha,\beta,\gamma})\tilde{K}_{1}(X,\xi)+K_{0}(\|u_{0}\|_{C^{\alpha}})(1+\|\xi\|_{C^{\alpha-2}})\Big{)},$

and

(2.20) $\displaystyle\|\Phi({\bf u}_{1})-\Phi({\bf
u}_{2})\|_{\alpha,\beta,\gamma}\leq MT^{\frac{\alpha+\beta-\gamma}{2}}K(\|{\bf
u}_{1}\|_{\alpha,\beta,\gamma},\|{\bf u}_{2}\|_{\alpha,\beta,\gamma})\|{\bf
u}_{1}-{\bf u}_{2}\|_{\alpha,\beta,\gamma}\tilde{K}_{2}(X,\xi),$

where $\beta\in(\frac{1}{3},\alpha-1)$, $\gamma\in(2\beta+1,\alpha+\beta)$,
$K(\lambda),K(\lambda,\lambda)$ denote the increasing and positive functions
in $\lambda>0$ introduced at the end of Section 2 in [7],
$\tilde{K}_{1}(X,\xi)$ and $\tilde{K}_{2}(X,\xi)$ are the positive polynomial
functions used in (3.48) and (3.50) in [7]. Let us determine $\lambda$ and
$T>0$ as follows.

(2.21) $\displaystyle\lambda=$ $\displaystyle
2M\Big{(}\tilde{K}_{1}(X,\xi)+K_{0}(\|u_{0}\|_{C^{\alpha}})(1+\|\xi\|_{C^{\alpha-2}})\Big{)},$
(2.22) $\displaystyle T=$
$\displaystyle\min\left\\{\Big{(}K(\lambda)+MK(\lambda,\lambda)\tilde{K}_{2}(X,\xi)\Big{)}^{-\frac{2}{\alpha+\beta-\gamma}},1\right\\},$

where $M$ is same as in (2.19). Then, Theorem 3.1 (i) of [7] can be restated
as follows.

######  Theorem 2.4 (Theorem 3.1 (i) of [7]).

Let $\lambda$ and $T$ be defined by (2.21) and (2.22), respectively. Then, for
any $u_{0}\in C^{\alpha}$ (or equivalently $u^{\prime}_{0}\in C^{\beta}$ and
$u_{0}^{\sharp}\in C^{\alpha}$), $\Phi$ is contractive from
$\mathcal{B}_{T}(\lambda)$ into itself. In particular, $\Phi$ has a unique
fixed point on $[0,T]$, which is the unique solution of the system (2.17) and
(2.18) in [7] and it solves the SPDE (1.1) on $[0,T]$ in the paracontrolled
sense.

###### Proof.

It is enough to show $\Phi$ is contractive from $\mathcal{B}_{T}(\lambda)$
into itself. By (2.19) and (2.21), we easily have

$\displaystyle\|\Phi({\bf
u})\|_{\alpha,\beta,\gamma}\leq\frac{\lambda}{2}\Big{(}T^{\frac{\alpha+\beta-\gamma}{2}}K(\|{\bf
u}\|_{\alpha,\beta,\gamma})+1\Big{)}.$

Then, noting that $T$ is given as (2.22), we have
$T^{\frac{\alpha+\beta-\gamma}{2}}K(\|{\bf u}\|_{\alpha,\beta,\gamma})<1$
whenever $\|{\bf u}\|_{\alpha,\beta,\gamma}\leq\lambda$ and therefore, $\Phi$
maps $\mathcal{B}_{T}(\lambda)$ into itself. The contractivity of the map
$\Phi$ on $\mathcal{B}_{T}(\lambda)$ is obvious by (2.20) and
$T^{\frac{\alpha+\beta-\gamma}{2}}MK(\lambda,\lambda)\tilde{K}_{2}(X,\xi)<1$.
∎

######  Remark 2.3.

From Theorem 2.4, we see that the time $T$ chosen as in (2.22) depends
continuously on the norm $\|u_{0}\|_{C^{\alpha}}$(or equivalently on
$\|u^{\prime}_{0}\|_{C^{\beta}}$ and $\|u_{0}^{\sharp}\|_{C^{\alpha}}$), which
is vital for the proof of the next Lemma 2.5.

Let us define the explosion time $T_{*}$ by

(2.23) $T_{*}\equiv T_{*}(\hat{\xi},u_{0}):=\sup\\{t\geq 0\,;\,\text{solution
}u(t)\in C^{\alpha}\text{ of }\eqref{eq:1.1}\ \text{starting from}\ u_{0}\
\text{exists}\\}.$

We know that $T_{*}>0$. Furthermore, we have the following result.

######  Lemma 2.5.

If $T_{*}<\infty$, we have

$\lim_{t\uparrow T_{*}}\|u(t)\|_{C^{\alpha}}=\infty.$

###### Proof.

By the definition of $T_{*}$, the solution exists and satisfies $u(\cdot)\in
C([0,T_{*}),C^{\alpha})$. If the conclusion does not hold, one can find $M>0$
and a sequence $t_{n}\uparrow T_{*}$ such that $\|u(t_{n})\|_{C^{\alpha}}\leq
M$. However, by Theorem 2.4 and Remark 2.3, there exists
$\varepsilon=\varepsilon_{M}>0$, which is uniform in $n$, such that one can
solve (1.1) starting from $u(t_{n})$ on the time interval
$[t_{n},t_{n}+\varepsilon]$. Since $t_{n}\uparrow T_{*}$, this shows that one
can solve (1.1) beyond $T_{*}$ and contradicts the definition of $T_{*}$. ∎

######  Remark 2.4.

In particular, let $C^{\alpha}\cup\\{\Delta\\}$ be the one point
compactification of $C^{\alpha}$ and define $u(t):=\Delta$ for $t\geq T_{*}$.
Then, $u(t)$ is defined for all $t\geq 0$ and $u(\cdot)\in
C([0,\infty),C^{\alpha}\cup\\{\Delta\\})$ by Lemma 2.5. Denoting $u(t)$ with
initial value $u_{0}\in C^{\alpha}\cup\\{\Delta\\}$ by $u(t,u_{0})$, it has
the flow property: $u(t,u(s,u_{0}))=u(t+s,u_{0})$ for all $t,s\geq 0$.

We extend the result of Proposition 2.2 to general noise $\xi\in
C^{\alpha-2}$, $\alpha\in(\frac{13}{9},\frac{3}{2})$. Recall that
$\mathcal{D}$ is the class of all functions $v\in C^{\alpha-1}$ satisfying
$\varphi(v)\theta^{-1}\in H^{1}$.

######  Proposition 2.6.

Assume that the initial value of the SPDE (1.6) satisfies
$v(0)\in\mathcal{D}$. Then, the solution $v(t)$ exists globally in time for
all $t\geq 0$ and $f(t)$ defined from $v(t)$ by (2.3) satisfies

(2.24) $\Phi(f(t))\leq e^{Ct}\Phi(f(0)),$

for some $C\in{\mathbb{R}}$. In particular, if $|\mu_{\xi}|$ is small enough,
one can take $C<0$.

###### Proof.

First assume that $\xi\in C^{\infty}({\mathbb{T}})$ and let $f(t,x)$ be as in
(2.3) and Proposition 2.2 defined from $v(t,x):=\nabla u(t,x)+m$, where
$m=\int_{\mathbb{T}}v(0,x)dx$ and $u(t,x)$ is a (classical) global-in-time
solution of (1.1) with initial value $u_{0}$ defined by (1.14),
$a=\varphi^{\prime}(\cdot+m)$ and $g=\varphi(\cdot+m)$. Then, we have the
estimate (2.9) for $\Phi(f(t))$, if $f(0)\in H_{\theta}^{1}$.

Now, let $\xi\in C^{\alpha-2},\alpha\in(\frac{13}{9},\frac{3}{2})$ be given
and take a sequence of $\xi_{n}\in C^{\infty}({\mathbb{T}})$ such that
$\hat{\xi}_{n}=(\xi_{n},\Pi(\nabla X_{n},\xi_{n}))$ converges to
$\hat{\xi}=(\xi,\Pi(\nabla X,\xi))$ in $C^{\alpha-2}\times C^{2\alpha-3}$ as
$n\to\infty$, see Lemma 5.2 in [7] for details. Let
$v_{n}(0):=\varphi^{-1}(f(0)\theta_{n})$,
$m_{n}=\int_{\mathbb{T}}v_{n}(0,x)dx$, $a_{n}=\varphi^{\prime}(\cdot+m_{n})$
and $g_{n}=\varphi(\cdot+m_{n})$, where $\theta_{n}=\theta_{\xi_{n}}$ is
defined as in (1.7) from the integral $\eta_{n}$ of $\xi_{n}$. Note that
$v_{n}(0)$ is chosen in such a way that
$f_{n}(0)(:=\varphi(v_{n}(0))/\theta_{n})=f(0)$ holds foe every $n$.

We consider the SPDE (1.6) with the initial value $v_{n}(0)$ and the SPDE
(1.1) with the initial value $u_{n}(0)$ by replacing $a,g$ by $a_{n},g_{n}$,
associated with $\xi_{n}$ respectively, where $u_{n}(0)$ is determined
similarly to $u(0)$ above, see (1.14). Then, we have smooth classical global-
in-time solutions $v_{n},u_{n}$ for such equations. Let $u$ and $v$ denote the
solutions of (1.1) and (1.6) in paracontrolled sense associated with
$\hat{\xi}$ for $t<T_{*}$. Noting that $\theta_{n}\to\theta$ in $C^{\alpha-1}$
and using assumptions on $\varphi^{\prime}$, we have, as $n\to\infty$,
$v_{n}(0)$ converges to $v(0)$ in $C^{\alpha-1}$, $m_{n}\to m$ and in
particular, $a_{n},g_{n}$ converge to $a,g$ on each compact set of
${\mathbb{R}}$. For the proof of $\theta_{n}\to\theta$ in $C^{\alpha-1}$, we
refer to the proof of Corollary 2.7, where a more complex case is dealt with.
Then, noting that $u_{n}(0)\to u(0)$ in $C^{\alpha}$ and using Remark 3.3 at
the end of this article, we know that $u_{n}$ converges to $u$ in
$\mathcal{L}_{T}^{\alpha}$ and therefore $v_{n}$ to $v$ in
$\mathcal{L}_{T}^{\alpha-1}$, $\alpha-1<\frac{1}{2}$ for $T<T_{*}$. Since the
initial values $v_{n}(0)$ move, we use the continuity of solutions in initial
values. Note that the coefficients $a_{n}$ and $g_{n}$ also move. So, we
require the condition $\alpha\in(\frac{13}{9},\frac{3}{2})$, see Theorem 1.3
or Remark 1.2 for explanation, and use Remark 3.3 in the proof.

In particular, recalling $\eta(x)=\langle\xi,1_{[0,x]}\rangle\in
C^{\alpha-1}([0,1])$ so that $\theta_{\xi}\in C^{\alpha-1}({\mathbb{T}})$,
$\theta_{\xi}>0$ and also $\varphi\in C^{4}({\mathbb{R}})$ satisfying (1.3),
$f_{n}(t):=\varphi(v_{n}(t))\theta^{-1}_{n}$ converges to
$f(t):=\varphi(v(t))\theta^{-1}$ in $\mathcal{L}_{T}^{\alpha-1}$ and
$f_{n}(0)=f(0)$ for all $n$. We note the lower semi-continuity of $\Phi(f)$ in
$C^{\alpha-1}$, which follows from the variational formula for the Dirichlet
form:

$\Phi(f)\equiv\Phi_{\theta}(f)=\frac{1}{2}\int_{\mathbb{T}}(\nabla
f)^{2}\theta
dx=\frac{1}{2}\sup\Big{\\{}-\int_{\mathbb{T}}\frac{\nabla(\theta\nabla
w)}{w}f^{2}dx;w\in C^{2}({\mathbb{T}})\Big{\\}},$

when $\theta\in C^{1}({\mathbb{T}})$, where $w(x)=0$ can happen only at $x$
such that $f(x)=0$. Indeed, we may use the integration by parts and note
$(\nabla f)^{2}-\nabla w\nabla(\frac{f^{2}}{w})=(\nabla f-f\frac{\nabla
w}{w})^{2}\geq 0$. Then, $f_{n}\to f$ in $C^{\alpha-1}$ implies
$\Phi_{\theta}(f)\leq\varliminf_{n\to\infty}\Phi_{\theta}(f_{n})$ if
$\theta\in C^{1}({\mathbb{T}})$, but in the definition of $\Phi_{\theta}(f)$,
we only have $\theta$ without its derivative so that this property holds also
for non-smooth $\theta$ by taking the limit $\theta_{m}\to\theta$ (in
$L^{\infty}$) or $\eta_{m}\to\eta$ (in $L^{\infty}$) introduced as above. More
precisely, noting that the constant $C(\theta)$ in (2.9) can be taken
uniformly in $n$: $C(\theta_{n})\leq C$, since the constant
$C(\theta)=C(\theta_{\xi})$ can be estimated only by $\min\theta$ and
$\max\theta$, for $t<T_{*}$, we see

$\displaystyle\Phi_{\theta}(f(t))$
$\displaystyle=\lim_{m\to\infty}\Phi_{\theta_{m}}(f(t))\leq\lim_{m\to\infty}\varliminf_{n\to\infty}\Phi_{\theta_{m}}(f_{n}(t))\leq
e^{Ct}\Phi_{\theta}(f(0)),$

at least if $f(0)\in H_{\theta}^{1}$; recall that $f_{n}(0)=f(0)$ for all $n$
and $\theta_{n}\to\theta$ in $C^{\alpha-1}$. For the last inequality, we use
$\Phi_{\theta_{n}}(f_{n}(t))\leq e^{Ct}\Phi_{\theta_{n}}(f_{n}(0))$ by noting
that, for arbitrary small $\varepsilon>0$,
$\theta_{m}/\theta_{n},\theta_{n}/\theta\leq 1+\varepsilon$ for large enough
$n,m$. In particular, if $|\mu_{\xi}|$ is small enough, one can take $C<0$.

Finally, we prove $T_{*}=\infty$, which shows the existence of $v(t)$ for all
$t\geq 0$. The above estimate on $\Phi(f(t))$ implies
$\|f(t)\|_{H_{\theta}^{1}}\leq M_{1}(e^{Ct/2}+1)$ for $t<T_{*}$, if $f(0)\in
H_{\theta}^{1}$, for some $M_{1}>0$. Indeed, this follows from
$\Phi(f)=\frac{1}{2}\|\nabla f\|_{L_{\theta}^{2}}^{2}$ and Lemma 2.8 below
(equivalence of norms). However, by Sobolev’s imbedding theorem and noting
that $H^{1}\equiv H^{1}({\mathbb{T}})\simeq H_{\theta}^{1}$ from $\theta\in
C({\mathbb{T}})$, we have $H_{\theta}^{1}\subset
C^{\alpha-1},\alpha-1<\frac{1}{2}$ and this shows $\|v(t)\|_{C^{\alpha-1}}\leq
M_{2}(e^{Ct/2}+1),t<T_{*}$. Therefore, noting that $\|u(t)\|_{C^{\alpha}}\leq
2\|v(t)\|_{C^{\alpha-1}}$, we see $\|u(t)\|_{C^{\alpha}}\leq
2M_{2}(e^{Ct/2}+1),t<T_{*}$. This proves $T_{*}=\infty$ by Lemma 2.5 at least
if $f(0)=\varphi(v(0))\theta^{-1}\in H_{\theta}^{1}\simeq H^{1}$, that is, if
$v(0)\in\mathcal{D}$. ∎

If $|\mu_{\xi}|$ is small enough, based on the estimate (2.24) obtained in
Proposition 2.6 with $c_{*}:=-C>0$, we can show the exponential decay of
$v(t)$ to the unique stationary solution $\bar{v}_{m}$ for each conserved
quantity $m$.

######  Corollary 2.7.

Assume $v(0)\in\mathcal{D}$ as in Proposition 2.6. Then, if $|\mu_{\xi}|$ is
small enough, $f(t):=\varphi(v(t))\theta^{-1}$ converges to the constant
$z_{m}$ in $H^{1}({\mathbb{T}})$ exponentially fast as $t\to\infty$:

(2.25) $\displaystyle\|f(t)-z_{m}\|_{H^{1}}\leq
Ce^{-c_{*}t/2}\|f(0)-z_{m}\|_{H^{1}},$

where $m=\int_{\mathbb{T}}v(0,x)dx$. We also have

(2.26) $\displaystyle\|v(t)-\bar{v}_{m}\|_{C^{\alpha-1}}\leq
Ce^{-c_{*}t/2}\|f(0)-z_{m}\|_{H^{1}}.$

###### Proof.

By Lemma 2.8 below, under the conservation law, we obtain

$\displaystyle\|f(t)-z_{m}\|_{H_{\theta}^{1}}^{2}$ $\displaystyle\leq
C\|\nabla f(t)\|_{L_{\theta}^{2}}^{2}=2C\Phi(f(t))$ $\displaystyle\leq
2Ce^{-c_{*}t}\Phi(f(0))\leq Ce^{-c_{*}t}\|f(0)-z_{m}\|_{H_{\theta}^{1}}^{2}.$

This shows the desired estimate on $\|f(t)-z_{m}\|_{H^{1}}$, since the norm of
$H_{\theta}^{1}$ is equivalent to that of $H^{1}$ due to the boundedness of
$\eta(x)$.

In order to give the estimate (2.26) on $v(t)$, we first show the uniform
boundedness of $\|v(t)\|_{C^{\alpha-1}}$ in $t\geq 0$, i.e.,

(2.27) $\displaystyle\sup_{t\geq 0}\|v(t)\|_{C^{\alpha-1}}<\infty.$

By the assumption (1.3) on $\varphi$, we have that
$\|\varphi^{-1}(v)\|_{C^{\alpha-1}}\leq C(1+\|v\|_{C^{\alpha-1}}),v\in
C^{\alpha-1}$. Therefore, recalling that $v(t)=\varphi^{-1}(f(t)\theta)$ and
using Sobolev’s imbedding theorem, we have that

$\displaystyle\|v(t)\|_{C^{\alpha-1}}$ $\displaystyle\leq
C(1+\|f(t)\theta\|_{C^{\alpha-1}})$ $\displaystyle\leq
C(1+\|f(t)\|_{C^{\alpha-1}}\|\theta\|_{C^{\alpha-1}})\leq
C(1+\|\theta\|_{C^{\alpha-1}}\|f(t)\|_{H^{1}}).$

Recall that the constant $C$ changes from line to line. So, by (2.25), we
obtain (2.27). Thus, by noting (2.27), we take a function
$\widetilde{\varphi^{-1}}\in C^{2}({\mathbb{R}})$ with compact support such
that $\widetilde{\varphi^{-1}}(f(t)\theta)=v(t)$ for all $t>0$ and
$\widetilde{\varphi^{-1}}(z_{m}\theta)=\bar{v}_{m}(=\varphi^{-1}(z_{m}\theta))$.
Then, Lemma 9 of [3] gives that

$\displaystyle\|v(t)-\bar{v}_{m}\|_{C^{\alpha-1}}$
$\displaystyle=\|\widetilde{\varphi^{-1}}(f(t)\theta)-\widetilde{\varphi^{-1}}(z_{m}\theta)\|_{C^{\alpha-1}}$
$\displaystyle\leq\|\widetilde{\varphi^{-1}}\|_{C^{2}}(1+\|z_{m}\theta\|_{C^{\alpha-1}})\|(f(t)-z_{m})\theta\|_{C^{\alpha-1}}$
$\displaystyle\leq\|\widetilde{\varphi^{-1}}\|_{C^{2}}(1+\|z_{m}\theta\|_{C^{\alpha-1}})\|\theta\|_{C^{\alpha-1}}\|(f(t)-z_{m})\|_{H^{1}}.$

Consequently, noting that $\|\widetilde{\varphi^{-1}}\|_{C^{2}}$ is
independent of $t\geq 0$, we obtain the desired result (2.26) by (2.25). ∎

The following lemma was used in the proof of Proposition 2.6 and Corollary
2.7.

######  Lemma 2.8.

Assume the following condition for $f=f(x)\in H_{\theta}^{1}$, which comes
from the conservation law:

(2.28)
$\int_{\mathbb{T}}\big{(}\varphi^{-1}(f\theta)-\varphi^{-1}(z_{m}\theta)\big{)}dx=0.$

Then, we have

(2.29) $\|f-z_{m}\|_{L^{2}_{\theta}}\leq C\|\nabla f\|_{L^{2}_{\theta}},$

for some $C>0$. In particular, under the condition (2.28), the Sobolev norm
$\|f-z_{m}\|_{H^{1}_{\theta}}$ is equivalent to $\|\nabla
f\|_{L^{2}_{\theta}}$.

###### Proof.

First, note that, by mean value theorem applied for (2.28), we see
$z_{m}=f(y_{*})$ for some $y_{*}\in{\mathbb{T}}$. Indeed, by noting the
monotone increasing property of $\varphi^{-1}$, we see

$\int_{\mathbb{T}}\varphi^{-1}(\min
f\cdot\theta)dx\leq\int_{\mathbb{T}}\varphi^{-1}(f\theta)dx\,\Big{(}\\!\\!\\!=\int_{\mathbb{T}}\varphi^{-1}(z_{m}\theta)dx\Big{)}\leq\int_{\mathbb{T}}\varphi^{-1}(\max
f\cdot\theta)dx$

and this implies $\min f\leq z_{m}\leq\max f$. Therefore, we have

$\displaystyle\|f-z_{m}\|_{L^{2}}$
$\displaystyle=\Big{(}\int_{\mathbb{T}}\big{(}f(x)-f(y_{*})\big{)}^{2}dx\Big{)}^{1/2}$
$\displaystyle=\Big{(}\int_{\mathbb{T}}dx\Big{\\{}\int_{y_{*}}^{x}\nabla
f(y^{\prime})dy^{\prime}\Big{\\}}^{2}\Big{)}^{1/2}$ $\displaystyle\leq\|\nabla
f\|_{L^{2}}.$

Since $0<\theta\in C({\mathbb{T}})$, this implies (2.29).

The equivalence of $\|\nabla f\|_{L^{2}_{\theta}}$ to the Sobolev norm is now
clear as

$\|\nabla
f\|_{L^{2}_{\theta}}\leq\|f-z_{m}\|_{H^{1}_{\theta}}=\big{(}\|\nabla(f-z_{m})\|_{L^{2}_{\theta}}^{2}+\|f-z_{m}\|_{L^{2}_{\theta}}^{2}\big{)}^{1/2}\leq
C\|\nabla f\|_{L^{2}_{\theta}}.$

∎

The next lemma gives an initial layer type result. This is used to remove the
condition $v(0)\in\mathcal{D}$ in Proposition 2.6 and Corollary 2.7. Recall
that we only assume $\xi\in
C^{\alpha-2}({\mathbb{T}}),\alpha\in(\frac{13}{9},\frac{3}{2})$.

######  Lemma 2.9.

For every initial value $v(0)\in C^{\alpha-1}$ and all $t\in(0,T_{*})$, the
solution $v(t)\in C^{\alpha-1}$ of the SPDE (1.6) in paracontrolled sense
satisfies $f(t):=\varphi(v(t))\theta^{-1}\in H^{1}$, that is,
$v(t)\in\mathcal{D}$. In other words, even if $f(0)\notin H^{1}$, immediately
after, we have $f(t)\in H^{1},t>0$ and this proves $T_{*}=\infty$ by
Proposition 2.6.

###### Proof.

If $\xi$ is smooth, taking any $u_{0}$ such that $\nabla u_{0}+m=v_{0}$ for
given $v_{0}$, we have from (1.1) with $a=\varphi^{\prime}(\cdot+m)$ and
$g=\varphi(\cdot+m)$

$\partial_{t}u=\varphi^{\prime}(v)\nabla v+\varphi(v)\xi=\theta\nabla f+\mu
f.$

This is an integrated form of (2.4). Therefore, we have

$u(t)-u(0)=\theta\int_{0}^{t}\nabla f(s)ds+\mu\int_{0}^{t}f(s)ds$

or we can rewrite this as

(2.30) $\displaystyle\int_{0}^{t}\nabla
f(s,x)ds=\theta(x)^{-1}\Big{\\{}u(t,x)-u(0,x)-\mu\int_{0}^{t}f(s,x)ds\Big{\\}}.$

The right hand side belongs to $C^{\alpha-1}$, since $\theta^{-1}\in
C^{\alpha-1}$, $u(t)\in C^{\alpha}$ and $f(s)\in C([0,T],C^{\alpha-1})$. On
the other hand, in the left hand side, $\nabla f(s)\in C([0,T],C^{\alpha-2})$,
$T<T_{*}$, since $\varphi(v(s))\in\mathcal{L}_{T}^{\alpha-1},\theta^{-1}\in
C^{\alpha-1}$. Therefore, taking the limit in $\hat{\xi}$, we have (2.30) for
$t<T_{*}$, if we interpret the left hand side as a Bochner integral in
$C^{\alpha-2}$.

First note that

$\int_{0}^{t}\nabla f(s,x)ds=\nabla\int_{0}^{t}f(s,x)ds,$

where the integrals are Bochner integrals in $C^{\alpha-2}$ for the left hand
side and $C^{\alpha-1}$ for the right hand side. (This can be shown by
regarding both sides as generalized functions and by multiplying test function
$\psi$.) Thus, (2.30) implies $\nabla\int_{0}^{t}f(s,x)ds\in
C([0,T],C^{\alpha-1})$, since the right hand side of (2.30) is in this class.
This shows, by also noting an obvious relation $\int_{0}^{t}f(s,x)ds\in
C([0,T],C^{\alpha-1})$, that

$\int_{0}^{t}f(s,x)ds\in C([0,T],C^{\alpha}).$

Recall that the left hand side is defined as a Bochner integral in
$C^{\alpha-1}$ and $\hat{\xi}$ is already general.

Since $f\in C([0,T],C^{\alpha-1})$, for every $t\in(0,T_{*})$, by mean value
theorem applied in the space $C^{\alpha-1}$, we see that there exists
$\tau=\tau_{t}\in(0,t)$ such that

(2.31) $\displaystyle\frac{1}{t}\int_{0}^{t}f(s,x)ds=f(\tau_{t})$

holds. However, since the left hand side belongs to $C^{\alpha}$ and
$\alpha\in(\frac{4}{3},\frac{3}{2})$, we see $f(\tau_{t})\in C^{\alpha}$ and,
in particular, $f(\tau_{t})\in H^{1}$. Once we have $f(\tau_{t})\in H^{1}$,
taking the initial value of $v$ as $v(\tau_{t})$, by Proposition 2.6, we see
$f(s)\in H^{1}$ for every $s\geq\tau_{t}$ and the solution exists globally in
time. Since we can take $t\in(0,T_{*})$ arbitrary small, this shows the
conclusion, recalling the flow property, see Remark 2.4. ∎

######  Remark 2.5.

We can decompose $v$ through $\varphi$ as $\varphi(v)=f\cdot\theta_{\xi}$ and
the bad regularity $v\in C^{\alpha-1}$ comes only from $\eta$ in
$\theta_{\xi}$ and, by Lemma 2.9, $f$ is a good part. Recall that Tsatsoulis
and Weber [17] decomposed the solution of $P(\phi)$-dynamics on
${\mathbb{T}}^{2}$ into the sum of OU-part and good part. Since we deal with
quasilinear equation, our decomposition is nonlinear, but in a sense, similar
to this.

Now we can easily complete the proof of Theorem 1.1.

###### Proof of Theorem 1.1.

The conclusion follows immediately by combining Proposition 2.6, Corollary 2.7
and Lemma 2.9. Here we only give a short explanation for (1.11). We first note
that, since $v\in C([0,\infty),C^{\alpha-1})$, we have

(2.32)
$\displaystyle\sup_{t\in[0,1]}\|v(t)-\bar{v}_{m}\|_{C^{\alpha-1}}<\infty.$

On the other hand, by Lemma 2.9, we have $f(1)=\varphi(v(1))\theta^{-1}\in
H^{1}$. In particular, $v(1)\equiv v(1,v_{0})\in\mathcal{D}$ so that the
condition of Corollary 2.7 is satisfied. Therefore, by the flow property and
(2.26), for $t>1$, we obtain

$\displaystyle\|v(t)-\bar{v}_{m}\|_{C^{\alpha-1}}$
$\displaystyle=\|v(t-1,v(1,v_{0}))-\bar{v}_{m}\|_{C^{\alpha-1}}$
$\displaystyle\leq Ce^{-c_{*}(t-1)/2}\|f(1)-\bar{v}_{m}\|_{H^{1}},\ t>1,$

which implies (1.11) by noting (2.32). ∎

###  2.3 Global existence and asymptotic behavior of $u(t)$ as $t\to\infty$

We can apply the result for $v(t)$ to study the global-in-time existence and
the asymptotic behavior of $u(t)$. First we note that $u(t)$ can be recovered
from $v(t):=\nabla u(t)$ at least when $\xi\in C^{\infty}({\mathbb{T}})$, cf.
Subsection 1.4. Note that $m=0$ under this choice.

######  Lemma 2.10.

Assume $\xi\in C^{\infty}({\mathbb{T}})$ and let the initial value $u_{0}\in
C^{\alpha},\alpha\in(\frac{4}{3},\frac{3}{2})$ of (1.1) be given. We determine
$v(t)$ by solving (1.4) with initial value $v_{0}:=\nabla u_{0}$, and set

(2.33)
$u(t,x):=\int_{0}^{x}v(t,y)dy+\int_{\mathbb{T}}u_{0}(y)dy-\int_{\mathbb{T}}(1-y)v(t,y)dy+\int_{0}^{t}ds\int_{\mathbb{T}}\chi(v(s,y))\xi(y)dy,$

where we regard ${\mathbb{T}}=[0,1)$ especially in the first and third terms
of the right hand side. Then, $u(t)$ solves the SPDE (1.1) with
$a=\varphi^{\prime}$ and $g=\chi$.

###### Proof.

First, note that $u(t,x)$ defined by (2.33) satisfies the initial condition:

$u(0,x)=\int_{0}^{x}\nabla
u_{0}(y)dy+\int_{\mathbb{T}}u_{0}(y)dy-\int_{\mathbb{T}}(1-y)\nabla
u_{0}(y)dy=u_{0}(x).$

To see that it satisfies the SPDE (1.1), writing the sum of the second to
fourth terms in the right hand of (2.33) as $A(t)$, we have

$\displaystyle\partial_{t}u(t,x)$
$\displaystyle=\int_{0}^{x}\partial_{t}v(t,y)dy+\partial_{t}A(t)$
$\displaystyle=\int_{0}^{x}[\Delta\\{\varphi(v)\\}+\nabla\\{\chi(v)\xi\\}]dy+\partial_{t}A(t)$
$\displaystyle=\nabla\\{\varphi(v)\\}(t,x)-\nabla\\{\varphi(v)\\}(t,0)+\chi(v(t,x))\xi(x)-\chi(v(t,0))\xi(0)+\partial_{t}A(t).$

Since $\nabla u=v$ holds for $u$ defined by (2.33), we have
$\nabla\\{\varphi(v)\\}=a(\nabla u)\Delta u$ and $\chi(v)\xi=g(\nabla u)\xi$.
Therefore, to complete the proof of the lemma, it is enough to show

(2.34) $\partial_{t}A(t)=\nabla\\{\varphi(v)\\}(t,0)+\chi(v(t,0))\xi(0).$

However,

$\displaystyle\partial_{t}A(t)$
$\displaystyle=-\int_{\mathbb{T}}(1-y)\partial_{t}v(t,y)dy+\int_{\mathbb{T}}\chi(v(t,y))\xi(y)dy$
$\displaystyle=-\int_{\mathbb{T}}(1-y)[\Delta\\{\varphi(v)\\}+\nabla\\{\chi(v)\xi\\}]dy+\int_{\mathbb{T}}\chi(v(t,y))\xi(y)dy$
$\displaystyle=-\int_{\mathbb{T}}[\nabla\\{\varphi(v)\\}+\chi(v)\xi]dy-(1-y)[\nabla\\{\varphi(v)\\}+\chi(v)\xi]\Big{|}_{0}^{1}+\int_{\mathbb{T}}\chi(v(t,y))\xi(y)dy$
$\displaystyle=[\nabla\\{\varphi(v)\\}+\chi(v)\xi](t,0),$

which shows (2.34). ∎

######  Remark 2.6.

When $\xi\in C^{\alpha-2}$, the last term in the right hand side of (2.33) is
well-defined even for general $\chi$ in the sense that the other four terms
are all well-defined. But, to look at it by itself especially without time
integral, the product $\chi(v(s))\xi$ is ill-posed in a classical sense, since
$\chi(v(s))\in C^{\alpha-1}$ and $\xi\in C^{\alpha-2}$ with
$\alpha\in(\frac{4}{3},\frac{3}{2})$. However, when $\chi=\varphi$, this
product turns out to be well-defined. Indeed, in this case,
$\varphi(v(t))=f(t)\theta$ with $f(t)\in H^{1}$ for every $t>0$ by Lemma 2.9.
Accordingly, the last term of (2.33) without time integral can be interpreted
as ${}_{H^{1}}\langle f(t),\theta\xi\rangle_{H^{-1}}$, $t>0$ if $\theta\xi$
can be regarded as an element of $H^{-1}$. Noting that $\theta\in
C^{\alpha-1}$, $\xi\in C^{\alpha-2}$ and $(\alpha-1)+(\alpha-2)<0$, in
general, the product $\theta\xi$ is ill-posed. However, in our special case,
we will see that $\theta\xi$ is well-defined as an element of $C^{\alpha-2}$
and then, in particular, $\theta\xi\in H^{-1}$ because of $C^{\alpha-2}\subset
H^{-1}$. This gives the motivation for the following proof of Theorem 1.2.

We now give the proof of Theorem 1.2.

###### Proof of Theorem 1.2.

The global-in-time existence of $u(t)$ is clear from $T_{*}=\infty$ noted in
Lemma 2.9. In the rest, we assume $|\mu_{\xi}|$ is sufficiently small as in
Theorem 1.1.

As we mentioned in Remark 2.6, we have to show that the term
${}_{H^{1}}\langle f(t),\theta\xi\rangle_{H^{-1}}$ (or equivalently the term
$\theta\xi$) is well-defined. For this, we apply Lemma 2.10 by introducing an
approximation of $u(t)$ as in the proof of Proposition 2.6. Since Lemma 2.10
holds for $v_{0}\in C^{\alpha-1}$, in the following, we consider the initial
values $u_{n}(0)=u_{0}\in C^{\alpha}$ for all $n$. Note that, we just have
$v(0)=\nabla u_{0}\in C^{\alpha-1}$, which does not imply
$v(0)\in\mathcal{D}$.

Take a sequence of enhanced noises $\hat{\xi}_{n}$ such that $\xi_{n}\in
C^{\infty}({\mathbb{T}})$ and $\hat{\xi}_{n}$ converges to $\hat{\xi}$ in
$C^{\alpha-2}\times C^{2\alpha-3}$ as $n\to\infty$. Then, the associated
solution $u_{n}$ converges to $u$ in $\mathcal{L}_{T}^{\alpha}$. Since
$T_{*}=\infty$, one can take $T>0$ arbitrarily large and, in particular, we
have for every $t\geq 0$,

(2.35) $\displaystyle\|u_{n}(t)-u(t)\|_{C^{\alpha}}\to 0\quad(n\to\infty).$

Since $\xi_{n}\in C^{\infty}({\mathbb{T}})$, by Lemma 2.10, we have the
formula (2.33) for $u_{n}(t,x)$ by replacing $v,\xi$ by $v_{n}:=\nabla
u_{n},\xi_{n}$, respectively, in the right hand side, which we denote as
$A_{1}^{n}(t,x)+A_{2}-A_{3}^{n}(t)+A_{4}^{n}(t)$. Recall that we take
$a=\varphi^{\prime}$ and $g=\chi=\varphi$ here. We also denote
$f_{n}(t,x):=\varphi(v_{n}(t,x))\theta_{n}^{-1}(x)$ and
$\theta_{n}:=\theta_{\xi_{n}}$. Then, we see $f_{n}(t,x)\to
f(t,x):=\varphi(v(t,x))\theta^{-1}(x)$ in $\mathcal{L}_{T}^{\alpha-1}$ as
before, where $\theta=\theta_{\xi}$.

Noting that $\xi_{n}\in C^{\infty}({\mathbb{T}})$ and (1.8), we have
$\theta_{n}\xi_{n}=\mu_{n}-\nabla\theta_{n}$, which are well-defined as the
elements of $C^{\alpha-2}$ and then

$\displaystyle\|\theta_{n}\xi_{n}-\theta_{m}\xi_{m}\|_{C^{\alpha-2}}=$
$\displaystyle\|\mu_{n}-\nabla\theta_{n}-(\mu_{m}-\nabla\theta_{m})\|_{C^{\alpha-2}}$
$\displaystyle\leq$
$\displaystyle|\mu_{n}-\mu_{m}|+\|\nabla\theta_{n}-\nabla\theta_{m}\|_{C^{\alpha-2}}$
$\displaystyle\leq$
$\displaystyle|\mu_{n}-\mu_{m}|+C\|\theta_{n}-\theta_{m}\|_{C^{\alpha-1}},$

where $\mu_{n}:=\mu_{\xi_{n}}$. Therefore, we see that $\theta_{n}\xi_{n}$ is
a Cauchy sequence of $C^{\alpha-2}$ by noting that $\theta_{n}\to\theta$ in
$C^{\alpha-1}$ and $\mu_{n}\to\mu=\mu_{\xi}$ as $n\to\infty$. In the sequel,
we denote by $\theta\xi$ the limit of the sequence $\theta_{n}\xi_{n}$ in
$C^{\alpha-2}$. So, we have $\theta\xi\in C^{\alpha-2}$ and in particular,
$\theta\xi\in H^{-1}$.

Using the above notation, we claim that

(2.36)
$\displaystyle\lim_{n\to\infty}\\{A_{1}^{n}(t,x)-A_{3}^{n}(t)+A_{4}^{n}(t)\\}$
$\displaystyle=$
$\displaystyle\int_{0}^{x}v(t,y)dy-\int_{\mathbb{T}}(1-y)v(t,y)dy+\int_{0}^{t}{}_{H^{1}}\langle
f(s),\theta\xi\rangle_{H^{-1}}ds$ $\displaystyle=:$ $\displaystyle
A_{1}(t,x)-A_{3}(t)+A_{4}(t).$

By Lemma 2.9, we know $f(t)\in H^{1}$ and $\int_{0}^{t}f(s,x)ds\in H^{1}$ for
$t>0$. So, ${}_{H^{1}}\langle f(t),\theta\xi\rangle_{H^{-1}},t>0$ is well-
defined. In particular, we see $A_{4}(t)$ is well-defined under the assumption
$a=g^{\prime}$. Let us first show

(2.37) $\displaystyle\lim_{n\to\infty}A_{4}^{n}(t)=A_{4}(t).$

In order to do it, we show $\int_{0}^{t}f_{n}(s,x)ds$ converges to
$\int_{0}^{t}f(s,x)ds$ in $H^{1}$ for $t>0$. By (2.30) and
$\theta_{n}\to\theta$ in $C^{\alpha-1}$ (in particular in $L^{\infty}$),
$\inf_{x,n}\theta_{n}(x)>0,\inf_{x}\theta(x)>0$, we have

$\displaystyle\left\|\nabla\int_{0}^{t}f_{n}(s)ds-\nabla\int_{0}^{t}f(s)ds\right\|_{L^{2}}$
$\displaystyle\leq$
$\displaystyle\|\theta_{n}^{-1}(u_{n}(t)-u(t))\|_{L^{2}}+\left\|(\theta_{n}^{-1}-\theta^{-1})\Big{(}u(t)-u_{0}+\mu\int_{0}^{t}f(s)ds\Big{)}\right\|_{L^{2}}$
$\displaystyle\quad+\left\|\theta_{n}^{-1}\mu_{n}\int_{0}^{t}f_{n}(s)ds-\theta_{n}^{-1}\mu\int_{0}^{t}f(s)ds\right\|_{L^{2}}$
$\displaystyle\leq$ $\displaystyle
C\left\\{\|u_{n}(t)-u(t)\|_{L^{2}}+\left\|(\theta_{n}-\theta)\Big{(}u(t)-u_{0}+\mu\int_{0}^{t}f(s)ds\Big{)}\right\|_{L^{2}}\right.$
$\displaystyle\left.\quad+\left\|\mu_{n}\int_{0}^{t}f_{n}(s)ds-\mu\int_{0}^{t}f(s)ds\right\|_{L^{2}}\right\\}.$

Since $f_{n}(t)\to f(t)\in\mathcal{L}_{T}^{\alpha-1},T>0$, we have
$\lim_{n\to\infty}\|\int_{0}^{t}f_{n}(s)ds-\int_{0}^{t}f(s)ds\|_{L^{2}}=0$.
Taking $n\to\infty$ in both sides of the above inequality, we have
$\lim_{n\to\infty}\|\nabla\int_{0}^{t}f_{n}(s)ds-\nabla\int_{0}^{t}f(s)ds\|_{L^{2}}^{2}=0$.
Therefore, we obtain the desired result, from which (2.37) follows by noting
that
$A_{4}^{n}(t)=\int_{0}^{t}ds\int_{\mathbb{T}}f_{n}(s,y)\theta_{n}(y)\xi_{n}(y)dy=\int_{0}^{t}{}_{H^{1}}\langle
f_{n}(s),\theta_{n}\xi_{n}\rangle_{H^{-1}}ds$.

For $A_{1}^{n}(t,x)$, we have $A_{1}^{n}(t,x)\to A_{1}(t,x)$ in
$C([0,T],C^{\alpha}),T>0$ as $n\to\infty$. Indeed, noting that $C^{\alpha}$
coincides with the usual $\alpha$-Hölder space for
$\alpha\in(\frac{13}{9},\frac{3}{2})$, we have

$\displaystyle\|A_{1}^{n}(t)-A_{1}(t)\|_{C^{\alpha}}=$
$\displaystyle\|A_{1}^{n}(t)-A_{1}(t)\|_{L^{\infty}}+\|\nabla
A_{1}^{n}(t)-\nabla A_{1}(t)\|_{L^{\infty}}$ $\displaystyle+\sup_{x\neq
y\in{\mathbb{T}}}\frac{|\nabla(A_{1}^{n}-A_{1})(t,x)-\nabla(A_{1}^{n}-A_{1})(t,y)|}{|x-y|^{\alpha-1}}$
$\displaystyle=$
$\displaystyle\|A_{1}^{n}(t)-A_{1}(t)\|_{L^{\infty}}+\|v_{n}(t)-v(t)\|_{C^{\alpha-1}}$
$\displaystyle\leq$
$\displaystyle\|v_{n}(t)-v(t)\|_{L^{\infty}}+\|v_{n}(t)-v(t)\|_{C^{\alpha-1}}$
$\displaystyle\leq$ $\displaystyle 2\|v_{n}(t)-v(t)\|_{C^{\alpha-1}},$

where $\nabla A_{1}^{n}(t)=v_{n}(t)$ and $\nabla A_{1}(t)=v(t)$ have been used
for the second equation. Therefore, noting that $v_{n}(t)\to v(t)$ in
$\mathcal{L}_{T}^{\alpha-1}$, we obtain the desired result. Finally for
$A_{3}^{n}(t)$, by similar arguments, it is easy to see
$|A_{3}^{n}(t)-A_{3}(t)|\leq\frac{1}{2}\|v_{n}(t)-v(t)\|_{L^{\infty}}\leq\frac{1}{2}\|v_{n}(t)-v(t)\|_{C^{\alpha-1}}$,
which implies that $A_{3}^{n}(t)$ converges uniformly to $A_{3}(t)$ on any
compact interval $[0,T]$. As a consequence, the proof of (2.36) is completed.

Combining (2.35) with (2.36), we have

$\displaystyle u(t,x)=A_{1}(t,x)+A_{2}-A_{3}(t)+A_{4}(t).$

We now show the uniform boundedness of $\|A_{1}(t)\|_{C^{\alpha}}+|A_{3}(t)|$
in $t\geq 0$, that is,

(2.38) $\displaystyle\sup_{t\geq
0}\\{\|A_{1}(t)\|_{C^{\alpha}}+|A_{3}(t)|\\}<\infty.$

Indeed, this can be easily shown by Theorem 1.1. By similar arguments for
convergences above, we easily have

$\|A_{1}(t)\|_{C^{\alpha}}+|A_{3}(t)|\leq C\|v(t)\|_{C^{\alpha-1}},\ t\geq 0.$

Noting that $\int_{\mathbb{T}}v_{0}(y)dy=\int_{\mathbb{T}}\nabla
u_{0}(y)dy=0\,(=m)$ and $|\mu|=|\mu_{\xi}|$ is sufficiently small, Theorem 1.1
gives that

$\|v(t)\|_{C^{\alpha-1}}\leq\|\bar{v}_{0}\|_{C^{\alpha-1}}+Ce^{-ct},$

where $\bar{v}_{0}$ is defined by (1.10) with $m=0$. Therefore, we have
$\sup_{t\geq 0}\|v(t)\|_{C^{\alpha}}<\infty$ and (2.38) is shown.

Noting that $A_{2}$ is a constant, in order to show (1.12), it is sufficient
to show

(2.39) $\displaystyle\sup_{t\geq 0}|A_{4}(t)-z_{0}\mu t|<\infty.$

Let us rewrite $A_{4}(t)$ as follows.

(2.40) $\displaystyle A_{4}(t)=z_{0}t{}_{H^{1}}\langle
1,\theta\xi\rangle_{H^{-1}}+\int_{0}^{t}r(s)ds,$

where $z_{0}$ is determined by (1.9) with $m=0$ and $r(s):={}_{H^{1}}\langle
f(s)-z_{0},\theta\xi\rangle_{H^{-1}}.$ By analogous arguments for (1.11), see
the proof of Theorem 1.1, we have

(2.41) $\displaystyle|r(s)|\leq
Ce^{-c_{*}s/2}\|\theta\xi\|_{H^{-1}}\|f(1)-z_{0}\|_{H^{1}},\ s\geq 1.$

holds for some $c_{*}>0$, where Corollary 2.7 and $f(1)\in H^{1}$ have been
used; recall that $|\mu_{\xi}|$ is sufficiently small. Noting that

$\left|\int_{0}^{1}r(s)ds\right|\leq\left\|\int_{0}^{1}(f(s)-z_{0})ds\right\|_{H^{1}}\|\theta\xi\|_{H^{-1}}$

and using (2.41), we see

$\displaystyle\sup_{t\geq 0}\left|\int_{0}^{t}r(s)ds\right|<\infty.$

Finally for the first term in the right hand side of (2.40), recalling that
$\theta\xi$ denotes the limit of $\theta_{n}\xi_{n}$ in $C^{\alpha-2}$ and
$\theta_{n}\xi_{n}=\mu_{n}-\nabla\theta_{n}$, we see

${}_{H^{1}}\langle
1,\theta\xi\rangle_{H^{-1}}=\lim_{n\to\infty}\int_{\mathbb{T}}\theta_{n}\xi_{n}dy=\lim_{n\to\infty}\int_{\mathbb{T}}(\mu_{n}-\nabla\theta_{n})dy=\lim_{n\to\infty}\mu_{n}=\mu,$

where the periodicity of $\theta_{n}$ has been used for the third equation.
Consequently, summarizing the above estimates, we have (2.39) and complete the
proof of Theorem 1.2. ∎

##  3 Continuity of the solution $u(t)$ of (1.1) in initial values

In this section, we give the proof of Theorem 1.3. As in [7] and stated in
Remark 3.2, Theorem 1.3 follows from Theorem 3.1 below, which is the main
result of this section. Since this part is a continuation of [7], the same
notations as in [7] will be mostly used. We assume
$\beta\in(\frac{1}{3},\alpha-1)$, $\gamma\in(2\beta+1,\alpha+\beta)$, and use
notations $K_{0}$, $K(\|{\bf u}\|_{\alpha,\beta,\gamma})$ and $K(\|{\bf
u}_{1}\|_{\alpha,\beta,\gamma},\|{\bf u}_{2}\|_{\alpha,\beta,\gamma})$
introduced at the end of Section 2 of [7], which may change from line to line.
We still write $a\lesssim b$ for two non-negative functions $a$ and $b$ if
there exists a constant $C>0$ independent of the variables under consideration
such that $a\leq Cb$. In addition, to emphasize the initial value, we
sometimes write $K_{0}(\|u_{0}\|_{C^{\alpha}})$,
$K_{0}(\|u_{0}^{1}\|_{C^{\alpha}},\|u_{0}^{2}\|_{C^{\alpha}})$ for $K_{0}$.

Noting that Theorem 1.3 is about the continuity of the local-in-time solutions
on initial values and recalling Lemma 1.4, for simplicity but without loss of
generality, we may assume the coefficients $a,g\in C_{b}^{3}({\mathbb{R}})$
with (1.2) in the sequel, which are same as in [7].

Let $\widetilde{\bf C}_{\alpha,\beta,\gamma}(X)$ be the family of functions
controlled by $X$, that is,

$\widetilde{\bf
C}_{\alpha,\beta,\gamma}(X):=\\{(u^{\prime},u^{\sharp});u=\bar{\Pi}_{u^{\prime}}X+u^{\sharp},\
\|(u^{\prime},u^{\sharp})\|_{\alpha,\beta,\gamma}<\infty\\},$

where $\bar{\Pi}_{u^{\prime}}X$ denotes the modified paraproduct of
$u^{\prime}$ and $X=(-\Delta)^{-1}Q\xi$, see Section 2 of [7] and

$\|(u^{\prime},u^{\sharp})\|_{\alpha,\beta,\gamma}:=\|u^{\prime}\|_{\mathcal{L}_{T}^{\beta}}+\|u^{\sharp}\|_{\mathcal{L}_{T}^{\alpha}}+\sup_{0<t\leq
T}t^{\frac{\gamma-\alpha}{2}}\|u^{\sharp}(t)\|_{C^{\gamma}}.$

Noting that $u$ is a function of $u^{\prime}$ and $u^{\sharp}$, we used the
space ${\bf C}_{\alpha,\beta,\gamma}(X)$ and the norm
$\|(u,u^{\prime})\|_{\alpha,\beta,\gamma}$ in [7], which are equivalent to the
above. For the given initial value $u_{0}$, we impose additional conditions
$u(0)=u_{0},\ u^{\prime}(0)=\frac{g(\nabla u_{0})}{a(\nabla u_{0})}$ and
define the (skew product) space $\mathcal{X}$ by

$\mathcal{X}:=\Big{\\{}(u_{0},u^{\prime},u^{\sharp})\in
C^{\alpha}\times\widetilde{\bf
C}_{\alpha,\beta,\gamma};(u^{\prime},u^{\sharp})\text{ satisfies
}u(0)=u_{0},u^{\prime}(0)=\frac{g(\nabla u_{0})}{a(\nabla u_{0})}\Big{\\}},$

which is viewed as a metric space embedded in the Banach space
$C^{\alpha}\times\widetilde{\bf C}_{\alpha,\beta,\gamma}$ equipped with the
norm

$\|(u_{0},{\bf u})\|_{C^{\alpha}\times\widetilde{\bf
C}_{\alpha,\beta,\gamma}}:=\|u_{0}\|_{C^{\alpha}}+\|{\bf u}\|_{\widetilde{\bf
C}_{\alpha,\beta,\gamma}},\ \ {\bf u}=(u^{\prime},u^{\sharp}).$

Let $\mathcal{L}^{0}:=\partial_{t}-a(\nabla u_{0}^{T})\Delta$,
$u_{0}^{T}:=e^{T\Delta}u_{0}$, where $e^{t\Delta}$ denotes the semigroup
generated by $\Delta$ on ${\mathbb{T}}$. Let us now define the map $\Phi$ from
$\mathcal{X}$ into itself by

(3.1)
$\displaystyle\Phi:(u_{0},u^{\prime},u^{\sharp})\in\mathcal{X}\mapsto(u_{0},v^{\prime},v^{\sharp})\in\mathcal{X},$

where

(3.2) $\displaystyle v^{\prime}=$ $\displaystyle\frac{g(\nabla
u)-\big{(}a(\nabla u)-a(\nabla u_{0}^{T})\big{)}u^{\prime}}{a(\nabla
u_{0}^{T})},$ (3.3) $\displaystyle\mathcal{L}^{0}v^{\sharp}=$
$\displaystyle\Pi_{a(\nabla
u_{0}^{T})v^{\prime}}\xi-\mathcal{L}^{0}(\bar{\Pi}_{z^{\prime}}X)+g^{\prime}(\nabla
u)\Pi(\nabla u^{\sharp},\xi)-a^{\prime}(\nabla u)\Pi(\nabla
u^{\sharp},\bar{\Pi}_{u^{\prime}}\xi)$ $\displaystyle\qquad+\big{(}a(\nabla
u)-a(\nabla u_{0}^{T})\big{)}\Delta u^{\sharp}+\zeta$ $\displaystyle=:$
$\displaystyle\Pi_{a(\nabla
u_{0}^{T})v^{\prime}}\xi-\mathcal{L}_{0}(\bar{\Pi}_{z}X)+\varepsilon_{1}(u,u^{\prime})+\varepsilon_{2}(u,u^{\prime})+\zeta(u,u^{\prime}),$

where $\varepsilon_{1}$ denotes the difference of the third and the fourth
terms, $\varepsilon_{2}$ denotes the fifth term in the right hand of (3.3),
and $\zeta$ denotes the remainder, see Lemma 3.3 and (2.32) of [7] for
details.

######  Remark 3.1.

The map $\Phi$ defined by (3.1) is a little different from the original one
introduced in [7], where the map $\Phi:(u,u^{\prime})\to(v,v^{\prime})$
defined on ${\bf
C}_{\alpha,\beta,\gamma}(X):=\\{(u,u^{\prime});u=\bar{\Pi}_{u^{\prime}}X+u^{\sharp},\|(u,u^{\prime})\|_{\alpha,\beta,\gamma}<\infty\\}$
(more precisely on the space $\mathcal{B}_{T}(\lambda)$, see Subsection 3.1 of
[7] or equivalently (3.4) below) is introduced, see (2.16)-(2.18) in [7]. As
we pointed out at the end of Section 3 of [7], the space ${\bf
C}_{\alpha,\beta,\gamma}(X)$ is actually identical to that of pairs
$(u^{\prime},u^{\sharp})$ with the norm
$\|(u,u^{\prime})\|_{\alpha,\beta,\gamma}$ defined in terms of
$(u^{\prime},u^{\sharp})$. In this way, ${\bf C}_{\alpha,\beta,\gamma}(X)$ is
identified with $\widetilde{\bf C}_{\alpha,\beta,\gamma}$. Therefore, Theorem
3.1 of [7] holds for $\Phi$ defined by (3.1) whenever the initial value is
fixed and we will not repeat this fact in the sequel.

For $\lambda>0$ and $T>0$, set

(3.4) $\displaystyle\mathcal{B}_{T}(\lambda):=\Big{\\{}({u}_{0},{\bf
u})\in\mathcal{X};\ u_{0}\in C^{\alpha},\ \|{\bf
u}\|_{\alpha,\beta,\gamma}\leq\lambda\Big{\\}}.$

We recall that for each fixed initial value $u_{0}$, in [7], it is proved that
the map $\Phi$ is contractive on $\mathcal{B}_{T}(\lambda)$ for a large enough
$\lambda$ and a small time $T>0$, and the unique fixed point on
$\mathcal{B}_{T}(\lambda)$ solves the paracontrolled SPDE (1.1) up to time
$T>0$, see Theorem 3.1 of [7] or Theorem 2.4 for details. In particular, one
explicit choice of $\lambda$ and $T$ is given in Theorem 2.4. Let us now give
the main result of this section.

######  Theorem 3.1.

Let $\alpha\in(\frac{13}{9},\frac{3}{2})$. Then, we have that the map $\Phi$
defined by (3.1) depends continuously on the initial value $u_{0}$ and its
contractivity on $\mathcal{B}_{T}(\lambda),\ T>0$ is locally uniform in
initial values. More precisely, there exists a unique continuous map $u_{0}\in
C^{\alpha}\mapsto(u^{\prime}({u}_{0}),u^{\sharp}({u}_{0}))\in\widetilde{\bf
C}_{\alpha,\beta,\gamma}$ up to time $T$ such that

$\Phi({u}_{0},u^{\prime}({u}_{0}),u^{\sharp}({u}_{0}))=({u}_{0},u^{\prime}({u}_{0}),u^{\sharp}(u_{0}))$

and $(u^{\prime}({u}_{0}),u^{\sharp}({u}_{0}))$ (or equivalently
$(u({u}_{0}),u^{\prime}({u}_{0}))$) solves the SPDE (1.1) starting from
${u}_{0}$ up to time $T$ in the paracontrolled sense. Moreover, $T$ can be
taken depending continuously on $\|u_{0}\|_{C^{\alpha}}$ and
$\|\hat{\xi}\|_{C^{\alpha-2}\times C^{2\alpha-3}}$.

######  Remark 3.2.

(i) Combining Theorem 3.1-(ii) of [7] with Theorem 3.1 above, we see that the
map $\Phi$ depends continuously on both the initial value $u_{0}\in
C^{\alpha}$ and the enhanced noise $\hat{\xi}$. In particular, the unique
fixed point of $\Phi$ in $\mathcal{B}_{T}(\lambda)$ inherits the continuity in
$(\hat{\xi},u_{0})$.
(ii) Theorem 1.3 immediately follows from Theorem 3.1 and the relation between
$\Phi$ and the solution $u(t)$, see the proof of Theorem 1.1 in [7] for
detailed explanation.
(iii) To obtain the estimate (3.13) in the proof of Lemma 3.8 below, which is
important for the proof of Theorem 3.1, the original assumption
$\alpha\in(\frac{4}{3},\frac{3}{2})$ in [7] is changed to
$\alpha\in(\frac{13}{9},\frac{3}{2})$ due to our techniques. Although the
assumption on $\alpha$ becomes slightly restrictive, the most important case
of the spatial white noise on ${\mathbb{T}}$ is covered.

We can obtain immediately the next result by Theorem 3.1.

######  Corollary 3.2.

The SPDE (1.1) is solvable up to time $T>0$ and one can take
$T=T(\|u_{0}\|_{C^{\alpha}},\|\hat{\xi}\|_{C^{\alpha-2}\times
C^{2\alpha-3}})$, which depends continuously on $\|u_{0}\|_{C^{\alpha}}$ and
$\|\hat{\xi}\|_{C^{\alpha-2}\times C^{2\alpha-3}}$.

To prove Theorem 3.1, thanks to Theorem 3.1 in [7] and the implicit function
theorem, the main task is to show the continuity of the map $\Phi$. We will
first give some lemmas as preparation and postpone the proof of Theorem 3.1 to
the end of this section.

Since the main purpose of this section is to prove the continuity of $u(t)$ in
its initial values, we have to consider different initial values. So, we first
generalize (2.15) in Lemma 2.9 of [7] to the next lemma.

######  Lemma 3.3.

Let $F\in C_{b}^{2}({\mathbb{R}})$, $\alpha\in(1,2)$ and
$\beta\in(0,\alpha-1]$. Then, for any $u,v\in\mathcal{L}_{T}^{\alpha}$
(without the restriction $u(0)=v(0)$), we have

$\displaystyle\|F(\nabla u)-F(\nabla v)\|_{\mathcal{L}_{T}^{\beta}}\lesssim$
$\displaystyle
T^{\frac{\alpha-\beta-1}{2}}\|F\|_{C^{2}}(1+\|u\|_{\mathcal{L}_{T}^{\alpha}})\|u-v\|_{\mathcal{L}_{T}^{\alpha}}$
$\displaystyle+\|F\|_{C^{2}}(1+\|u(0)\|_{C^{\alpha}})\|u(0)-v(0)\|_{C^{\alpha}}.$

###### Proof.

This lemma can be easily shown by modifying the proof of Lemma 9 of [3].
Although we do not assume $u(0)=v(0)$, refining the proof of Lemma 9 of [3],
we deduce that

$\displaystyle\|F(\nabla u)-F(\nabla v)\|_{\mathcal{L}_{T}^{\beta}}\lesssim$
$\displaystyle T^{\frac{\alpha-\beta-1}{2}}\|F\|_{C^{2}}(1+\|\nabla
u\|_{\mathcal{L}_{T}^{\alpha-1}})\|\nabla u-\nabla
v\|_{\mathcal{L}_{T}^{\alpha-1}}$ $\displaystyle+\|F(\nabla u(0))-F(\nabla
v(0))\|_{C^{\beta}},$

which gives the desired result by Lemmas 2.3 and 2.9 of [7]. ∎

As an application of Lemma 3.3, we have the following lemma, which will be
used frequently.

######  Lemma 3.4.

For any
$(u_{0}^{i},u_{i}^{\prime},u_{i}^{\sharp})\in\mathcal{B}_{T}(\lambda),i=1,2$,
we have

(3.5) $\displaystyle\left\|a(\nabla u_{1})-a(\nabla
u_{0}^{1,T})-\Big{(}a(\nabla u_{2})-a(\nabla
u_{0}^{2,T})\Big{)}\right\|_{\mathcal{L}_{T}^{\beta}}$ $\displaystyle\lesssim$
$\displaystyle T^{\frac{\alpha-\beta-1}{2}}K(\|{\bf
u}_{1}\|_{\alpha,\beta,\gamma})(1+\|\xi\|_{C^{\alpha-2}})^{2}\|{\bf
u}_{1}-{\bf
u}_{2}\|_{\alpha,\beta,\gamma}+K_{0}(\|u_{0}^{1}\|_{C^{\alpha}})\|u_{0}^{1}-u_{0}^{2}\|_{C^{\alpha}}.$

###### Proof.

By Lemma 2.9-(i) and Lemma 3.2-(i) of [7], we have that for any
$\beta\in(0,\alpha-1]$,

(3.6) $\displaystyle\|a(\nabla u_{0}^{1,T})-a(\nabla
u_{0}^{2,T})\|_{C_{\beta}}$
$\displaystyle\lesssim(1+\|u_{0}^{1,T}\|_{C^{\beta+1}})\|u_{0}^{1,T}-u_{0}^{2,T}\|_{C^{\beta+1}}$
$\displaystyle\lesssim(1+\|u_{0}^{1}\|_{C^{\alpha}})\|u_{0}^{1}-u_{0}^{2}\|_{C^{\alpha}}.$

On the other hand, Lemma 3.3 gives that

$\displaystyle\left\|a(\nabla u_{1})-a(\nabla
u_{2})\right\|_{\mathcal{L}_{T}^{\beta}}\lesssim$ $\displaystyle
T^{\frac{\alpha-\beta-1}{2}}(1+\|u_{1}\|_{\mathcal{L}_{T}^{\alpha}})\|u_{1}-u_{2}\|_{\mathcal{L}_{T}^{\alpha}}+(1+\|u_{0}^{1}\|_{C^{\alpha}})\|u_{0}^{1}-u_{0}^{2}\|_{C^{\alpha}}.$

Therefore, by the estimate
$\|u\|_{\mathcal{L}_{T}^{\alpha}}\lesssim(1+\|X\|_{C^{\alpha}})\|{\bf
u}\|_{\alpha,\beta,\gamma}$, see (3.9) in [7], together with
$\|X\|_{C^{\alpha}}\lesssim\|\xi\|_{C^{\alpha-2}}$, we immediately obtain
(3.5) with $K(\|{\bf u}_{1}\|_{\alpha,\beta,\gamma})=(1+\|{\bf
u}_{1}\|_{\alpha,\beta,\gamma})$ and
$K_{0}(\|u_{0}^{1}\|_{C^{\alpha}})=(1+\|u_{0}^{1}\|_{C^{\alpha}})$. ∎

According to Theorem 3.1 of [7] and its proof, we have the following result.

######  Lemma 3.5.

Let $\Phi$ be the map on $\mathcal{B}_{T}(\lambda)$ defined by (3.1), that is,
$\Phi(u_{0},u^{\prime},u^{\sharp})=(u_{0},v^{\prime},v^{\sharp})$ for
$(u_{0},u^{\prime},u^{\sharp})\in\mathcal{B}_{T}(\lambda)$. Then, we have

$\displaystyle\|(v^{\prime},v^{\sharp})\|_{\alpha,\beta,\gamma}\lesssim
T^{\frac{\alpha+\beta-\gamma}{2}}K(\|{\bf
u}\|_{\alpha,\beta,\gamma})\tilde{K}_{1}(X,\xi)+K_{0}(\|u_{0}\|_{C^{\alpha}})(1+\|\xi\|_{C^{\alpha-2}}),$

where $v^{\prime}$ are $v^{\sharp}$ are determined by (3.2) and (3.3)
respectively, and $\tilde{K}_{1}(X,\xi)$ denotes the same constant as that
introduced in Proposition 3.7 of [7].

By Lemma 3.5, in particular, we know that $v^{\prime}$ is well-defined as an
element of $\mathcal{L}_{T}^{\beta}$ for each
$(u_{0},u^{\prime},u^{\sharp})\in\mathcal{B}_{T}(\lambda)$. In the following,
we will show the local Lipschitz continuity of $v^{\prime}$ in
$(u_{0},u^{\prime},u^{\sharp})$. For
$(u_{0}^{i},u_{i}^{\prime},u_{i}^{\sharp})\in\mathcal{B}_{T}(\lambda),\
i=1,2$, we set
$\Phi(u_{0}^{i},u_{i}^{\prime},u_{i}^{\sharp})=(u_{0}^{i},v_{i}^{\prime},v_{i}^{\sharp})$
in the following.

######  Lemma 3.6.

We have that $v^{\prime}$ is locally Lipschitz in
$(u_{0},u^{\prime},u^{\sharp})\in\mathcal{B}_{T}(\lambda)$. More precisely, we
have

$\displaystyle\|v_{1}^{\prime}-v_{2}^{\prime}\|_{\mathcal{L}_{T}^{\beta}}\lesssim$
$\displaystyle T^{\frac{\alpha-\beta-1}{2}}K(\|{\bf
u}_{1}\|_{\alpha,\beta,\gamma},\|{\bf
u}_{2}\|_{\alpha,\beta,\gamma})(1+\|\xi\|_{C^{\alpha-2}})^{2}\|{\bf
u}_{1}-{\bf u}_{2}\|_{\alpha,\beta,\gamma}$ $\displaystyle+K(\|{\bf
u}_{1}\|_{\alpha,\beta,\gamma})K_{0}(\|u_{0}^{1}\|_{C^{\alpha}},\|u_{0}^{2}\|_{C^{\alpha}})\|u_{0}^{1}-u_{0}^{2}\|_{C^{\alpha}},$

where ${\bf u}_{i}=(u^{\prime}_{i},u^{\sharp}_{i})$ for $i=1,2$.

###### Proof.

For simplicity of notation, we set

(3.7) $\displaystyle b(u_{i})=a(\nabla u_{i})-a(\nabla u_{0}^{i,T}),\ i=1,2.$

Then, it is easy to know that
$\|v_{1}^{\prime}-v_{2}^{\prime}\|_{\mathcal{L}_{T}^{\beta}}$ is bounded from
above by the sum of the following three terms:

$\displaystyle I_{1}=$ $\displaystyle\left\|\big{(}g(\nabla
u_{1})-b(u_{1})u_{1}^{\prime}\big{)}\Big{(}\frac{1}{a(\nabla
u_{0}^{1,T})}-\frac{1}{a(\nabla
u_{0}^{2,T})}\Big{)}\right\|_{\mathcal{L}_{T}^{\beta}},$ $\displaystyle
I_{2}=$ $\displaystyle\left\|\frac{1}{a(\nabla u_{0}^{2,T})}\Big{(}g(\nabla
u_{1})-g(\nabla
u_{2})-\big{(}b(u_{1})-b(u_{2})\big{)}u_{1}^{\prime}\Big{)}\right\|_{\mathcal{L}_{T}^{\beta}},$
$\displaystyle I_{3}=$
$\displaystyle\left\|\frac{b(u_{2})(u_{1}^{\prime}-u_{2}^{\prime})}{a(\nabla
u_{0}^{2,T})}\right\|_{\mathcal{L}_{T}^{\beta}}.$

Let us first deal with the first term $I_{1}$. From the proof of Lemma 3.4 of
[7], it easily follows that

$\displaystyle\|g(\nabla
u_{1})-b(u_{1})u_{1}^{\prime}\|_{\mathcal{L}_{T}^{\beta}}\lesssim
T^{\frac{\alpha-\beta-1}{2}}K(\|{\bf
u}_{1}\|_{\alpha,\beta,\gamma})(1+\|X\|_{C^{\alpha}})+K_{0}(\|u_{0}^{1}\|_{C^{\alpha}}).$

Recalling that $a\in C_{b}^{3}({{\mathbb{R}}})$ satisfies (1.2) and using
(3.6) together with Lemma 2.9 of [7], we have that

$\displaystyle\left\|\frac{1}{a(\nabla u_{0}^{1,T})}-\frac{1}{a(\nabla
u_{0}^{2,T})}\right\|_{C^{\beta}}\lesssim$
$\displaystyle\left\|\frac{1}{a(\nabla u_{0}^{1,T})a(\nabla
u_{0}^{2,T})}\right\|_{C^{\beta}}\|a(\nabla u_{0}^{1,T})-a(\nabla
u_{0}^{2,T})\|_{C_{\beta}}$ $\displaystyle\lesssim$
$\displaystyle(1+\|u_{0}^{1}\|_{C^{\alpha}})^{2}(1+\|u_{0}^{2}\|_{C^{\alpha}})\|a(\nabla
u_{0}^{1,T})-a(\nabla u_{0}^{2,T})\|_{C_{\beta}}$ $\displaystyle\lesssim$
$\displaystyle
K_{0}(\|u_{0}^{1}\|_{C^{\alpha}},\|u_{0}^{2}\|_{C^{\alpha}})\|u_{0}^{1}-u_{0}^{2}\|_{C^{\alpha}},$

where we have been used the estimate

(3.8) $\displaystyle\left\|\frac{1}{a(\nabla
u_{0}^{i,T})}\right\|_{C^{\beta}}\lesssim(1+\|u_{0}^{i}\|_{C^{\alpha}}),\
i=1,2,$

see (3.23) of [7], for the second inequality.

Therefore, by Lemma 2.1 of [7] and the above estimates, we obtain

$\displaystyle I_{1}\lesssim$ $\displaystyle\left\|g(\nabla
u_{1})-b(u_{1})u_{1}^{\prime}\right\|_{\mathcal{L}_{T}^{\beta}}\left\|\frac{1}{a(\nabla
u_{0}^{1,T})}-\frac{1}{a(\nabla u_{0}^{2,T})}\right\|_{C^{\beta}}$
$\displaystyle\lesssim$
$\displaystyle\Big{(}T^{\frac{\alpha-\beta-1}{2}}K(\|{\bf
u}_{1}\|_{\alpha,\beta,\gamma})(1+\|\xi\|_{C^{\alpha-2}})+1\Big{)}K_{0}(\|u_{0}^{1}\|_{C^{\alpha}},\|u_{0}^{2}\|_{C^{\alpha}})\|u_{0}^{1}-u_{0}^{2}\|_{C^{\alpha}}.$

Next, let us evaluate the second term $I_{2}$. By Lemma 3.4, we have

$\displaystyle\left\|\big{(}b(u_{1})-b(u_{2})\big{)}u_{1}^{\prime}\right\|_{\mathcal{L}_{T}^{\beta}}$
$\displaystyle\lesssim$ $\displaystyle T^{\frac{\alpha-\beta-1}{2}}K(\|{\bf
u}_{1}\|_{\alpha,\beta,\gamma})(1+\|\xi\|_{C^{\alpha-2}})^{2}\|{\bf
u}_{1}-{\bf
u}_{2}\|_{\alpha,\beta,\gamma}+K_{0}(\|u_{0}^{1}\|_{C^{\alpha}})\|{\bf
u}_{1}\|_{\alpha,\beta,\gamma}\|u_{0}^{1}-u_{0}^{2}\|_{C^{\alpha}}.$

In addition, we have the following estimate more easily.

$\displaystyle\|g(\nabla u_{1})-g(\nabla
u_{2})\|_{\mathcal{L}_{T}^{\beta}}\lesssim$ $\displaystyle
T^{\frac{\alpha-\beta-1}{2}}K(\|{\bf
u}_{1}\|_{\alpha,\beta,\gamma})(1+\|\xi\|_{C^{\alpha-2}})^{2}\|{\bf
u}_{1}-{\bf u}_{2}\|_{\alpha,\beta,\gamma}$
$\displaystyle+K_{0}(\|u_{0}^{1}\|_{C^{\alpha}})\|u_{0}^{1}-u_{0}^{2}\|_{C^{\alpha}}.$

Therefore, noting (3.8), we have

$\displaystyle I_{2}\lesssim$ $\displaystyle
T^{\frac{\alpha-\beta-1}{2}}K(\|{\bf
u}_{1}\|_{\alpha,\beta,\gamma},\|u_{0}^{2}\|_{C^{\alpha}})(1+\|\xi\|_{C^{\alpha-2}})^{2}\|{\bf
u}_{1}-{\bf u}_{2}\|_{\alpha,\beta,\gamma}$
$\displaystyle+K_{0}(\|u_{0}^{1}\|_{C^{\alpha}},\|u_{0}^{2}\|_{C^{\alpha}})(1+\|{\bf
u}_{1}\|_{\alpha,\beta,\gamma})\|u_{0}^{1}-u_{0}^{2}\|_{C^{\alpha}}.$

Finally, using (3.17) of [7] together with (3.8), we easily have

$\displaystyle I_{3}\lesssim T^{\frac{\alpha-\beta-1}{2}}K(\|{\bf
u}_{2}\|_{\alpha,\beta,\gamma})(1+\|\xi\|_{C^{\alpha-2}})\|{\bf u}_{1}-{\bf
u}_{2}\|_{\alpha,\beta,\gamma}.$

Consequently, the proof of this lemma is completed by the above estimates. ∎

In the next two lemmas, we give the bounds for the terms involving both
$u_{0}^{T}$ or (and) $v^{\sharp}$, which should be evaluated in the weighted
space in time.

######  Lemma 3.7.

We have

$\displaystyle\sup_{0<t\leq
T}t^{\frac{\gamma-\alpha}{2}}\Big{\|}\Big{(}\big{(}a(\nabla
u_{0}^{1,T})-a(\nabla u_{0}^{2,T})\big{)}\Delta
v_{1}^{\sharp}\Big{)}(t)\Big{\|}_{C^{\gamma-2}}\lesssim
K_{0}(\|u_{0}^{1}\|_{C^{\alpha}})\|(v_{1}^{\prime},v_{1}^{\sharp})\|_{\alpha,\beta,\gamma}\|u_{0}^{1}-u_{0}^{2}\|_{C^{\alpha}}.$

###### Proof.

By Lemma 3.5, we know $(v_{1}^{\prime},v_{1}^{\sharp})\in\widetilde{\bf
C}_{\alpha,\beta,\gamma}$. In particular, we have that for $t>0$,
$v_{1}^{\sharp}\in C^{\gamma}$. Then, noting that $0<\beta+\gamma-2<\beta$ and
using Lemma 2.4 of [7], we have

$\displaystyle\Big{\|}\Big{(}\big{(}a(\nabla u_{0}^{1,T})-a(\nabla
u_{0}^{2,T})\big{)}\Delta
v_{1}^{\sharp}\Big{)}(t)\Big{\|}_{C^{\gamma-2}}\lesssim$
$\displaystyle\Big{\|}\Big{(}\big{(}a(\nabla u_{0}^{1,T})-a(\nabla
u_{0}^{2,T})\big{)}\Delta
v_{1}^{\sharp}\Big{)}(t)\Big{\|}_{C^{\beta+\gamma-2}}$ $\displaystyle\lesssim$
$\displaystyle\Big{\|}a(\nabla u_{0}^{1,T})-a(\nabla
u_{0}^{2,T})\Big{\|}_{C^{\beta}}\|v_{1}^{\sharp}(t)\|_{C^{\gamma}}.$

Now, we conclude our proof by (3.6) together with the fact $\sup_{0<t\leq
T}t^{\frac{\gamma-\alpha}{2}}\|v_{1}^{\sharp}(t)\|_{C^{\gamma}}\leq\|(v_{1}^{\prime},v_{1}^{\sharp})\|_{\alpha,\beta,\gamma}.$
∎

######  Lemma 3.8.

Suppose further $\alpha\in(\frac{13}{9},\frac{3}{2})$. Then we have

(3.9) $\displaystyle\sup_{0<t\leq
T}t^{\frac{\gamma-\alpha}{2}}\Big{\|}\Big{(}\big{(}a(\nabla
u_{0}^{1,T})-a(\nabla
u_{0}^{2,T})\big{)}\Delta\bar{\Pi}_{v_{1}^{\prime}}X+{\Pi}_{(a(\nabla
u_{0}^{1,T})-a(\nabla
u_{0}^{2,T}))v_{1}^{\prime}}\xi\Big{)}(t)\Big{\|}_{C^{\alpha+\beta-2}}$
$\displaystyle\lesssim$ $\displaystyle
K_{0}(\|u_{0}^{1}\|_{C^{\alpha}})\|(v_{1}^{\prime},v_{1}^{\sharp})\|_{\alpha,\beta,\gamma}\|\xi\|_{C^{\alpha-2}}\|u_{0}^{1}-u_{0}^{2}\|_{C^{\alpha}}.$

###### Proof.

Set $\widetilde{a_{0}}:=a(\nabla u_{0}^{1,T})-a(\nabla u_{0}^{2,T})$ for the
sake of brevity. Using the the commutator
$R_{2}(v^{\prime}_{1},X)=[\Delta,\bar{\Pi}_{v_{1}^{\prime}}]X$ and $\Delta
X=-Q\xi=-(\xi-\xi({\mathbb{T}}))$, see (2.1) of [7], we have

(3.10)
$\displaystyle\|R_{2}(v^{\prime}_{1},X)\|_{C_{T}C^{\alpha+\beta-2}}\lesssim\|v_{1}^{\prime}\|_{C_{T}C^{\beta}}\|\xi\|_{C^{\alpha-2}}$

by Lemma 2.6 of [7] and

$\displaystyle\widetilde{a_{0}}\Delta\bar{\Pi}_{v_{1}^{\prime}}X=\widetilde{a_{0}}\big{(}\bar{\Pi}_{v_{1}^{\prime}}(\Delta
X)+R_{2}(v^{\prime}_{1},X)\big{)}=$
$\displaystyle-\widetilde{a_{0}}\bar{\Pi}_{v_{1}^{\prime}}\xi+\widetilde{a_{0}}R_{2}(v^{\prime}_{1},X).$

Then, an analogous argument for (2.26) of [7] shows that

$\displaystyle\widetilde{a_{0}}\Delta\bar{\Pi}_{v_{1}^{\prime}}X=-{\Pi}_{\widetilde{a_{0}}v_{1}^{\prime}}\xi-\Pi(\widetilde{a_{0}},\bar{\Pi}_{v_{1}^{\prime}}\xi)-R(\widetilde{a_{0}},v_{1}^{\prime};\xi)-\Pi_{\bar{\Pi}_{v_{1}^{\prime}}\xi}\widetilde{a_{0}}+\widetilde{a_{0}}R_{2}(v^{\prime}_{1},X),$

see Lemma 2.8 of [7] for the meaning of
$R(\widetilde{a_{0}},v_{1}^{\prime};\xi)$. Therefore, the term inside of the
norm $\|\cdot\|_{C^{\alpha+\beta-2}}$ of the left hand side of (3.9) equals to

(3.11)
$\displaystyle-\Pi(\widetilde{a_{0}},\bar{\Pi}_{v_{1}^{\prime}}\xi)-R(\widetilde{a_{0}},v_{1}^{\prime};\xi)-\Pi_{\bar{\Pi}_{v_{1}^{\prime}}\xi}\widetilde{a_{0}}+\widetilde{a_{0}}R_{2}(v^{\prime}_{1},X).$

Let us first deal with the first term of (3.11), i.e., the resonant term
$\Pi(\widetilde{a_{0}},\bar{\Pi}_{v_{1}^{\prime}}\xi)$. By Lemmas 2.9 and 3.2
of [7], we have for $\alpha<\gamma^{\prime}\leq\gamma$

(3.12) $\displaystyle\|\widetilde{a_{0}}\|_{C^{\gamma^{\prime}-1}}\lesssim$
$\displaystyle(1+\|u_{0}^{1,T}\|_{C^{\gamma^{\prime}}})\|u_{0}^{1,T}-u_{0}^{2,T}\|_{C^{\gamma^{\prime}}}$
$\displaystyle\lesssim$ $\displaystyle
T^{-\frac{\gamma^{\prime}-\alpha}{2}}(1+T^{-\frac{\gamma^{\prime}-\alpha}{2}}\|u_{0}^{1}\|_{C^{\alpha}})\|u_{0}^{1}-u_{0}^{2}\|_{C^{\alpha}}$
$\displaystyle\lesssim$ $\displaystyle
T^{-(\gamma^{\prime}-\alpha)}(T^{\frac{\gamma^{\prime}-\alpha}{2}}+\|u_{0}^{1}\|_{C^{\alpha}})\|u_{0}^{1}-u_{0}^{2}\|_{C^{\alpha}}.$

Let $\gamma^{\prime}=\frac{\gamma+\alpha}{2}$. Then, we have
$\alpha+\gamma^{\prime}-3>0$ and $\gamma^{\prime}>\alpha$ by noting that
$\alpha\in(\frac{13}{9},\frac{3}{2})$. Using Lemmas 2.1 and 2.4 of [7] and
(3.12) with $\gamma^{\prime}=\frac{\gamma+\alpha}{2}$, we have

(3.13) $\displaystyle\sup_{0<t\leq
T}t^{\frac{\gamma-\alpha}{2}}\|\Pi(\widetilde{a_{0}},\bar{\Pi}_{v_{1}^{\prime}}\xi)(t)\|_{C^{\alpha+\beta-2}}\lesssim$
$\displaystyle\sup_{0<t\leq
T}t^{\frac{\gamma-\alpha}{2}}\|\Pi(\widetilde{a_{0}},\bar{\Pi}_{v_{1}^{\prime}}\xi)(t)\|_{C^{\alpha+\gamma^{\prime}-3}}$
$\displaystyle\lesssim$ $\displaystyle\sup_{0<t\leq
T}t^{\frac{\gamma-\alpha}{2}}\|\widetilde{a_{0}}\|_{C^{\gamma^{\prime}-1}}\|\bar{\Pi}_{v_{1}^{\prime}}\xi(t)\|_{C^{\alpha-2}}$
$\displaystyle\lesssim$
$\displaystyle(1+\|u_{0}^{1}\|_{C^{\alpha}})\|v_{1}^{\prime}\|_{\mathcal{L}_{T}^{\beta}}\|\xi\|_{C^{\alpha-2}}\|u_{0}^{1}-u_{0}^{2}\|_{C^{\alpha}},$

where we have used $\alpha+\gamma^{\prime}-3>0$ for the first inequality and
the relation $\gamma^{\prime}-\alpha=\frac{\gamma-\alpha}{2}>0$ for the last
inequality.

Next, we deal with the last three terms of (3.11). By (3.6) and the similar
arguments to Lemma 4.3 of [7], we easily have

(3.14)
$\displaystyle\|-R(\widetilde{a_{0}},v_{1}^{\prime},\xi)-\Pi_{\bar{\Pi}_{u^{\prime}}\xi}\widetilde{a_{0}}+\widetilde{a_{0}}R_{2}(v^{\prime}_{1},X)\|_{C_{T}C^{\alpha+\beta-2}}$
$\displaystyle\lesssim$
$\displaystyle(1+\|u_{0}^{1}\|_{C^{\alpha}})\|v_{1}^{\prime}\|_{\mathcal{L}_{T}^{\beta}}\|\xi\|_{C^{\alpha-2}}\|u_{0}^{1}-u_{0}^{2}\|_{C^{\alpha}}.$

We omit the details and just give the estimate on the last term
$\widetilde{a_{0}}R_{2}(v^{\prime}_{1},X)$ as an example. Noting the relation
$0<2\alpha+\beta-3<\alpha-1$ and using (3.10), we have

$\displaystyle\|\widetilde{a_{0}}R_{2}(v_{1}^{\prime},X)\|_{C_{T}C^{\alpha+\beta-2}}\lesssim$
$\displaystyle\|\widetilde{a_{0}}R_{2}(v_{1}^{\prime},X)\|_{C_{T}C^{2\alpha+\beta-3}}$
$\displaystyle\lesssim$
$\displaystyle\|\widetilde{a_{0}}\|_{C^{\alpha-1}}\|R_{2}(v_{1}^{\prime},X)\|_{C_{T}C^{\alpha+\beta-2}}$
$\displaystyle\lesssim$
$\displaystyle(1+\|u_{0}^{1}\|_{C^{\alpha}})\|v_{1}^{\prime}\|_{\mathcal{L}_{T}^{\beta}}\|\xi\|_{C^{\alpha-2}}\|u_{0}^{1}-u_{0}^{2}\|_{C^{\alpha}},$

where (3.6) has been used for the last inequality. As a consequence, we
conclude the proof of this lemma by (3.13) and (3.14) together with
$\gamma>\alpha$. ∎

Next, we reevaluate the term $\varepsilon_{2}$ involving $u^{\sharp}$
according to our purpose.

######  Lemma 3.9.

We have the local Lipschitz estimate for $\varepsilon_{2}$:

(3.15) $\displaystyle\sup_{0<t\leq
T}t^{\frac{\gamma-\alpha}{2}}\|\varepsilon_{2}({\bf
u}_{1})-\varepsilon_{2}({\bf u}_{2})\|_{C^{\gamma-2}}$ $\displaystyle\lesssim$
$\displaystyle T^{\frac{\alpha-\beta-1}{2}}K(\|{\bf
u}_{1}\|_{\alpha,\beta,\gamma},\|{\bf
u}_{2}\|_{\alpha,\beta,\gamma})(1+\|\xi\|_{C^{\alpha-2}})^{2}\|{\bf
u}_{1}-{\bf u}_{2}\|_{\alpha,\beta,\gamma}$
$\displaystyle+K_{0}(\|u_{0}^{1}\|_{C^{\alpha}})\|{\bf
u}_{1}\|_{\alpha,\beta,\gamma}\|u_{0}^{1}-u_{0}^{2}\|_{C^{\alpha}}.$

###### Proof.

This lemma can be shown essentially by analogous arguments to Lemma 3.3 of
[7]. However, in that proof, the initial value is fixed and the explicit
relation between the constants and its initial value is not written down.
Hence, for the reader’s convenience, we give proof of (3.15) briefly. In the
following, we use the notation $b(u_{i})$ introduced in the proof of Lemma
3.6, see (3.7). By the definition of $\varepsilon_{2}$, we have

$\displaystyle\|\big{(}\varepsilon_{2}({\bf u}_{1})-\varepsilon_{2}({\bf
u}_{2})\big{)}(t)\|_{C^{\gamma-2}}\lesssim$
$\displaystyle\|\big{(}\varepsilon_{2}({\bf u}_{1})-\varepsilon_{2}({\bf
u}_{2})\big{)}(t)\|_{C^{\beta+\gamma-2}}$ $\displaystyle\lesssim$
$\displaystyle\|b(u_{1})-b(u_{2})\|_{\mathcal{L}_{T}^{\beta}}\|u_{1}^{\sharp}(t)\|_{C^{\gamma}}+\|b(u_{2})\|_{\mathcal{L}_{T}^{\beta}}\|\big{(}u_{1}^{\sharp}-u_{2}^{\sharp}\big{)}(t)\|_{C^{\gamma}}.$

Therefore, the estimate (3.15) is immediately obtained by Lemma 3.4 above and
(3.17) of [7]. ∎

As the final preparation, we give the estimate on the remainder $\zeta$ in
(3.3).

######  Lemma 3.10.

We have that $\zeta$ is locally Lipschitz continuous in
$(u_{0},u^{\prime},u^{\sharp})\in\mathcal{B}_{T}(\lambda)$. In fact, we have
the following estimate

$\displaystyle\sup_{0<t\leq T}t^{\frac{\gamma-\alpha}{2}}\|\big{(}\zeta({\bf
u}_{1})-\zeta({\bf u}_{2})\big{)}(t)\|_{C^{\alpha+\beta-2}}\lesssim$
$\displaystyle K(\|{\bf u}_{1}\|_{\alpha,\beta,\gamma},\|{\bf
u}_{2}\|_{\alpha,\beta,\gamma})\tilde{K}_{2}(X,\xi)\|{\bf u}_{1}-{\bf
u}_{2}\|_{\alpha,\beta,\gamma}$
$\displaystyle+K_{0}(\|u_{0}^{1}\|_{C^{\alpha}})\|{\bf
u}_{1}\|_{\alpha,\beta,\gamma}\|\xi\|_{C^{\alpha-2}}\|u_{0}^{1}-u_{0}^{2}\|_{C^{\alpha}},$

where $\tilde{K}_{2}(X,\xi)$ is the constant introduced in Proposition 3.7 of
[7], which depends on $\|\xi\|_{C^{\alpha-2}}$ and $\|\Pi(\nabla
X,\xi)\|_{C^{2\alpha-3}}$.

###### Proof.

This can be shown by similar arguments to Proposition 3.7 of [7]. Although
many terms should be dealt with, from the proof of Proposition 3.7 of [7], we
only have to reevaluate the terms involving $u_{0}^{T}$, that is,
$\Pi(a(\nabla u_{0}^{T}),\bar{\Pi}_{u^{\prime}}\xi)$ and the term $A_{2}$
defined by (2.27) of [7]. Since we use the norm of $\|{\bf u}_{1}-{\bf
u}_{2}\|_{\alpha,\beta,\gamma}$, the initial value will only affect our
estimate when the estimates of $\|b(u_{i})\|_{\mathcal{L}_{T}^{\beta}}$ and
$\|b(u_{1})-b(u_{2})\|_{\mathcal{L}_{T}^{\beta}}$ are used in the proof of
Proposition 3.7 of [7], where $b(u_{i})$ is defined by (3.7). However, the
terms $\Pi(a(\nabla u_{0}^{T}),\bar{\Pi}_{u^{\prime}}\xi)$ and $A_{2}$ can be
handled by the analogous arguments to Lemma 3.8 above, and Lemma 4.3 of [7].
Noting $u^{\prime}\in\mathcal{L}_{T}^{\beta}$, a similar argument to (3.13)
yields that

$\displaystyle\sup_{0<t\leq
T}t^{\frac{\gamma-\alpha}{2}}\Big{\|}\Big{(}\Pi(a(\nabla
u_{0}^{1,T}),\bar{\Pi}_{u_{1}^{\prime}}\xi)-\Pi(a(\nabla
u_{0}^{2,T}),\bar{\Pi}_{u_{1}^{\prime}}\xi)\Big{)}(t)\Big{\|}_{C^{\alpha+\beta-2}}$
$\displaystyle\lesssim$
$\displaystyle(1+\|u_{0}^{1}\|_{C^{\alpha}})\|u_{1}^{\prime}\|_{\mathcal{L}_{T}^{\beta}}\|\xi\|_{C^{\alpha-2}}\|u_{0}^{1}-u_{0}^{2}\|_{C^{\alpha}}.$

Then, recalling the estimate deduced in the proof of Lemma 4.1 of [7] and the
bilinearity of the resonant term, we easily have

$\displaystyle\sup_{0<t\leq
T}t^{\frac{\gamma-\alpha}{2}}\Big{\|}\Big{(}\Pi(a(\nabla
u_{0}^{1,T}),\bar{\Pi}_{u_{1}^{\prime}}\xi)-\Pi(a(\nabla
u_{0}^{2,T}),\bar{\Pi}_{u_{2}^{\prime}}\xi)\Big{)}(t)\Big{\|}_{C^{\alpha+\beta-2}}$
$\displaystyle\lesssim$ $\displaystyle K_{0}(\|u_{0}^{1}\|_{C^{\alpha}})\|{\bf
u}_{1}\|_{\alpha,\beta,\gamma}\|\xi\|_{C^{\alpha-2}}\|u_{0}^{1}-u_{0}^{2}\|_{C^{\alpha}}.$

On the other hand, $A_{2}$ can be essentially evaluated by the arguments for
Lemma 4.3 of [7] thanks to Lemma 3.4. Here, we give the estimate on the first
term of $A_{2}$ as an example. By bilinearity of $R$, we have

$\displaystyle\|R(b(u_{1}),u_{1}^{\prime};\xi)-R(b(u_{2}),u_{2}^{\prime};\xi)\|_{C_{T}C^{\alpha+\beta-2}}$
$\displaystyle\lesssim$
$\displaystyle\|R(b(u_{1})-b(u_{2}),u_{1}^{\prime};\xi)\|_{C_{T}C^{\alpha+\beta-2}}+\|R(b(u_{2}),u_{1}^{\prime}-u_{2}^{\prime};\xi)\|_{C_{T}C^{\alpha+\beta-2}}$
$\displaystyle\lesssim$ $\displaystyle K(\|{\bf
u}_{1}\|_{\alpha,\beta,\gamma},\|{\bf
u}_{2}\|_{\alpha,\beta,\gamma})(1+\|\xi\|_{C^{\alpha-2}})^{2}\|\xi\|_{C^{\alpha-2}}\|{\bf
u}_{1}-{\bf u}_{2}\|_{\alpha,\beta,\gamma}$
$\displaystyle+K_{0}(\|u_{0}^{1}\|_{C^{\alpha}})\|{\bf
u}_{1}\|_{\alpha,\beta,\gamma}\|\xi\|_{C^{\alpha-2}}\|u_{0}^{1}-u_{0}^{2}\|_{C^{\alpha}},$

where Lemma 3.4 and $\alpha-\beta-1>0$ have been used for the last inequality.
Consequently, we complete the proof. ∎

In the end, let us give the proof of Theorem 3.1 based on the above lemmas.

###### Proof of Theorem 3.1.

According to the proof of Theorem 3.1 of [7], we also know that the choice of
$T$ depends locally uniformly on $\|\tilde{u}_{0}-u_{0}\|_{C^{\alpha}}$
according to the estimates (3.48) and (3.50) of [7]. In fact, noting that
constants $K(\|{\bf u}\|_{\alpha,\beta,\gamma})$ in (3.48) of [7] and
$K(\|{\bf u}_{1}\|_{\alpha,\beta,\gamma},\|{\bf
u}_{1}\|_{\alpha,\beta,\gamma})$ in (3.50) of [7] can be controlled by some
polynomial, we can choose the time $T$ as the function $T=T(r)$ for all
$\|\tilde{u}_{0}-u_{0}\|_{C^{\alpha}}<r$ with a small enough $r>0$. In
particular, we know that $\Phi$ is contractive on $\mathcal{B}_{T}(\lambda)$
for a large enough $\lambda$ and a small enough $T>0$, that is, there exists
$\kappa\in(0,1)$ such that

$\|\Phi({u}_{0},u^{\prime}_{1},u_{1}^{\sharp})-\Phi({u}_{0},u^{\prime}_{2},u_{2}^{\sharp})\|_{\alpha,\beta,\gamma}\leq\kappa\|{\bf
u}_{1}-{\bf u}_{2}\|_{\alpha,\beta,\gamma},\ (u_{0},{\bf
u}_{i})\in\mathcal{B}_{T}(\lambda).$

Note that $u_{0}$ in $\Phi$ is the same in this estimate. Therefore, thanks to
the implicit function theorem, it is enough for us to show the continuity of
the map $\Phi$ on $\mathcal{B}_{T}(\lambda)$. In fact, we can show that the
map $\Phi$ is locally Lipschitz continuous on $\mathcal{B}_{T}(\lambda)$. Let
us take two elements $(u_{0}^{i},{\bf u}_{i})\in\mathcal{B}_{T}(\lambda),\
i=1,2$ and denote their images by $(u_{0}^{i},v_{i}^{\prime},v_{i}^{\sharp})$
under the map $\Phi$. Then by (3.1)-(3.3), we have that the different
$v_{1}^{\sharp}-v_{2}^{\sharp}$ satisfies the equation

$\displaystyle\mathcal{L}^{0}_{1}v_{1}^{\sharp}-\mathcal{L}^{0}_{2}v_{2}^{\sharp}=$
$\displaystyle{\Pi}_{a(\nabla u_{0}^{1,T})v_{1}^{\prime}}\xi-{\Pi}_{a(\nabla
u_{0}^{2,T})v_{2}^{\prime}}\xi-\big{(}\mathcal{L}_{1}^{0}(\bar{\Pi}_{z_{1}^{\prime}}X)-\mathcal{L}_{2}^{0}(\bar{\Pi}_{z_{2}^{\prime}}X)\big{)}$
$\displaystyle+\varepsilon_{1}({\bf u}_{1})-\varepsilon_{1}({\bf
u}_{2})+\varepsilon_{2}({\bf u}_{1})-\varepsilon_{2}({\bf u}_{2})+\zeta({\bf
u}_{1})-\zeta({\bf u}_{2})$

with the initial value
$(v_{1}^{\sharp}-v_{2}^{\sharp})(0)=u_{0}^{1}-u_{0}^{2}-\Pi_{u_{1}^{\prime}(0)-u_{2}^{\prime}(0)}X\in{C}^{\alpha};$
recall that
$v_{0}^{i}=u_{0}^{i},v_{i}^{\prime}(0)=u_{i}^{\prime}(0)=\frac{g(\nabla
u_{0}^{i})}{a(\nabla u_{0}^{i})},\ i=1,2$, where
$\mathcal{L}_{i}^{0}:=\partial_{t}-a(\nabla u_{0}^{i,T})\Delta$. We easily see
that it can be rewritten to the next equation.

(3.16) $\displaystyle\mathcal{L}^{0}_{2}(v_{1}^{\sharp}-v_{2}^{\sharp})=$
$\displaystyle{\Pi}_{a(\nabla
u_{0}^{2,T})(v_{1}^{\prime}-v_{2}^{\prime})}\xi-\mathcal{L}_{2}^{0}(\bar{\Pi}_{v_{1}^{\prime}-v_{2}^{\prime}}X)+(a(\nabla
u_{0}^{1,T})-a(\nabla u_{0}^{2,T}))\Delta v_{1}^{\sharp}$
$\displaystyle+\big{(}a(\nabla u_{0,1}^{T})-a(\nabla
u_{0,2}^{T}))\Delta\bar{\Pi}_{v_{1}^{\prime}}X+{\Pi}_{(a(\nabla
u_{0}^{1,T})-a(\nabla u_{0}^{2,T}))v_{1}^{\prime}}\xi$
$\displaystyle+\varepsilon_{1}({\bf u}_{1})-\varepsilon_{1}({\bf
u}_{2})+\varepsilon_{2}({\bf u}_{1})-\varepsilon_{2}({\bf u}_{2})+\zeta({\bf
u}_{1})-\zeta({\bf u}_{2}).$

By Lemma 2.10 of [7] and
$v_{1}^{\prime}-v_{2}^{\prime}\in\mathcal{L}_{T}^{\beta}$, we see the first
two terms of (3.16) can be estimated as follows:

(3.17) $\displaystyle\sup_{0<t\leq
T}t^{\frac{\gamma-\alpha}{2}}\|\big{(}{\Pi}_{a(\nabla
u_{0}^{2,T})(v_{1}^{\prime}-v_{2}^{\prime})}\xi-\mathcal{L}_{2}^{0}(\bar{\Pi}_{v_{1}^{\prime}-v_{2}^{\prime}}X)\big{)}(t)\|_{C^{\alpha+\beta-2}}$
$\displaystyle\lesssim$
$\displaystyle(1+\|u_{0}^{2}\|_{C^{\alpha}})\|v_{1}^{\prime}-v_{2}^{\prime}\|_{\mathcal{L}_{T}^{\beta}}\|X\|_{C^{\alpha}}.$

As we explained in Lemma 3.10, the estimate (3.7) in Lemma 3.3 of [7] still
holds for $\varepsilon_{1}({\bf u}_{1})-\varepsilon_{1}({\bf u}_{2})$ in the
framework of this theorem. More precisely, we have

(3.18) $\displaystyle\sup_{0<t\leq
T}t^{\frac{\gamma-\alpha}{2}}\|\big{(}\varepsilon_{1}({\bf
u}_{1})-\varepsilon_{1}({\bf u}_{2})\big{)}(t)\|_{C^{2\alpha-3}}$
$\displaystyle\lesssim$ $\displaystyle K(\|{\bf
u}_{1}\|_{\alpha,\beta,\gamma},\|{\bf
u}_{2}\|_{\alpha,\beta,\gamma})(1+\|\xi\|_{C^{\alpha-2}})^{2}\|\xi\|_{C^{\alpha-2}}\|{\bf
u}_{1}-{\bf u}_{2}\|_{\alpha,\beta,\gamma}.$

Now let us denote by $\phi_{1}(t)$ the sum of $(a(\nabla u_{0}^{1,T})-a(\nabla
u_{0}^{2,T}))\Delta v_{1}^{\sharp}(t)$ and $\big{(}\varepsilon_{2}({\bf
u}_{1})-\varepsilon_{2}({\bf u}_{2})\big{)}(t)$, and by $\phi_{2}(t)$ all of
the other terms in the right hand side of (3.16). Then by (3.17), (3.18) and
Lemmas 3.5-3.10, we easily known that $\phi_{1}\in C((0,T],C^{\gamma-2})$ and
$\phi_{2}\in C((0,T],C^{\alpha+\beta-2})$ satisfy the assumptions formulated
in Lemma 3.5 of [7]. So we obtain

(3.19) $\displaystyle\sup_{0<t\leq
T}t^{\frac{\gamma-\alpha}{2}}\|v_{1}^{\sharp}(t)-v_{2}^{\sharp}(t)\|_{C^{\gamma}}+\|v_{1}^{\sharp}-v_{2}^{\sharp}\|_{\mathcal{L}_{T}^{\alpha}}$
$\displaystyle\lesssim$
$\displaystyle\|u_{0}^{1}-u_{0}^{2}\|_{C^{\alpha}}+\|\Pi_{u_{1}^{\prime}(0)-u_{2}^{\prime}(0)}X\|_{C^{\alpha}}+T^{\frac{\alpha+\beta-\gamma}{2}}(1+\|u_{0}^{2}\|_{C^{\alpha}})\|v_{1}^{\prime}-v_{2}^{\prime}\|_{\mathcal{L}_{T}^{\beta}}\|\xi\|_{C^{\alpha-2}}$
$\displaystyle+K_{0}(\|u_{0}^{1}\|_{C^{\alpha}})\big{(}\|{\bf
u}_{1}\|_{\alpha,\beta,\gamma}+\|(v_{1}^{\prime},v_{1}^{\sharp})\|_{\alpha,\beta,\gamma}\big{)}(1+\|\xi\|_{C^{\alpha-2}})\|u_{0}^{1}-u_{0}^{2}\|_{C^{\alpha}}$
$\displaystyle+T^{\frac{\alpha+\beta-\gamma}{2}}K(\|{\bf
u}_{1}\|_{\alpha,\beta,\gamma},\|{\bf
u}_{2}\|_{\alpha,\beta,\gamma})\tilde{K}_{2}(X,\xi)\|{\bf u}_{1}-{\bf
u}_{2}\|_{\alpha,\beta,\gamma}.$

On the other hand, using Lemmas 2.1 and 2.9 of [7], we have

$\displaystyle\|\Pi_{u_{1}^{\prime}(0)-u_{2}^{\prime}(0)}X\|_{C^{\alpha}}\lesssim$
$\displaystyle\left\|\frac{1}{a(\nabla u_{0}^{1})a(\nabla
u_{0}^{2})}\right\|_{C^{\beta}}\|g(\nabla u_{0}^{1})a(\nabla
u_{0}^{2})-g(\nabla u_{0}^{2})a(\nabla
u_{0}^{1})\|_{C^{\beta}}\|X\|_{C^{\alpha}}$ $\displaystyle\lesssim$
$\displaystyle
K_{0}(\|u_{0}^{1}\|_{C^{\alpha}},\|u_{0}^{2}\|_{C^{\alpha}})\|u_{0}^{1}-u_{0}^{2}\|_{C^{\alpha}}\|\xi\|_{C^{\alpha-2}}.$

Therefore, thanks to Lemmas 3.5 and 3.6, by (3.19) together with
$0<\alpha+\beta-\gamma<\alpha-\beta-1$, we have that

$\displaystyle\|\Phi(u_{0}^{1},u_{1}^{\prime},u_{1}^{\sharp})-\Phi(u_{0}^{2},u_{2}^{\prime},u_{2}^{\sharp})\|_{\alpha,\beta,\gamma}$
$\displaystyle\lesssim$ $\displaystyle K(\|{\bf
u}_{1}\|_{\alpha,\beta,\gamma})\big{(}\tilde{K}_{1}(X,\xi)+K_{0}(\|u_{0}^{1}\|_{C^{\alpha}},\|u_{0}^{2}\|_{C^{\alpha}})\big{)}(1+\|\xi\|_{C^{\alpha}})\|u_{0}^{1}-u_{0}^{2}\|_{C^{\alpha}}$
$\displaystyle+T^{\frac{\alpha+\beta-\gamma}{2}}K(\|{\bf
u}_{1}\|_{\alpha,\beta,\gamma},\|{\bf
u}_{2}\|_{\alpha,\beta,\gamma})\tilde{K}_{2}(X,\xi)\|{\bf u}_{1}-{\bf
u}_{2}\|_{\alpha,\beta,\gamma}$

which implies the local Lipschitz continuity of $\Phi$ on
$\mathcal{B}_{T}(r,\lambda)$. Consequently, the proof is completed. ∎

######  Remark 3.3.

In the proof of Proposition 2.6, we used the continuity of the solutions in
$(m,u_{0},\hat{\xi})$ of the SPDE

(3.20) $\partial_{t}u=a(\nabla u+m)\Delta u+g(\nabla u+m)\cdot\xi,$

where $m\in{\mathbb{R}}$. Although we took $m=0$ and fixed it in Theorem 3.1,
the continuity of the solutions in $(m,u_{0},\hat{\xi})$ of the SPDE (3.20)
can be shown by similar arguments. Roughly speaking, instead of the map $\Phi$
defined by (3.1), it is natural to study the map
$\Phi:(m,u_{0},u^{\prime},u^{\sharp})\mapsto(v^{\prime},v^{\sharp})$, where
$v^{\prime},v^{\sharp}$ are determined by (3.2), (3.3) by replacing
$a(\cdot),g(\cdot)$ by $a(\cdot+m),g(\cdot+m)$ respectively. Then, all of the
estimates in Lemmas 3.4-3.10 still hold if we replace $\|{\bf
u}_{i}\|_{\alpha,\beta,\gamma},\|{\bf u}_{1}-{\bf
u}_{2}\|_{\alpha,\beta,\gamma},\|u_{0}^{i}\|_{C^{\alpha}},\|u_{0}^{1}-u_{0}^{2}\|_{C^{\alpha}}$
by $\|{\bf u}_{i}\|_{\alpha,\beta,\gamma}+|m^{i}|,\|{\bf u}_{1}-{\bf
u}_{2}\|_{\alpha,\beta,\gamma}+|m^{1}-m^{2}|,\|u_{0}^{i}\|_{C^{\alpha}}+|m^{i}|,\|u_{0}^{1}-u_{0}^{2}\|_{C^{\alpha}}+|m^{1}-m^{2}|$
respectively. For example, if
$(m^{i},u_{0}^{i},u_{i}^{\prime},u_{i}^{\sharp}),i=1,2$ are given, then
instead of (3.5) in Lemma 3.4, we have

$\displaystyle\left\|a(\nabla u_{1}+m^{1})-a(\nabla
u_{0}^{1,T}+m^{1})-\Big{(}a(\nabla u_{2}+m^{2})-a(\nabla
u_{0}^{2,T}+m^{2})\Big{)}\right\|_{\mathcal{L}_{T}^{\beta}}$
$\displaystyle\lesssim$ $\displaystyle T^{\frac{\alpha-\beta-1}{2}}K(\|{\bf
u}_{1}\|_{\alpha,\beta,\gamma},|m_{1}|)(1+\|\xi\|_{C^{\alpha-2}})^{2}(\|{\bf
u}_{1}-{\bf u}_{2}\|_{\alpha,\beta,\gamma}+|m^{1}-m^{2}|)$
$\displaystyle+K_{0}(\|u_{0}^{1}\|_{C^{\alpha}},|m_{1}|)(\|u_{0}^{1}-u_{0}^{2}\|_{C^{\alpha}}+|m^{1}-m^{2}|),$

because we just replaced the functions $a(\nabla u_{i}),a(\nabla u_{0}^{i,T})$
by $a(\nabla u_{i}+m^{i}),a(\nabla u_{0}^{i,T}+m^{i})$ and $a(\cdot+m)$
satisfies (1.2) for all $m\in{\mathbb{R}}$. Consequently, we can testify that
$\Phi$ satisfies the conditions of the implicit function theorem by the
analogous arguments to the proof of Theorem 3.1 and therefore show the desired
result.

One can interpret the continuity in $m$ as that in the boundary condition,
modified as in (1.6) of [7], for the original SPDE (1.1).

## Acknowledgements

T. Funaki was supported in part by JSPS KAKENHI, Grant-in-Aid for Scientific
Researches (A) 18H03672 and (S) 16H06338, and B. Xie was supported in part by
JSPS KAKENHI, Grant-in-Aid for Scientific Research (C) 20K03627.

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