equation (5.2), we study the asymptotic properties of a few important
functionals.

###### Important functionals

Here we aim to study a dynamical way to calculate the solution, asymptotic
behavior, and stability of non-local and nonlinear differential equations. We
introduce two functionals, $E$ and $F$, which provides different gradient
structure to describe concentration-dispersion dynamics (5.2). Let

$\displaystyle E(u)$ $\displaystyle:=\frac{1}{2p}\int u^{p}K\ast u^{p}dx,$
$\displaystyle F(u)$ $\displaystyle:=e^{-\frac{2p}{p+1}\int u^{p+1}dx}E(u).$

Note that $c(t)=2pE(u)$.

###### Gradient structure on the sphere in $L^{p+1}$

Let

$\mathcal{M}:=\\{u>0:\int u(x)^{p+1}dx=1\\}\subset
H^{1}(\mathds{T}_{L})=:\mathcal{V}.$

$\mathcal{M}$ is an invariant manifold of (5.2), and, moreover, $\mathcal{M}$
attracts every solution: for any $L\in(0,\infty]$,

$\frac{1}{p+1}(\int u^{p+1}dx)^{\prime}=\int u^{p}K\ast u^{p}dx\big{(}1-\int
u^{p+1}dx\big{)}.$

We will discuss the detailed proof later in Proposition 5.1. In this case,
(5.2) is a gradient flow of $E$ with respect to the Riemannian structure given
as follows,

$\langle v,w\rangle_{u}:=\int vwu^{p-1}dx,\quad\forall u\in\mathcal{M},\quad
v,w\in T_{u}\mathcal{M},$

where the tangent manifold at $u\in\mathcal{M}$ is given as follows,

$T_{u}\mathcal{M}:=\\{v\in\mathcal{V}:\int vu^{p}dx=0\\}.$

It is interesting to see that with respect to the given metric
$T_{u}\mathcal{M}$ is “orthogonal” to $u$, i.e., $\langle u,v\rangle_{u}=0$,
as if $\mathcal{M}$ is spherical. It is easy to check that (5.2) on
$\mathcal{M}$ is equivalent to the Gram-Schmidt projection from $X$ onto the
manifold, i.e.,

$u^{\prime}=K*u^{p}-\frac{\langle K*u^{p},u\rangle_{u}}{\langle
u,u\rangle_{u}}u.$ (5.3)

Indeed, (5.2) is a projected gradient flow onto $\mathcal{M}$. If we perturb
$u$ with $v\in T_{u}\mathcal{M}$

$\displaystyle DE(u)(v)$ $\displaystyle=\int u^{p-1}(K\ast u^{p})vdx$ (5.4)
$\displaystyle=:\int\nabla E(u)vdx$ $\displaystyle=\langle\nabla
E(u),v\rangle_{L^{2}}$ $\displaystyle=\int(K\ast u^{p}-c(t)u)vu^{p-1}dx$
$\displaystyle=\langle\text{grad}E(u),v\rangle_{u}$

so that

$\displaystyle u^{\prime}$ $\displaystyle=\text{grad}E(u),$
$\displaystyle\frac{d}{dt}E(u(t))$
$\displaystyle=\|\text{grad}E(u)\|^{2}_{u}.$

######  $L^{2}$-gradient structure

For a fixed $L\in(0,\infty]$, let $\mathcal{H}:=L^{2}(\mathds{T}_{L})$ and
$\mathcal{V}:=H^{1}(\mathds{T}_{L})$, so that

$\mathcal{V}\subset\mathcal{H}\subset\mathcal{V}^{\prime}.$

As long as $E$ is well defined, we can extend the gradient structure on
$\mathcal{M}$ to $L^{2}$ by the proper scaling of the energy $E$. Let the
scaling factor be

$\alpha(t):=e^{\frac{2p}{p+1}\int u^{p+1}dx},$

so that

$F(u):=E(u)/\alpha(t).$

Then we can check the $L^{2}$ gradient-like structure given by $F$,

$\displaystyle\alpha(t)DF(u)(v)$
$\displaystyle=DE(u)(v)-E(u)\frac{2p}{p+1}D(\int u^{p+1}dx)(v)$
$\displaystyle=\int u^{p-1}K*u^{p}vdx-2pE(u)\int u^{p}vdx$ $\displaystyle=\int
u^{p-1}(K*u^{p}-2pE(u)u)vdx.$ $\displaystyle=\int u^{p-1}(K*u^{p}-c(t)u)vdx.$
$\displaystyle DF(u)(v)$ $\displaystyle=e^{-\frac{2p}{p+1}\int u^{p+1}dx}\int
u^{p-1}(K*u^{p}-c(t)u)vdx,$ $\displaystyle=:\langle\nabla
F(u),v\rangle_{L^{2}}.$

Therefore, (5.2) takes the form

$u^{\prime}=e^{\frac{2p}{p+1}\int u^{p+1}dx}u^{1-p}\nabla F(u).$

We finish the chapter by proving some convergence properties of the
functionals. It is easy to see that $u(t,\cdot)$ of (5.2) stays positive when
$u(0,\cdot)>0$. The concrete proof along with well-posedness can be found in
Theorem 6.1.

######  Proposition 5.1.

Let $u(x,t)\in L^{p+1}(\mathds{T}_{L})$ be a solution of (5.2) such that
$u(x,0)>0$ for all $x\in\mathds{T}_{L}$, then the following functionals have
convergent properties as $t\to\infty:$

  1. (1)

$F(u)=E(u)/\alpha(t)$ is nondecreasing,

  2. (2)

$\|u\|_{L^{p+1}}$ converges monotonically to 1,

  3. (3)

$E(u)=\frac{1}{2p}\int u^{p}K\ast u^{p}dx$ converges to a positive constant.

Additionally, if $\|u(0)\|_{L^{p+1}}\leq 1$ then $E(u)$ is nondecreasing as
well.

###### Proof.

$F$ is nondecreasing because the time derivative of $F$ is nonnegative, i.e.,

$\displaystyle F(u(t))^{\prime}e^{\frac{2p}{p+1}\int u^{p+1}dx}$
$\displaystyle=E(u)^{\prime}-E(u)\frac{2p}{p+1}(\int u^{p+1}dx)^{\prime}$
$\displaystyle=\int u^{p-1}K*u^{p}u^{\prime}dx-2pE(u)\int u^{p}u^{\prime}dx$
$\displaystyle=\int u^{p-1}(K*u^{p}-2pE(u)u)u^{\prime}dx.$ $\displaystyle=\int
u^{p-1}(K*u^{p}-c(t)u)^{2}dx\geq 0.$

Since

$F(u)=E(u)/\alpha(t)\leq E(u)=2pc(t),$

we have $\inf_{t\geq 0}c(t)>0$. This yields the monotonic convergence of
$L^{p+1}$ norm together with taking time-differentiation,

$\frac{1}{p+1}(\int u^{p+1}dx)^{\prime}=c(t)\big{(}1-\int u^{p+1}dx\big{)}.$
(5.5)

Therefore,

$\lim_{t\to\infty}\|u(t)\|_{L^{p+1}}\to 1.$

Additionally, if $\|u(0)\|_{L^{p+1}}\leq 1$, $E$ is nondecreasing as well
because $\|u(0)\|_{L^{p+1}}$ is nondecreasing.

Finally, we show that $E(u)$ is bounded above. From (5.5) we have

$\|u\|_{L^{p+1}}\leq 1+\|u_{0}\|_{L^{p+1}}.$

So we use Hölder’s inequality and Young’s convolution inequality to bound $E$,

$\displaystyle 2pE(u)$ $\displaystyle=\int u^{p}K\ast u^{p}dx$
$\displaystyle\leq\|u\|^{p}_{L^{p+1}}\|K*u^{p}\|_{L^{p+1}}$
$\displaystyle\leq\|K\|_{L^{\frac{p+1}{2}}}\|u\|^{2p}_{L^{p+1}}$
$\displaystyle\leq(1+\|u_{0}\|_{L^{p+1}})^{2p}\|K\|_{L^{\frac{p+1}{2}}}<\infty.$
(5.6)

This implies $F$ is bounded. Therefore $F>0$ converges monotonically to a
positive constant, and thus $E=\alpha(t)F>0$ also converges to a positive
constant. ∎

###  Chapter 6 Well-posedness, compactness and nontrivial wave profiles

In this chapter, we explore properties of the concentration-dispersion
equation (5.2) on the torus
$\mathds{T}_{L}=\mathbb{R}/L=[-\frac{L}{2},\frac{L}{2}]$ of length
$L\in(0,\infty]$.

Recall that the equation (5.2) is motivated by solitary waves and the solution
$u$ evolves in a way to balance the concentration effect and the dispersion
effect. We use Picard iteration to show that the solution is well-posed in
$\mathcal{V}:=H^{1}(\mathds{T}_{L})$. Simultaneously, we show that if the
initial data $u_{0}$ is “bell-shaped”, then the solution stays bell-shaped
along the evolution.

We also show that the solution is bounded above, and below away from $0$.
Directly, by Arzela-Ascoli’s theorem, the solution is precompact in the space
of continuous functions. Actually, we can say much more. Recall that
dispersive effect on the semi-linear heat equations provides the smoothing
effect, which makes the solution smoother than it initially was, so we have
compactness (Section 3.3 of Chapter 3). Even if the convolution operator
cannot make the solution smoother than the initial data, but we can show that
the solution is a combination of the scalar multiple of the initial data and
the smoother part from the dispersion effect by $K\in W^{1,1}$. So the
solution is precompact in $H^{1}(\mathds{T}_{L})$.

Note that we aim to approximate nontrivial solitary wave solutions. We show
the constant equilibrium is unstable and, moreover, use the gradient structure
to prove that nontrivial equilibria exist. The convergence result, however,
cannot be proven using the Łojasiewicz framework. We will discuss the issue at
the end of this chapter.

####  6.1. Well-posedness of solutions and consistency of bell-shape

We use the Banach fixed point theorem to show well-posedness of the
concentration-dispersion equation. Later in the section, we continue the
argument to prove that the solution with bell-shaped initial value stays bell-
shaped. At a first glance at (5.2), it is not immediate to believe this to be
true, since $u^{\prime}$ is a subtraction of two bell-shaped function which,
in general, might not be bell-shaped. However, using simple ODE trick we show
that the bell-shape of $u$ persists

######  Theorem 6.1 (Well-posedness).

For any initial $u_{0}\in\mathcal{V}$, there exists unique solution $u(t,x)\in
C^{1}(\mathbb{R}_{+};\mathcal{V})$ of (5.2). Additionally, if $u_{0}(\cdot)>0$
then $u(t,\cdot)>0$ for all $t\geq 0$.

###### Proof.

To use Picard iteration, we modify (5.2) as follows,

$\displaystyle u^{\prime}+c(t)u$ $\displaystyle=K\ast u^{p}$ $\displaystyle
u(t)$
$\displaystyle=e^{-\int_{0}^{t}c(s)ds}u(0)+\int_{0}^{t}e^{-\int_{s}^{t}c(r)dr}K\ast
u^{p}ds.$

For fixed $u_{0}\in\mathcal{V}$ we define
$I:C\big{(}[0,T],\overline{B_{+}(R)}\big{)}\to
C\big{(}[0,T],\overline{B_{+}(R)}\big{)}$, as follows

$I(u)(x,t):=e^{-\int_{0}^{t}c(s)ds}u_{0}(x)+\int_{0}^{t}e^{-\int_{s}^{t}c(r)dr}K\ast
u^{p}ds,$

where $B_{+}(R)=\\{u\in\mathcal{V}:\|u\|_{\mathcal{V}}<R\text{ and
}u(x)>0,\forall x\in\mathds{T}_{L}\\}.$ Claim that if
$\|u_{0}(x)\|_{H^{1}}\leq\frac{R}{2}$ for some $R$ then there exists $T>0$
such that,

$I:C\big{(}[0,T],\overline{B_{+}(R)}\big{)}\to
C\big{(}[0,T],\overline{B_{+}(R)}\big{)}$

with the induced uniform norm

$\|u\|_{\infty}:=\sup_{t\in[0,T]}\|u(x,t)\|_{H^{1}}.$

Let $u(x,t)\in C([0,T],\overline{B_{+}(R)})$,

$\displaystyle\|I(u)\|_{\infty}$
$\displaystyle\leq\|e^{-\int_{o}^{T}c(t)dt}u_{0}\|_{\infty}+\int_{0}^{T}\|K\ast
u^{p}\|_{\infty}dt$ $\displaystyle\leq\|u_{0}\|_{H^{1}}+T\|K\ast
u^{p}\|_{\infty}.$

If $1<p<2$, we use Young’s convolution inequality to analyze $K\ast u^{p}$

$\displaystyle\|K\ast u^{p}\|_{H^{1}}$ $\displaystyle\leq\|K\ast
u^{p}\|_{L^{2}}+\|K_{x}\ast u^{p}\|_{L^{2}}$
$\displaystyle\leq(\|K\|_{L^{q}}+\|K_{x}\|_{L^{q}})\|u\|_{L^{2}}^{p}$
$\displaystyle=\|K\|_{W^{q,1}}R^{p},$

where $q$ satisfies, $\frac{p}{2}+\frac{1}{q}=1+\frac{1}{2}$, i.e.
$q=2/(3-p)$.

If $p\geq 2$, $\|K\ast
u^{p}\|_{H^{1}}=\|K*u^{2}u^{p-2}\|_{L^{2}}+\|K_{x}*u^{2}u^{p-2}\|_{L^{2}}$ so
we can use Morrey’s embedding theorem and Young’s convolution inequality to
get the bound.

$\displaystyle\|K\ast u^{p}\|_{H^{1}}$
$\displaystyle=\|K*(u^{2}u^{p-2})\|_{L^{2}}+p\|K\ast(u_{x}uu^{p-2})\|_{L^{2}}$
$\displaystyle\leq
R^{p-2}(\|K\|_{L^{2}}\|u\|_{2}^{2}+p\|K\|_{L^{2}}\|u\|_{L^{2}}\|u_{x}\|_{L^{2}})$
$\displaystyle\leq pC\|K\|_{L^{2}}R^{p}$

Therefore,

$\displaystyle\|I(u)\|_{\infty}$ $\displaystyle\leq\frac{R}{2}+C(K,p)R^{p}T$
$\displaystyle\leq R$

by letting $T\leq\frac{1}{2CR^{p-1}}$.

Also we can check that $I$ is Lipschitz on
$C\big{(}[0,T],\overline{B(0,R)}\big{)}$, since $x\mapsto e^{-x}$ and
$y\mapsto y^{p}$ are Lipschitz when $x\geq 0$ and when $y$ is bounded,
respectively. We use Picard iteration, or Banach fixed point theorem to
conclude the well-posedness for short-time. To extend this to long-time, we
build time independent $H^{1}$ bound (Proposition 6.2) later in this chapter,
and use the Picard iteration again. ∎

###### Bell-shape consistency

In this section, in addition to prove well-posedness of the equation (5.2), we
show that the solution with bell-shaped initial value stays bell-shaped. We
consider the equation (5.2) on compact domain $\mathds{T}_{L}$ with the same
kernel $K$ for $L\in(0,\infty]$. We start with making an observation on the
shape of the periodization of $K$ and $K*u$.

######  Definition 6.1.

A real-valued function $u$ on $\mathds{T}_{L}$ is bell-shaped if $u$ is even
and monotone decreasing on $(0,L/2)$.

######  Definition 6.2.

We say $K_{L}$ is the $L$-periodization of a function $K$ on $\mathbb{R}$ when

$\displaystyle K_{L}(x)$ $\displaystyle=\sum_{k=-\infty}^{\infty}K(x-kL).$

Also the convolution of $K$ with a $L$-periodic function
$u:\mathbb{R}\to\mathbb{R}$ is as follows,

$\displaystyle(K\ast u)(x)$
$\displaystyle:=\int_{-\infty}^{\infty}u(y)K(x-y)dy$
$\displaystyle=\int_{x_{0}}^{x_{0}+L}u(y)K_{L}(x-y)dy,$

for any $x_{0}\in\mathbb{R}.$

######  Lemma 6.1.

Let $K$ be a bell-shaped function. If $K$ is convex in $[0,\infty)$ then
$K_{L}$ is bell-shaped on $\mathds{T}_{L}$.

###### Proof.

Enough to show that $K_{L}$ is monotone non-increasing in $[0,\frac{L}{2}]$.

$\displaystyle K_{L}(x)=$ $\displaystyle\sum_{n=-\infty}^{\infty}K(x-nL)$
$\displaystyle=\sum_{n=0}^{\infty}K(x+nL)+\sum_{n=1}^{\infty}K(x-nL)$
$\displaystyle=\sum_{n=0}^{\infty}K(x+nL)+K((n+1)L-x)$

Let $h>0$, $x\in[0,\frac{L}{2})$.

$\displaystyle K_{L}(x+h)-K_{L}(x)=\sum_{n=0}^{\infty}$
$\displaystyle[K(x+h+nL+K((n+1)L-x-h]$
$\displaystyle\quad-[K(x+hL)+K((n+1)L-x)]$

Claim that $\forall n\in\mathbb{N}\cup\\{0\\}$,

$K(x+h+nL)+K((n+1)L-x-h)-K(x+hL)+K((n+1)L-x)\leq 0.$

In the interval $[x+nL,(n+1)L-x]$, let $s=\frac{h}{L-2x}$ and use convexity.

$\displaystyle K(x+h+nL)+K((n+1)L-x-h)$
$\displaystyle\leq(1-s)K(x+nL)+sK((n+1)L-x)$
$\displaystyle+sK(x+nL)+(1-s)K((n+1)L-x)$ $\displaystyle\leq
K(x+nL)+K((n+1)L-x)$

as claimed. ∎

######  Remark 6.1.

Note as long as $K$ does not resemble “stairs”, i.e. $K_{x}$ does not vanish
except at $0$, we can generalize the previous result as follows. If $K$ is a
bell-shaped function which is strictly concave inside a compact interval and
convex outside, then $K_{L}$ is bell-shaped for big enough $L$. For example
suppose that $K\approx o(x^{-m})$ for $m>1$ then

$K_{L}(x)=K(x)+\sum_{n=1}^{\infty}\frac{1}{(nL+x)^{m}}+\frac{1}{(nL-x)^{m}}\approx
K(x)+\frac{1}{L^{m}}\sum_{n=1}^{\infty}\frac{1}{(n+x/L)^{m}}+\frac{1}{(n-x/L)^{m}},$

so that for big enough $L$ and near $0$, $K_{L}$ resembles the shape of $K$.

######  Lemma 6.2.

For such $K$, where $K_{L}$ is bell-shaped, if $u$ is a $L$-periodic bell-
shaped function, then $K\ast u$ is $L$-periodic bell-shaped.

###### Proof.

$\displaystyle K\ast
u(x)=\int_{\mathbb{R}}K(y)u(y-x)dx=\int_{-\frac{L}{2}}^{\frac{L}{2}}K_{L}(y)u(y-x)dy.$

On the inverval of half period such that $u(x)$ is monotone, $u_{x}(x)$ exists
for almost every $x\in[0,\frac{L}{2}]$. Note that $u_{x}$ is odd and
$u_{x}(x)\leq 0$ when $x\geq 0$. Claim that $\forall
x\in[0,\frac{L}{2}]\quad(K\ast u)_{x}(x)\leq 0$. We start with partitioning
$\mathds{T}_{L}$ with the intervals of length $x$,$x$,$(\frac{L}{2}-x)$ and
$(\frac{L}{2}-x)$.

$\displaystyle(K\ast u)_{x}(x)$
$\displaystyle=\int_{-\frac{L}{2}}^{\frac{L}{2}}K_{L}(y)\frac{d}{dx}u(y-x)dy$
$\displaystyle=\int_{-\frac{L}{2}}^{\frac{L}{2}}-K_{L}(y)u_{x}(y-x)dy$
$\displaystyle=(\int_{-\frac{L}{2}}^{-\frac{L}{2}+x}+\int_{-\frac{L}{2}+x}^{\frac{L}{2}+2x})+(\int_{-\frac{L}{2}+2x}^{x}+\int_{x}^{\frac{L}{2}})(-K_{L}(y)u_{x}(y-x))dy$
$\displaystyle=I+II$

Now on each part, we reflect to another side.

$\displaystyle I$
$\displaystyle=\int_{-\frac{L}{2}}^{-\frac{L}{2}+x}+\int_{-\frac{L}{2}+x}^{\frac{L}{2}+2x}(-K_{L}(y)u_{x}(y-x))dy$
$\displaystyle=\int_{-\frac{L}{2}}^{-\frac{L}{2}+x}(-K_{L}(y)u_{x}(y-x))-K_{L}(-L+2x-y)u_{x}(-L+x-y)dy$
$\displaystyle=\int_{-\frac{L}{2}}^{-\frac{L}{2}+x}(-K_{L}(y)u_{x}(y-x))-K_{L}(y^{\prime})u_{x}(x-y)dy$
$\displaystyle=\int_{-\frac{L}{2}}^{-\frac{L}{2}+x}-(K_{L}(y)-(K_{L}(y^{\prime}))(u_{x}(y-x))dy<0$

Same for II. where $\frac{y^{\prime}+y}{2}=-\frac{L}{2}+x$. ∎

######  Remark 6.2.

With similar observations, we can prove the same for $L=\infty$, i.e. if $u$
and $K$ are bell-shaped on $\mathbb{R}$ then $K*u$ is bell-shaped.

######  Theorem 6.2 (Bell-shape consistency).

If $u_{0}\in\mathcal{V}$ is bell-shaped, $u(t)$ is bell-shaped for all $t\geq
0.$

###### Proof.

We continue from the proof of Theorem 6.1. Let

$\mathcal{K}:=\\{u\in H^{1}(\mathds{T}_{L}):\text{u is bell-shaped}\\}$

and suppose $u_{0}\in\mathcal{K}.$ Due to Lemma 6.2 if $u\in\mathcal{K}$ then
$I(u)\in\mathcal{K}$. Again, by Picard iteration, $u(t)$ stays bell-shaped for
$0\leq t<T$ if $u_{0}\in\mathcal{K}$. Furthermore by the similar argument from
the proof of Theorem 6.1 we can let $T\to\infty.$∎

####  6.2. Compactness of the solutions

###### Asymptotic bound on $u$ and $u_{x}$

We investigate asymptotic pointwise bounds on the solution. It turns out the
solution stays bounded not only above but also below, as long as $K$ stays
away from 0. We can expect that the lower bound will diminish as $L\to\infty$.

######  Proposition 6.1.

Assume that $K(x)\in W^{1,1}\cap W^{1,p+1}(\mathds{T}_{L})$. Let
$u\in\mathcal{V}$ be a solution of (5.2). Then there exists $M(K,u_{0})>0$
such that $\sup_{t\geq 0}\|u(x,t)\|_{\infty}<M$. Additionally, there exists
$m>0$, for any $x$ such that $\frac{\partial}{\partial x}u_{0}$ exists,
$\lim\sup_{t\to\infty}\|u_{x}(x,t)\|_{\infty}<m$.

###### Proof.

We want to find a pointwise threshold where $u$ and $u_{x}$ stop growing.
Start from (5.2),

$\displaystyle u^{\prime}$ $\displaystyle=K\ast u^{p}-c(t)u$ (6.1)
$\displaystyle=\int K(x-y)u(y)^{p}dy-c(t)u$ (6.2) $\displaystyle\leq(\int
K(x-y)^{p+1}dy)^{1/p+1}(\int u^{p+1}dy)^{p/(p+1)}-\underline{c}u,$ (6.3)

where $\underline{c}=\inf_{t\geq 0}c(t)$. Note that $\underline{c}>0$ because
$u>0$ and $F(u)=\frac{1}{2p}e^{-\frac{2p}{p+1}\int u^{p+1}dx}c(t)>0$ is
increasing according to the Proposition 5.1. This means

$u^{\prime}(x,t)<0\quad\text{if
}u(x,t)>\frac{1}{\underline{c}}\|K\|_{L^{p+1}}\|u\|^{p}_{L^{p+1}}.$

By Morrey’s embedding theorem, $u\in L^{\infty}$. Due to monotonic convergence
of $\|u\|_{L^{p+1}}\to 1$ as $t\to\infty$, we can conclude that

$\max_{t\geq 0}\|u(x,t)\|_{\infty}\leq
M=\|u_{0}\|_{\infty}+\frac{1}{\underline{c}}(1+\|u_{0}\|^{p}_{L^{p+1}})\|K\|_{L^{p+1}}.$

Similarly, for any point where $u_{x}$ exists and is positive,

$\displaystyle u_{x}^{\prime}$ $\displaystyle=(K\ast u^{p})_{x}-c(t)u_{x}$
$\displaystyle=\int K_{x}(x-y)u(y)^{p}dy-c(t)u_{x}$
$\displaystyle\leq\|K_{x}\|_{L^{p+1}}\|u\|^{p}_{L^{p+1}}-\underline{c}u_{x},$

and a similar argument holds for any point where $u_{x}$ exists and is
negative. ∎

######  Remark 6.3.

In the case when $\|u(0)\|_{L^{p+1}}\leq 1$, $c(t)$ is increasing. So we can
obtain the explicit upper bound,

$\|u(t)\|_{\infty}\leq M=\max\\{\|u(x,0)\|_{\infty},\|K\|_{L^{p+1}}/c(0)\\}.$

In the compact spatial domain, we can use the upper bound to calculate the
lower bound. This will be used to control the Riemannian metric. Naturally,
the lower bound vanishes as $L\to\infty.$

######  Corollary 6.1.

For fixed $L<\infty$, let $u\in\mathcal{V}$ be a solution of (5.2). Assume
that $K_{L}(x)\in W^{1,1}\cap W^{1,p+1}(\mathds{T}_{L})$ and there is $k>0$
such that $K_{L}(x)\geq k,$ $\forall x\in\mathds{T}_{L}$. Then there exist
$m(K,L,u_{0})>0$ such that $\min_{x}u(x,t)\geq m$.

###### Proof.

From Proposition 6.1, there exists $M(u_{0},K)>0$ such that $u(x,t)<M$ for all
$t\geq 0$. Now for any $\epsilon>0$ we can choose large enough $T>0$ so that

$1-\epsilon\leq\int u^{p+1}dx\leq M\int u^{p}dx.$

Note that $\overline{c}:=\sup_{t\geq 0}c(t)$ is well defined since it
converges. Then,

$\displaystyle u^{\prime}$ $\displaystyle=K\ast u^{p}-c(t)u$
$\displaystyle\geq k\int u^{p}dx-\overline{c}u$
$\displaystyle\geq\frac{(1-\epsilon)k}{M}-\overline{c}u>0,$

when $u<(1-\epsilon)k/M\overline{c}$. So the result follows. ∎

######  Remark 6.4.

If $\|u(0)\|_{L^{p+1}}\leq 1$, we can obtain a more explicit lower bound using
Remark 6.3 and (5.6). For any $\epsilon$ there exists $T>0$ such that for all
$t>T$, we have

$\min_{x}u(t,x)\geq m=\frac{(1-\epsilon)k}{M\|K\|_{L^{\frac{p+1}{2}}}}.$

######  $L^{2}$-compactness of the solutions

Since the Rellich–Kondrachov theorem is only valid on compact domain, we need
more careful analysis to achieve compactness on the real line. We will take
advantage of the bell-shaped solutions to show that the solution is tight in
$L^{p}$-spaces for any $p>1$.

######  Proposition 6.2 (uniform $H^{1}$-norm).

Let $u\in H^{1}(\mathds{T}_{L})$ be a (maximal) solution of (5.2). Then there
exists time independent $C>0$ such that $\|u(t,\cdot)\|_{H^{1}}<C.$

###### Proof.

We used the fixed point formula (6.5),

$\displaystyle u(x,t)$
$\displaystyle=e^{-\int_{0}^{t}c(\tau)d\tau}u_{0}+\int_{0}^{t}e^{-\int_{s}^{t}c(\tau)d\tau}K*u^{p}ds.$
(6.4)

Since we have uniform $L^{p+1}$ and $L^{\infty}$ norm (Proposition 5.1,
Proposition 6.1), we have uniform $L^{2}$ bound on the integrand as follows,

$\displaystyle\|K*u^{p}\|_{H^{1}}\leq\|K\|_{W^{1,1}}\|u^{p}\|_{L^{2}}\leq\|K\|_{W^{1,1}}\|u\|_{L^{p+1}}^{\frac{p+1}{2}}\|u\|_{L^{\infty}}^{\frac{p-1}{2}}<\infty.$

Therefore,

$\displaystyle\|u\|_{H^{1}}$
$\displaystyle\leq\|u_{0}\|_{H^{1}}+\|\int_{0}^{t}e^{-\int_{s}^{t}c(\tau)d\tau}K*u^{p}ds\|_{H^{1}}$
$\displaystyle\leq\|u_{0}\|_{H^{1}}+\int_{0}^{t}e^{-\underline{c}(t-s)}\|K*u^{p}\|_{H^{1}}ds$
$\displaystyle=\|u_{0}\|_{H^{1}}+\underline{c}^{-1}(1-e^{-\underline{c}t})\|K*u^{p}\|_{H^{1}}<\infty,$

where $\underline{c}=\inf_{t>0}c(t)>0$. ∎

When $L<\infty$, we can use the Rellich-Kondrachov theorem to directly infer
that the solution is precompact in $L^{2}$. Indeed, we can show the solution
is precompact in $H^{1}$. We will discuss in the next subsection with detail.
On the other hands, when $L=\infty$, we need further restriction on $u$ to
derive the compactness result. The following lemma will be useful to show the
tightness of the bell-shaped solution.

######  Lemma 6.3 (Uniform $L^{1}$-norm).

Suppose $u_{0}\in H^{1}\cap L^{1}$ then there exists $C>0$ such that

$\|u(t,\cdot)\|_{L^{1}}<C.$

###### Proof.

The argument can be much simpler when $p\geq 2$. We claim that $u(t,x)$ is
uniformly bounded in $L^{p}$ using the uniform $L^{2}$ and $L^{\infty}$ bound,

$\int u^{p}dx\leq\|u\|_{\infty}^{p-2}\int u^{2}dx<\infty.$

Now we use Grönwall type estimate on $L^{1}$ norm of $u$ as follows,

$\displaystyle\frac{d}{dt}\int udx=\int K*u^{p}dx-c(t)\int
udx\leq\|K\|_{L^{1}}\|u(t,x)\|_{L^{p}}^{p}-\underline{c}\int udx.$

The general proof starts from (6.5),

$\displaystyle u(x,t)$
$\displaystyle=e^{-\int_{0}^{t}c(\tau)d\tau}u_{0}+\int_{0}^{t}e^{-\int_{s}^{t}c(\tau)d\tau}K*u^{p}ds.$

Let $n\in\mathbb{N}$ be such that $p^{n}\geq 2$ so that we have uniform
control in $L^{p^{n}}$, then

$\displaystyle\|K*u^{p}\|_{L^{p^{n-1}}}$
$\displaystyle\leq\|K\|_{L^{1}}\|u^{p}\|_{L^{p^{n-1}}}$
$\displaystyle\leq\|K\|_{L^{1}}\|u\|_{L^{p^{n}}}^{p}.$

Therefore,

$\displaystyle\|u\|_{L^{p^{n-1}}}\leq\|e^{-\int_{0}^{t}c(\tau)d\tau}u_{0}\|_{L^{p^{n-1}}}+\int_{0}^{t}e^{-\int_{s}^{t}c(\tau)d\tau}\|K*u^{p}\|_{L^{p^{n-1}}}ds$
$\displaystyle\leq
e^{-\underline{c}t}\|u_{0}\|_{L^{p^{n-1}}}+\underline{c}^{-1}(1-e^{-\underline{c}t})\|K*u^{p}\|_{L^{p^{n-1}}},$

where $\underline{c}:=\inf_{t\geq 0}c(t)>0$, constructed by Proposition 5.1.
So we have uniform control in $L^{p^{n-1}}$. We recur the argument until we
reach $n=0$ to get $L^{1}$ control. ∎

Using the previous lemma, we will show that the solution is precompact in
$L^{2}$. Prior to that, we show some interesting consequences of the lemma.
Directly, we can show that the solution cannot converge uniformly to $0$.

######  Corollary 6.2.

Suppose $u_{0}\in H^{1}\cap L^{1}$ then
$\lim\inf_{t\to\infty}\|u(t,\cdot)\|_{\infty}>0$.

###### Proof.

For the sake of contradiction, let’s assume that
$\lim\inf_{t\to\infty}\|u(t,\cdot)\|_{\infty}=0$.

$\displaystyle c(t)=\int u^{p}K*u^{p}dx\leq\|u(t,\cdot)\|_{\infty}^{p}\int
K*u^{p}dx\leq\|u(t,\cdot)\|_{\infty}^{p}\|K\|_{L^{1}}\|u(t,\cdot)\|_{L^{p}}^{p}.$

Since we achieve the uniform control on $L^{p}$ norm and $c(t)$ converges to a
positive constant this leads to the contradiction. ∎

The next corollary will be useful to show $L^{2}$-compactness.

######  Corollary 6.3 ($L^{q}$ tightness).

Let $q>1$ and $u(t,x)$ be a solution of (5.2) such that $u_{0}\in H^{1}\cap
L^{1}$ is bell shaped. Then for any $\epsilon>0$ there exists $R>0$ such that

$\int_{|x|>R}u^{q}dx<\epsilon.$

###### Proof.

Note that $u(t,x)$ is bell-shaped. Using the previous lemma and the
Chebyshev’s inequality, for $x>0$

$\displaystyle 2xu(t,x)\leq\|u(t,x)\|_{L^{1}}<C.$

Therefore,

$u(t,x)\leq C/2x$

$\int_{|x|>R}u^{q}dx<(\frac{C}{2})^{q}\int_{|x|>R}1/|x|^{q}dx=\tilde{C}R^{-q+1}<\epsilon.$

∎

Now we show that any bell-shaped solutions are precompact in
$L^{2}(\mathbb{R})$.

######  Theorem 6.3 (Compactness on $\mathbb{R}$).

Let $u(t,x)$ be a solution of (5.2). If $u_{0}\in H^{1}(\mathbb{R})$ is bell-
shaped then $\\{u(t,x)\\}_{t\geq 0}$ is precompact in $L^{2}(\mathbb{R})$.

###### Proof.

Since $u(t,\cdot)$ is bell-shaped for all $t\geq 0$ by Theorem 6.2, we can use
Helly’s selection theorem to have a (further) subsequence
$\\{u(t_{k})\\}_{k\in\mathbb{N}}$ and the pointwise limit $\tilde{u}$. Recall
that we have uniform bound on $u$ and $u_{x}$ by Proposition 6.1. This means,
by Arzela-Ascoli theorem, on any compact interval, the (subsequential)
convergence is uniform and $\tilde{u}$ is continuous on $\mathbb{R}$.

Now we claim that $\|u(t_{k})-\tilde{u}\|_{L^{2}}\to 0$ as $k\to\infty$. For
given $\epsilon>0$, we use Corollary 6.3 to find $R>0$ to have small
$L^{2}$-tail bound and we choose $k\in\mathbb{N}$ large enough to have small
$L^{2}$-bound, i.e.,

$\|u(t_{k})-\tilde{u}\|_{L^{2}}^{2}\leq\int_{|x|\leq
R}(u(t_{k})-\tilde{u})^{2}dx+\int_{|x|>R}(u(t_{k})-\tilde{u})^{2}dx<\epsilon.\qed$

######  $H^{1}$-compactness of the solutions ($L<\infty$)

In the nice case, when $L<\infty$ and $u_{x}$ is uniformly bounded, we can use
Arzela-Ascoli’s theorem to directly infer the compactness of solutions.

######  Corollary 6.4.

Assume $L<\infty$ and let $u\in\mathcal{V}$ be a solution of (5.2) such that
$u_{x}\in L^{\infty}$. Then $\\{u(x,t)\\}_{t\geq 0}$ is precompact in
$C_{0}(\mathds{T}_{L})$.

Note that we can relax the assumption of uniform bound on $u_{x}$, using usual
diagonalization trick. In fact, we can choose another approach to prove
compactness of the solution, even with $L^{2}$-initial data.

Recall that in semi-linear heat equation, the Laplacian operator provides
smoothing action resulting the solution trajectory to be precompact. Even
though not as powerful as the Laplacian operator, convolution with a nice
function is renowned for regularization. From the proof of Theorem 6.1 we have

$\displaystyle u(x,t)$
$\displaystyle=e^{-\int_{0}^{t}c(\tau)d\tau}u_{0}+\int_{0}^{t}e^{-\int_{s}^{t}c(\tau)d\tau}K*u^{p}ds.$
(6.5)

Although we cannot claim that $u(x,t)$ is smoother than the initial data
$u_{0}$, we can see that $u(x,t)$ has a smoother part. Indeed, the fixed point
formula (6.5) suggests that $u$ consists of $u_{0}$ with a scaling factor and
a smoother trajectory created by the convolution with $K$. It is immediate to
see that the first part of (6.5) converges to $0$ as $t\to\infty$, since
$c(t)$ converges to a positive constant as $t\to\infty$. So the first part of
(6.5) is precompact. Therefore once we prove the second part of (6.5) is
precompact, we prove that $\\{u(t,x)\\}_{t\geq 0}$ is precompact.

######  Theorem 6.4.

Let $L<\infty$ and $u(t,x)$ be a solution of (5.2). If $u_{0}\in
H^{1}(\mathds{T}_{L})$ then $\\{u(t,x)\\}_{t\geq 0}$ is precompact in
$H^{1}(\mathds{T}_{L})$.

###### Proof.

from (6.5) and the previous discussion, it is enough to show that the
trajectory $t\mapsto\int_{0}^{t}e^{-\int_{s}^{t}2pE(u)d\tau}K*u^{p}ds$ is
compact in $H^{1}(\mathds{T}_{L})$. From Proposition 5.1 we know that $c(t)$
stays positive and converges, i.e., there exists $\underline{c}:=\inf_{t\geq
0}c(t)>0$. Without specifying the norm, we have

$\displaystyle\|\int_{0}^{t}e^{-\int_{s}^{t}c(\tau)d\tau}K*u^{p}ds\|$
$\displaystyle\leq\int_{0}^{t}e^{-\underline{c}(t-s)}\|K*u^{p}(s,\cdot)\|ds$
$\displaystyle=\underline{c}^{-1}(1-e^{-\underline{c}t})\sup_{\tau\geq
0}\|K*u^{p}(\tau,\cdot)\|.$

So once we achieve time independent control on the integrand, we can control
the integral term likewise.

Firstly we use Proposition 5.1 and Proposition 6.1 to achieve uniform $H^{1}$
bound,

$\displaystyle\|K*u^{p}\|_{H^{1}}$
$\displaystyle=\|K*u^{p}\|_{L^{2}}+\|K_{x}*u^{p}\|_{L^{2}}$
$\displaystyle\leq\|K\|_{W^{1,1}}\|u^{p}\|_{L^{2}}$
$\displaystyle\leq\|K\|_{W^{1,1}}\|u\|_{L^{p+1}}^{\frac{p+1}{2}}\|u\|_{\infty}^{\frac{p-1}{2}}<\infty.$

Since $u_{0}\in\mathcal{V}$, by (6.5) we have uniform $H^{1}$ bound on $u$.

Finally, we claim that $\|K*u^{p}\|_{H^{2}}<C$ for a time independent $C>0$.
It is enough to check $\|K_{x}*(u^{p-1}u_{x})\|_{L^{2}}.$

$\displaystyle\|K_{x}*(u^{p-1}u_{x})\|_{L^{2}}$ $\displaystyle=\|\int
K_{x}(x-y)u^{p-1}(y)u_{x}(y)dy\|_{L^{2}}$
$\displaystyle\leq\|K\|_{W^{1,1}}\|\|u\|_{\infty}^{p-1}\|u\|_{H^{1}}<\infty.$

So the trajectory $t\mapsto\int_{0}^{t}e^{-\int_{s}^{t}2pE(u)d\tau}K*u^{p}ds$
is uniformly bounded in $H^{2}$, and compact in $H$. Therefore by the
Rellich–Kondrachov theorem we have the result. ∎

####  6.3. Nontrivial wave profiles

The famous LaSalle’s invariance principle states that any precompact of
gradient flow of $C^{1}$ energy $E$ approaches to the set of critical points
of $E$. Since we achieve the compactness, one can expect that the
subsequential limit point to be a critical point. Recall that the uniform
lower bound holds only on the compact domain, so LaSalle’s principle cannot be
directly applied. So we will perform careful analysis to show the limit point
on $\mathbb{R}$ satisfies the fixed point, which shows the existence of the
solitary wave profile. Additionally, we show the instability of the constant
solution, so that we have nontrivial limit point with the choice of the
initial data.

###### Instability of constant solutions

Note that $\bar{u}$ is a constant satisfying $L\bar{u}^{p+1}=1$, $\bar{u}$ is
a fixed point of (5.2). Considering that we aim to calculate nontrivial
equilibria of (5.2), we need to investigate the stability of $\bar{u}.$

Since (5.2) is linear in $K$, without loss of generality we assume that $\int
K(y)dy=1.$ In this case,

$2pE(\bar{u})=\int_{\mathds{T}_{L}}\bar{u}^{p}K*\bar{u}^{p}=L\bar{u}^{2p}=\bar{u}^{p-1},$

Recall that

$\displaystyle DF(u)(v)=e^{-\frac{2p}{p+1}\int u^{p+1}dx}\int
u^{p-1}(K*u^{p}-c(t)u)vdx.$

The Hessian at an equilibrium $\tilde{u}$ is the following,

$\displaystyle D^{2}F(\tilde{u})(v,w)$
$\displaystyle=D(e^{-\frac{2p}{p+1}\int\tilde{u}^{p+1}dx})(w)\int\tilde{u}^{p-1}\cancelto{0}{(K*\tilde{u}^{p}-c(t)u)}vdx$
$\displaystyle+e^{-\frac{2p}{p+1}\int\tilde{u}^{p+1}dx}D(\int\tilde{u}^{p-1}(K*\tilde{u}^{p}-c(t)\tilde{u})vdx)(w)$
$\displaystyle=e^{-\frac{2p}{p+1}\int\tilde{u}^{p+1}dx}\int(p-1)\tilde{u}^{p-2}w\cancelto{0}{(K*\tilde{u}^{p}-c(t)\tilde{u})}vdx$
$\displaystyle+e^{-\frac{2p}{p+1}\int\tilde{u}^{p+1}dx}\int\tilde{u}^{p-1}\Big{(}pK*(\tilde{u}^{p-1}w)$
$\displaystyle-2p(\int\tilde{u}^{p-1}(K*\tilde{u}^{p})wdx)\tilde{u}-2pE(\tilde{u})w\Big{)}vdx$
$\displaystyle=\langle v,D^{2}F(\tilde{u})w\rangle_{L^{2}}.$ $\displaystyle
D^{2}F(\bar{u})(v)=e^{-\frac{2p}{p+1}}\bar{u}^{2p-2}(pK*v-\frac{2p}{L}\int_{\mathds{T}_{L}}vdy-v).$
(6.6)

Note that, the constant function $\mathds{1}$ is an eigenvector with negative
eigenvalue,

$D^{2}F(\bar{u})(\mathds{1})=-e^{-\frac{2p}{p+1}}\bar{u}^{2p-2}(p+1)\mathds{1}.$

In fact, other possible eigenvectors pointing outside of the invariant
manifold $\mathcal{M}$ have the same negative eigenvalue. Let’s take a look at
the following eigenvalue problem,

$\displaystyle K*v-\frac{2}{L}\int_{\mathds{T}_{L}}vdy$ $\displaystyle=\lambda
v$
$\displaystyle\int_{\mathds{T}_{L}}(K*v-\frac{2}{L}\int_{\mathds{T}_{L}}vdy)dx$
$\displaystyle=\lambda\int_{\mathds{T}_{L}}vdx$
$\displaystyle-\int_{\mathds{T}_{L}}vdx$
$\displaystyle=\lambda\int_{\mathds{T}_{L}}vdx.$

So we can conclude that, either

$\lambda=-1\text{ or }\int_{\mathds{T}_{L}}vdx=0.$

Note that the set of mean zero functions corresponds to the tangent manifold
at the constant equilibrium $\bar{u}$, i.e.,
$T_{\bar{u}}\mathcal{M}=\\{v\in\mathcal{V}:\int_{\mathds{T}_{L}}vdx=0\\}$. So
for such $v\in T_{\bar{u}}\mathcal{M}$,

$\displaystyle D^{2}F(\bar{u})(v)=e^{-\frac{2p}{p+1}}\bar{u}^{2p-2}(pK*v-v).$
(6.7)

We claim that $D^{2}F(\bar{u})$ has positive eigenvalues, which implies the
instability of the constant solution $\bar{u}.$

###### Spectrum of convolution operator

Note that for $\lambda\in\mathbb{R}$, $e^{i\lambda x}$ is an eigenvector of
convolution operators,

$\displaystyle K*e^{i\lambda x}$ $\displaystyle=\int e^{i\lambda(x-y)}K(y)dy$
$\displaystyle=e^{i\lambda x}\int e^{-i\lambda y}K(y)dy$
$\displaystyle=\hat{K}(\lambda)e^{i\lambda x}.$

Recall that the perturbation $\phi$ we chose has mean zero, i.e. $\int\phi
dx=0$. In order to satisfy $\int_{\mathds{T}_{L}}e^{i\lambda y}dy=0$, or
$e^{i\lambda x}\in H^{1}(\mathds{T}_{L})$, $\lambda$ must be resonant with
$L$, i.e., $\lambda=\frac{2n\pi}{L}$ for $n\in\mathbb{N}$.

Since $K$ is even, we can check that the corresponding eigenvalue
$\hat{K}(\lambda)$ of the eigenvector $e^{i\lambda x}$ is positive,

$\displaystyle\hat{K}(\lambda)$ $\displaystyle=\int(\cos(\lambda
y)-i\sin(\lambda y))K(y)dy$ $\displaystyle=\int\cos(\lambda y)K(y)dy>0.$

Therefore, the eigenvalues of $D^{2}F(\bar{u})$ on $\mathcal{M}$ is
$e^{-\frac{2p}{p+1}}\bar{u}^{2p-2}(p\hat{K}(\frac{2n\pi}{L})-1)$. Because
$\hat{K}(\frac{2n\pi}{L})\to 1$ as $L\to\infty$, for any $p>1$ there exists
large $L>0$ such that $\bar{u}$ is a saddle point. Moreover, every eigenvalues
of $D^{2}E(\bar{u})$ becomes positive as $L\to\infty$, which corresponds to
the fact that $0$ is a source when $L=\infty$.

###### Existence of Nontrivial Wave Profiles

One of the benefits of a gradient system is that by LaSalle’s invariance
principle for any bounded solution $u(t)$ of gradient flow of $C^{1}$ energy,
we have $\lim_{t\to\infty}\|u^{\prime}(t)\|_{\mathcal{H}}=0$. So by showing
the compactness of trajectory, we infer the existence of periodic and solitary
wave profiles. Furthermore, we can show that there is a nontrivial wave
profile by starting with $u_{0}=\bar{u}+\delta\cos(\frac{2\pi}{L}x)$ for a
small enough $\delta>0$, we can conclude that there is a nontrivial
equilibrium of (5.2).

######  Theorem 6.5 (Existence of nontrivial wave profiles).

For large enough $L\in(0,\infty]$, there exists nontrivial equilibrium
$\tilde{u}\in\mathcal{M}$ of (5.2) such that
$K\ast\tilde{u}^{p}=\tilde{c}\tilde{u}$ for some $\tilde{c}>0$ .

###### Proof.

1\. $L<\infty$: Let $\bar{u}$ be the constant equilibrium such that
$L\bar{u}^{p+1}=1$ and $\nabla F(\bar{u})=0$. We continue the argument from
the discussion about the instability of $\bar{u}.$ Let $L>0$ be large enough
such that $p\hat{K}(\frac{2\pi}{L})-1\geq 0$ and let the perturbation be
$\phi(x)=\cos(\frac{2\pi}{L}x)$ so that $\nabla^{2}F(\bar{u})(\phi,\phi)>0$.
Using Taylor’s theorem for small enough $\delta>0$ we have

$\displaystyle
F(\bar{u}+\delta\phi)=F(\bar{u})+\frac{1}{2}\delta^{2}\nabla^{2}F(\bar{u})(\phi,\phi)+o(\delta^{2})>F(\bar{u}).$

Now we initiate the evolution starting with
$u_{0}=\bar{u}+\delta\cos(\frac{2\pi}{L}x)$. Then by the compactness result,
Theorem 6.4, we can find a limit point $\tilde{u}.$

Note that when $L<\infty$, the solution is uniformly bounded above, and below
away from $0$ by Proposition 6.1 and Corollary 6.1. So the norm defined by the
metric $\langle\cdot,\cdot\rangle_{u}$ is equivalent to $L^{2}$ norm, and we
can use LaSalle’s invariance principle. Technically, we have

$\|\nabla F(u)\|_{L^{2}}\geq C\|u^{\prime}\|_{L^{2}},$

which implies $F$ is a strict Lyapunov function for $\\{u(t)\\}_{t\geq 0}$
(Definition 8.4.5 of [27]). We use LaSalle’s invariance principle (Theorem
8.4.6 of [27]) to deduce that

$\|\nabla F(\tilde{u})\|_{\mathcal{H}}=0.$

Note that since $F$ is increasing along the evolution we have
$F(\tilde{u})>F(\bar{u})$, which implies $\tilde{u}\not=\bar{u}.$

2\. $L\leq\infty$: Here we provide more careful analysis for $L=\infty$, since
$F$ might not be a strict Lyapunov function, or the inequality

$\|\nabla F(u)\|_{L^{2}}\geq C\|u^{\prime}\|_{L^{2}},$

might not hold. We will proceed with the arguments as follows,

  1. (1)

$t\mapsto u^{q}(t,\cdot)$ is Hölder continuous in $L^{2}$ for any
$q\geq{\frac{p+1}{2}}$.

  2. (2)

$t\mapsto\frac{d}{dt}(u^{p+1})(t,\cdot)$ is uniformly continuous in $L^{2}$.

  3. (3)

When $u_{0}$ is bell-shaped, $\|\frac{d}{dt}(u^{p+1})(t,\cdot)\|_{L^{2}}\to 0$
as $t\to\infty$.

  4. (4)

For any $\tilde{u}\in\omega(u)$,
$\|\tilde{u}^{p}(K*\tilde{u}^{p}-\tilde{c}\tilde{u})\|_{L^{2}}=0$.

  5. (5)

$K*\tilde{u}^{p}=\tilde{c}\tilde{u}$.

  6. (6)

$\tilde{u}$ is nontrivial.

Step (1): Claim that for any $q\geq{\frac{p+1}{2}}$

$\|u^{q}(t)-u^{q}(s)\|_{L^{2}}\leq C|t-s|^{1/2}.$

Note that by the dissipation of $F$,

$\alpha(t)\frac{d}{dt}F(u(t))=\int_{\mathbb{R}}u^{p-1}(K*u^{p}-c(t)u)^{2}dx=\frac{4}{(p+1)^{2}}\|\frac{d}{dt}u^{\frac{p+1}{2}}(t)\|_{L^{2}}^{2}$

Therefore,

$\displaystyle\|u^{\frac{p+1}{2}}(t)-u^{\frac{p+1}{2}}(s)\|_{L^{2}}$
$\displaystyle\leq\int_{s}^{t}\|\frac{d}{d\tau}u^{\frac{p+1}{2}}(\tau)\|_{L^{2}}d\tau$
$\displaystyle=|t-s|^{1/2}\int_{s}^{t}\|\frac{d}{dt}u^{\frac{p+1}{2}}(t)\|_{L^{2}}^{2}d\tau$
$\displaystyle\leq\sup_{\tau\geq
0}\alpha(\tau)\frac{(p+1)^{2}}{4}\int_{s}^{t}\frac{d}{dt}F(u(\tau))d\tau|t-s|^{1/2}$
$\displaystyle\leq C|t-s|^{1/2}.$

Note that since $u$ is bounded by Proposition 6.1, $u\mapsto u^{\gamma}$ for
any $\gamma>1$ is (locally) Lipschitz, i.e.,

$|u^{\gamma}(t)-u^{\gamma}(s)|\leq C|u(t)-u(s)|,$ (6.8)

where the Lipschitz constant $C>0$ may depend on $\|u\|_{L^{\infty}}$. So for
any $q\geq{\frac{p+1}{2}}$ we choose $\gamma=\frac{2q}{p+1}$ so that
$u^{q}=(u^{\frac{p+1}{2}})^{\gamma}$. Then

$\|u^{q}(t)-u^{q}(s)\|_{L^{2}}^{2}\leq
C\int_{\mathbb{R}}|u^{\frac{p+1}{2}}(t)-u^{\frac{p+1}{2}}(s)|^{2}dx\leq
C|t-s|.$

Step (2): Claim that $t\mapsto\frac{d}{dt}(u^{p+1})(t,\cdot)$ is uniformly
continuous in $L^{2}$. Because,

$\frac{1}{p+1}\frac{d}{dt}(u^{p+1})=u^{p}K*u^{p}-c(t)u^{p+1},$

and $u^{p}(t)$ and $u^{p+1}(t)$ are uniformly continuous in $L^{2}$ and
bounded uniformly in $L^{\infty}$, it is enough to show that $t\mapsto c(t)$
is Hölder continuous.

$\displaystyle c(t)-c(s)$
$\displaystyle=\int(u^{p}(t)-u^{p}(s))K*u^{p}(t)dx+\int(u^{p}(t)-u^{p}(s))K*u^{p}(s)dx$
$\displaystyle\leq 2\|u^{p}(t)-u^{p}(s)\|_{L^{2}}\|K*u^{p}\|_{L^{2}}$
$\displaystyle\leq
2\|u^{p}(t)-u^{p}(s)\|_{L^{2}}\|K\|_{L^{1}}\|u\|_{L^{p+1}}^{\frac{p+1}{2}}\|u\|_{\infty}^{\frac{p-1}{2}}.$

By Proposition 5.1 and the previous step, we have the claim.

Step (3): Let $u_{0}$ be bell-shaped and claim that
$\|\frac{d}{dt}(u^{p+1})(t,\cdot)\|_{L^{2}}\to 0$ as $t\to\infty$. We use the
uniformly continuity of $t\mapsto\frac{d}{dt}(u^{p+1})(t,\cdot)$ in $L^{2}$.
For any $\epsilon>0$ and $t>0$, we can find $\delta>0$ such that for any
$h\in[0,\delta]$

$\|\frac{d}{dt}(u^{p+1})(t,\cdot)\|_{L^{2}}\leq\epsilon+\|\frac{d}{dt}(u^{p+1})(t+h,\cdot)\|_{L^{2}}.$

Therefore, by averaging

$\displaystyle\|\frac{d}{dt}(u^{p+1})(t,\cdot)\|_{L^{2}}$
$\displaystyle\leq\epsilon+\frac{1}{\delta}\int_{0}^{\delta}\|\frac{d}{dt}(u^{p+1})(t+h,\cdot)\|_{L^{2}}dh$
$\displaystyle\leq\epsilon+\delta^{-1/2}(\int_{t}^{t+\delta}\|\frac{d}{dt}(u^{p+1})(s,\cdot)\|_{L^{2}}^{2}ds)^{1/2}$
$\displaystyle\leq\epsilon+\delta^{-1/2}(\int_{t}^{\infty}\|2u^{\frac{p+1}{2}}(s,\cdot)\frac{d}{dt}(u^{\frac{p+1}{2}})(s,\cdot)\|_{L^{2}}^{2}ds)^{1/2}$
$\displaystyle\leq\epsilon+C(\int_{t}^{\infty}\|\frac{d}{dt}(u^{\frac{p+1}{2}})(s,\cdot)\|_{L^{2}}^{2}ds)^{1/2}$
$\displaystyle\leq\epsilon+C(F(\infty)-F(t))^{1/2}.$

which implies the claim.

Step (4): Claim that
$\|\tilde{u}^{p}(K*\tilde{u}^{p}-\tilde{c}\tilde{u})\|_{L^{2}}=0$ where
$\tilde{c}=\int_{\mathbb{R}}\tilde{u}^{p}K*\tilde{u}^{p}dx$. Since the
$L^{2}$-compactness, Theorem 6.3, we have a increasing sequence time
$\\{t_{k}\\}_{k\in\mathbb{N}}\subset\mathbb{R}_{+}$ and the limit
$\tilde{u}\in L^{2}(\mathbb{R})$ such that

$\lim_{k\to\infty}\|u(t_{k})-\tilde{u}\|_{L^{2}}=0.$

Using uniform $L^{\infty}$-bound, one can show that $u\mapsto u^{p},u^{p+1}$
is continuous in $L^{2}$, and we have the claim by continuity. Note that as we
discussed in the proof of Theorem 6.3, $\tilde{u}$ is a also continuous
subsequential limit.

Step (5): Claim that either $\tilde{u}\equiv 0$ or $\tilde{u}>0$ so that
$K*\tilde{u}^{p}=\tilde{c}\tilde{u}$. Recall that $\tilde{u}$ is continuous
and bell-shaped. Suppose $\tilde{u}$ is not identically $0$ but not strictly
positive. Since $K*\tilde{u}^{p}>0$, the minimum of $K*\tilde{u}^{p}$ on the
support of $\tilde{u}$ is positive and let’s call it $m$. Now we can find an
open interval in the support of $\tilde{u}$ such that $m-2\tilde{c}\epsilon>0$
and $\epsilon<\tilde{u}<2\epsilon$, so that in the interval

$|\tilde{u}^{p}(K*\tilde{u}^{p}-\tilde{c}\tilde{u})|\geq\epsilon^{p}(m-2\tilde{c}\epsilon)>0,$

which leads to the contradiction to the acquired Step (4).

Step (6): Claim that $\tilde{u}>0$. We can conclude using the continuity of
$\tilde{u}$, Corollary 6.2 and the fact that $\tilde{u}$ is also a pointwise
limit of $u(t_{k})$. ∎

###### Discussion

In fact, there is an issue with applying the Łojasiewicz convergence criterion
to (5.2). So at present we are unable to conclude convergence of solutions as
$t\to\infty.$ Let $H^{1}_{+}(\mathds{T}_{L})$ denote the positive cone in
$H^{1}(\mathds{T}_{L})$, i.e.,

$H^{1}_{+}(\mathds{T}_{L}):=\\{u\in H^{1}(\mathds{T}_{L}):u(x)>0,\;\;\forall
x\in H^{1}(\mathds{T}_{L})\\}.$

Note that $H^{1}(\mathds{T}_{L})$ is a Banach algebra. So the Lyapunov
functional,

$F(u)=\frac{1}{2p}e^{-\frac{2p}{p+1}\int u^{p+1}dx}\int u^{p}K*u^{p}dx,$

is analytic in the neighborhood of $H^{1}_{+}(\mathds{T}_{L}).$

The issue, however, arise from failure of the Hessian of $F$

$D^{2}F(\tilde{u}):w\mapsto
e^{-\frac{2p}{p+1}}\tilde{u}^{p-1}\left({pK*(\tilde{u}^{p-1}w)}-{4p^{2}E(\tilde{u})(\int\tilde{u}^{p}wdx)\tilde{u}}-2pE(\tilde{u})w\right)$

being semi-Fredholm.

Firstly, on $\mathbb{R}$ there is no chance for $D^{2}E(\tilde{u})$ to have
closed range since $\tilde{u}(x)\to 0$ as $x\to\pm\infty$. For example, for
any $u\in\mathcal{V}$,
$\big{\\{}u^{p-1}\mathds{1}_{\\{-L,L\\}}\big{\\}}_{L>0}$ is Cauchy in
$\mathcal{V}^{\prime}$ but $\mathds{1}\not\in\mathcal{V}$. So let’s assume
that $L<\infty$. Now since $\tilde{u}$ is bounded above and below from $0$,
the multiplication with $\tilde{u}$, $v\mapsto\tilde{u}v$ is an isomorphism
from $\mathcal{V}$ to itself. By Theorem 1.4 one can check that is semi-
Fredholm from $\mathcal{V}$ to itself, but not from $\mathcal{V}$ to
$\mathcal{V}^{\prime}$.

On the contrary, the key to expand the Łojasiewicz inequality to infinite
dimensional domain is to construct the finite dimensional copy of the energy,
which can be approximated by $\|\nabla E(u)\|_{\mathcal{H}}$, like in
Proposition 1.5. When $\Pi+A:\mathcal{V}\to\mathcal{V}^{\prime}$ is an
isomorphism, we represent $u\sim(\Pi u,\nabla E(u))$ using the isomorphism
$\mathcal{N}:\mathcal{V}\to\mathcal{V}^{\prime}$ such that

$\displaystyle\mathcal{N}u=\Pi u+\nabla E(u).$

The issue with (5.2) for applying the infinite dimensional framework in
Section 1.2 of Chapter 1 is $R(\Pi+A)$ is too regular to be closed in
$\mathcal{V}^{\prime}$. Moreover, even if we identify $\Pi+A$ as an
isomorphism from $\mathcal{V}$ to itself, we will end up having
$\|\nabla{E}(u)\|_{\mathcal{V}}$, not in control of $\mathcal{H}.$

###  Chapter 7 Regularized concentration-dispersion dynamics

In this chapter, we propose and analyze a modification of concentration-
dispersion dynamics with small $\epsilon>0$, as follows.

###### Regularized concentration-dispersion equation

$\left\\{\begin{aligned} \;\;\frac{d}{dt}u(t,x)&=\epsilon(u^{p})_{xx}+K\ast
u^{p}-c(t)u\\\ c(t)&=\int u^{p}K\ast
u^{p}-\epsilon[(u^{p})_{x}]^{2}dx,\end{aligned}\right.$ (7.1)

where $K(x)>0$ for all $x\in\mathds{T}_{L}$ and $K(x)\in W^{1,1}\cap
W^{1,p+1}(\mathds{T}_{L})$. We will use the gradient structure of the
regularized concentration-dispersion equation (7.1) to prove convergence of
solutions to a nontrivial equilibrium. From now on we consider only
$L<\infty.$

###### Well-posedness

For local well-posendess with positive initial data, we refer Chapter 8 of
[46], and set $X=C(\mathds{T}_{L})$ and $D=C^{2}(\mathds{T}_{L})$. Considering
singular aspect of solutions of porous medium equations, global well-posedness
is not trivial. By combining the local well-posedness, regularity (Theorem
8.1.1, Theorem 8.3.4 of [46]) and the $C^{2}$-compactness result in the
section 2, we deduce regularity up to $C^{5}(\mathds{T}_{L})$. Thus for any
$u_{0}\in C^{5}(\mathds{T}_{L})$ initial data, we can choose sufficiently
small $\epsilon>0$ to achieve global well-posedness in $C^{2}(\mathds{T}_{L})$
by pointwise maximum principle and maximal solution argument (Proposition
8.2.1 of [46]).

####  7.1. Gradient structure

Based on those regularity properties, we show the asymptotic bound of the
solution $u(t,x)$. Prior to show that the solution stay above from $0$
uniformly, we show the solution $u(t,x)$ stays positive. In this way, (7.1) is
well-defined.

######  Proposition 7.1 (Positivity).

Let $u(t,x)\in C^{2}(\mathds{T}_{L})$ be a solution of (7.1). If $u(0,x)>0$
for all $x\in\mathds{T}_{L}$ then $u(t,x)>0$ for all $x\in\mathds{T}_{L}$ and
$t>0.$

###### Proof.

Let $\beta(t):=\exp{(\int_{0}^{t}c(s)ds)}$ and $v(t,x):=\beta(t)u(t,x)$. We
claim that for all $t\geq 0$,

$v(t,x)\geq\min_{x}v(0,x).$

Suppose at $t_{0}>0$, $x_{0}=\arg\min_{x}v(t_{0},x)$, then

$\displaystyle\frac{dv}{dt}(t_{0},x_{0})=\beta^{1-p}(t)\left(\epsilon(v^{p})_{xx}+K\ast
v^{p}\right)\geq\beta^{1-p}(t)K\ast v^{p}>0.$

In particular, $\\{v\in C^{2}(\mathds{T}_{L}):v>0\\}$ is an invariant set of
(7.1). So for all $t>0$,

$u(t,x)=\beta^{-1}(t)v(t,x)>0.\qed$

###### Important functionals

We will show that (7.1) has a nice $L^{2}$-gradient structure. We use the
following functionals.

$\displaystyle E(u)$ $\displaystyle:=\frac{1}{2p}\int
u^{p}K*u^{p}-\epsilon[(u^{p})_{x}]^{2}dx,$ (7.2) $\displaystyle F(u)$
$\displaystyle:=\frac{1}{2p}e^{-\frac{2p}{p+1}\int u^{p+1}dx}\int
u^{p}K*u^{p}-\epsilon[(u^{p})_{x}]^{2}dx,$ (7.3)
$\displaystyle=:E(u)/\alpha(t)$ (7.4)

The derivative of $F$ shows that (7.1) is a gradient-like system of the energy
$F.$

$\displaystyle\alpha(t)DF(u)(v)$
$\displaystyle=DE(u)(v)-E(u)\frac{2p}{p+1}D(\int u^{p+1}dx)(v)$
$\displaystyle=\int
u^{p-1}vK*u^{p}-\epsilon(u^{p-1}v)_{x}(u^{p})_{x}dx-2pE(u)\int
u^{p}u^{\prime}dx$ $\displaystyle=\int
u^{p-1}(\epsilon(u^{p})_{xx}+K*u^{p}-c(t)u)vdx.$ (7.5)

We show the convergence properties of the functionals.

######  Proposition 7.2.

Let $u(t,x)\in L^{p+1}(\mathds{T}_{L})$ be a solution of (5.2) such that
$u_{0}(x)>0$ for all $x\in\mathds{T}_{L}$. Assume that $E(u_{0})>0$, then the
following functionals have convergent properties as $t\to\infty:$

  1. (1)

$F(u)=E(u)/\alpha(t)$ is nondecreasing,

  2. (2)

$\|u\|_{L^{p+1}}$ converges monotonically to 1,

  3. (3)

$E(u)$ stays positive and converges to a positive constant.

Additionally, if $\|u_{0}\|_{L^{p+1}}\leq 1$ then $E(u)$ is nondecreasing as
well.

###### Proof.

Firstly, we prove monotonic convergence of $F$. $F$ is nondecreasing because
the time derivative of $F$ is nonnegative, i.e.,

$\displaystyle F^{\prime}(u(t))e^{\frac{2p}{p+1}\int u^{p+1}dx}$
$\displaystyle=E^{\prime}(u)-E(u)\frac{2p}{p+1}(\int u^{p+1}dx)^{\prime}$
$\displaystyle=\int
u^{p-1}(K*u^{p}+\epsilon(u^{p})_{xx}u^{\prime}dx-2pE(u)\int u^{p}u^{\prime}dx$
$\displaystyle=\int u^{p-1}(\epsilon(u^{p})_{xx}+K*u^{p}-c(t)u)^{2}dx\geq 0.$

Since $E(u_{0})>0$ implies $F(u_{0})>0$ and

$F(u)=e^{\frac{-2p}{p+1}\int u^{p+1}dx}E(u)\leq E(u)=2pc(t),$ (7.6)

we have $\inf_{t\geq 0}c(t)>0$. So the monotonic convergence of $L^{p+1}$ norm
can be shown by taking time-differentiation,

$\frac{1}{p+1}(\int u^{p+1}dx)^{\prime}=c(t)\big{(}1-\int u^{p+1}dx\big{)}.$
(7.7)

We can conclude

$\lim_{t\to\infty}\|u(t)\|_{L^{p+1}}\to 1.$

Now we claim that $E(u)$ is bounded above.

$\displaystyle 2pE(u)$ $\displaystyle\leq\int u^{p}K\ast u^{p}dx$
$\displaystyle\leq\|u\|^{p}_{L^{p+1}}\|K*u^{p}\|_{L^{p+1}}$
$\displaystyle\leq\|K\|_{L^{\frac{p+1}{2}}}\|u\|^{2p}_{L^{p+1}}$
$\displaystyle\leq(1+\|u_{0}\|_{L^{p+1}})^{2p}\|K\|_{L^{\frac{p+1}{2}}}<\infty.$
(7.8)

Therefore $E(u(t))$ converges to a positive constant as $t\to\infty$ and $F$
converges monotonically. ∎

####  7.2. Asymptotic bound on $u$ and $C^{2}$-compactness

We open this section with doing the same pointwise estimate as in the original
concentration-dispersion dynamics.

######  Proposition 7.3.

Let $u\in\mathcal{V}$ be a solution of (5.2). Then there exists $M(K,u_{0})>0$
such that $\sup_{t\geq 0}\|u(t,\cdot)\|_{L^{\infty}}<M$.

###### Proof.

We use similar argument in the proof of Proposition 6.1. Since
$(u^{p})_{xx}\leq 0$ at the maximizer,

$\displaystyle u^{\prime}$ $\displaystyle=\epsilon(u^{p})_{xx}+K\ast
u^{p}-c(t)u$ (7.9) $\displaystyle\leq\int K(x-y)u(y)^{p}dy-c(t)u$ (7.10)
$\displaystyle\leq(\int K(x-y)^{p+1}dy)^{1/p+1}(\int
u^{p+1}dy)^{p/(p+1)}-\underline{c}u.$ (7.11)

where $\underline{c}=\inf_{t\geq 0}c(t)$. Note that $\underline{c}>0$ because
$u>0$ and $F(u)=\frac{1}{2p}e^{-\frac{2p}{p+1}\int u^{p+1}dx}c(t)>0$ is
increasing according to the Proposition 7.2. This means

$u^{\prime}(t,x)<0\quad\text{if
}u(t,x)>\frac{1}{\underline{c}}\|K\|_{L^{p+1}}\|u\|^{p}_{L^{p+1}}.$

Due to monotonic convergence of $\|u\|_{L^{p+1}}\to 1$ as $t\to\infty$, we can
conclude that

$\max_{t\geq
0}\|u(t,x)\|_{\infty}\leq\|u_{0}\|_{\infty}+\frac{1}{\underline{c}}(1+\|u_{0}\|^{p}_{L^{p+1}})\|K\|_{L^{p+1}}.$

∎

In the compact spatial domain, we can use the upper bound to calculate the
lower bound. This will be used to control the Riemannian metric. Naturally,
the lower bound vanishes as $L\to\infty.$

######  Corollary 7.1.

For fixed $L<\infty$, let $u$ be a solution of (5.2) with $u_{0}\in
C^{2}(\mathds{T}_{L})$. Assume that there is $k>0$ such that $K(x)\geq k,$
$\forall x\in\mathds{T}_{L}$. Then there exist $m(K,L,u_{0})>0$ such that
$\min_{x}u(x,t)\geq m$.

###### Proof.

From Proposition 7.3, there exists $M(u_{0},K)>0$ such that $u(x,t)<M$ for all
$t\geq 0$. Now for any $\delta>0$ we can choose large enough $T>0$ so that

$1-\delta\leq\int u^{p+1}dx\leq M\int u^{p}dx.$

Note that $\overline{c}:=\sup_{t\geq 0}c(t)$ is well-defined since it
converges. Then at the minimizer,

$\displaystyle u^{\prime}$ $\displaystyle=\epsilon(u^{p})_{xx}+K\ast
u^{p}-c(t)u$ $\displaystyle\geq k\int u^{p}dx-\overline{c}u$
$\displaystyle\geq\frac{(1-\delta)k}{M}-\overline{c}u>0,$

when $u<(1-\delta)k/M\overline{c}$. So the result follows. ∎

######  $C^{2}$-compactness

Recall that the original concentration dispersion equations (5.2), provide
precompact solution even without smoothing effect. For the regularized
equations, we can derive the following: for any regular enough initial data,
we can make $\epsilon$ small enough to make the trajectory precompact in
$C^{2}$. Recall that we need $C^{2}$-compactness to achieve global well-
posedness (cf., [46]). We assume $L<\infty$ and analyze asymptotic pointwise
bound of $u_{x}$, $u_{xx}$ and $u_{xxx}$.

######  Theorem 7.1.

Let $L<\infty$. For any $u_{0}\in C^{5}(\mathds{T}_{L})$ with
$u_{0}(t,\cdot)>0$ and $\|u_{0}\|_{L^{p+1}}\leq 1$, there exists
$\epsilon_{0}>0$ such that for any $\epsilon\in(0,\epsilon_{0}]$ the solution
$\\{u(t)\\}_{t\geq 0}$ of (7.1) with $u(0)=u_{0}$ has uniformly bounded
$C^{3}$-norm. In particular, $\\{u(t)\\}_{t\geq 0}$ precompact in
$C^{2}(\mathds{T}_{L}).$

###### Proof.

We will firstly calculate pointwise time evolution of $u_{x}$ and $u_{xx}$ and
later argue how to deduce the conclusion.

The derivatives of each term on (7.1) are as follows,

$\displaystyle\partial_{x}(u^{p})$ $\displaystyle=pu^{p-1}u_{x}$
$\displaystyle\partial^{2}_{x}(u^{p})$
$\displaystyle=p(p-1)u^{p-2}u_{x}^{2}+pu^{p-1}u_{xx}$
$\displaystyle\partial^{3}_{x}(u^{p})$
$\displaystyle=p(p-1)(p-2)u^{p-3}u_{x}^{3}+3p(p-1)u^{p-2}u_{x}u_{xx}+pu^{p-1}u_{xxx}$
$\displaystyle\partial^{4}_{x}(u^{p})$
$\displaystyle=p(p-1)(p-2)(p-3)u^{p-4}u_{x}^{4}+6p(p-1)(p-2)u^{p-3}u_{x}^{2}u_{xx}$
$\displaystyle\quad+3p(p-1)u^{p-2}u_{xx}^{2}+4p(p-1)u^{p-2}u_{x}u_{xxx}+pu^{p-1}u_{xxxx}.$
$\displaystyle\partial^{5}_{x}(u^{p})$
$\displaystyle=p(p-1)(p-2)(p-3)(p-4)u^{p-5}u_{x}^{5}+10p(p-1)(p-2)u^{p-4}u_{x}^{3}u_{xx}$
$\displaystyle\quad+15p(p-1)(p-2)u^{p-3}u_{x}u_{xx}^{2}+10p(p-1)(p-2)u^{p-3}u_{x}^{2}u_{xxx}$
$\displaystyle\quad+10p(p-1)u^{p-2}u_{xx}u_{xxx}+5p(p-1)u^{p-2}u_{x}u_{xxxx}+pu^{p-1}u_{xxxxx}.$
$\displaystyle|\partial_{x}(K*u^{p})|$
$\displaystyle=|K_{x}*u^{p}|\leq\|K\|_{W^{1}}\|u^{p}\|_{L^{\infty}}$
$\displaystyle|\partial^{2}_{x}(K*u^{p})|$
$\displaystyle=|K_{x}*(u^{p-1}u_{x})|\leq\|K\|_{W^{1}}\|u^{p-1}\|_{L^{\infty}}\|u_{x}\|_{L^{\infty}}.$

Since $\|u_{0}\|_{L^{p+1}}\leq 1$, $c(t)$ is increasing, so

$c(t)=\int u^{p}K\ast u^{p}-\epsilon[(u^{p})_{x}]^{2}dx\geq c(0).$

Let $c(0)=\alpha-\epsilon\beta$ where,

$\alpha=\int u_{0}^{p}K\ast
u_{0}^{p}dx,\qquad\beta=\int[(u_{0}^{p})_{x}]^{2}dx.$

We use Proposition 7.3 and Proposition 7.1 to control the powers of $u$.

1\. $u_{x}$ control: Suppose for a fixed $t>0$, there is a point
$x_{0}\in\mathds{T}_{L}$, $u_{x}(t,x_{0})$ is positive and reached maximum so
that $u_{xx}=(u_{x})_{x}=0$ and $u_{xxx}=(u_{x})_{xx}\leq 0$. Then from (7.1)
there exist constants $A,B\in\mathbb{R}$ such that

$\partial_{t}u_{x}\leq\epsilon Au_{x}^{3}+B-(\alpha-\epsilon\beta)u_{x}.$

Note that as $\epsilon$ gets smaller, pointwise invariant interval for $u_{x}$
gets larger. Thus we can find small enough $\epsilon_{0}>0$ so that
$\limsup_{t}u_{x}(t,\cdot)$ is bounded. We can use the similar argument when
$u_{x}$ is negative and reached minimum. Therefore, for
$\epsilon\in(0,\epsilon_{0}]$ we can say
$u_{x}(t)\leq\max\\{\|u_{x}(0,\cdot)\|_{L^{\infty}},1+B/\alpha\\}$.

2\. $u_{xx}$ control: Suppose $u_{xx}$ is positive and reached maximum so that
$u_{xxx}=(u_{xx})_{x}=0$ and $u_{xxxx}=(u_{xx})_{xx}\leq 0$. Then from (7.1)
there exist constants, possibly depend on $\|u_{x}\|_{L^{\infty}}$,
$A^{\prime},B^{\prime},D\in\mathbb{R}$ such that

$\partial_{t}u_{xx}\leq\epsilon A^{\prime}u_{xx}^{2}-(\alpha-\epsilon
B^{\prime})u_{xx}+D.$

Since we already achieved $\epsilon$ and $t$ independent control on
$\|u_{x}\|_{L^{\infty}}$ we continue the same argument. As $\epsilon$ gets
smaller, pointwise invariant interval for $u_{xx}$ gets larger. Thus we can
find, possibly smaller, $\epsilon_{0}>0$ so that $\limsup_{t}u_{xx}(t,\cdot)$
is bounded. We can use the similar argument when $u_{xx}$ is negative and
reached minimum. In particular, since the domain is compact, we have uniform
$H^{2}$-norm and by Rellich–Kondrachov theorem.

3\. $u_{xxx}$ control: Suppose $u_{xxx}$ is positive and reached maximum so
that $u_{xxxx}=(u_{xxx})_{x}=0$ and $u_{xxxxx}=(u_{xxx})_{xx}\leq 0$. Then
from (7.1) there exist constants, depend on $\|u_{x}\|_{L^{\infty}}$ and
$\|u_{xx}\|_{L^{\infty}}$ , $A^{\prime\prime},B^{\prime\prime}\in\mathbb{R}$
such that

$\partial_{t}u_{xxx}\leq A^{\prime\prime}-(\alpha-\epsilon
B^{\prime\prime})u_{xxx}.$

We continue with the similar argument as before to make conclusion. ∎

####  7.3. Convergence analysis

Let $L<\infty$ and $\mathcal{V}=H^{1}(\mathds{T}_{L})$,
$\mathcal{H}=L^{2}(\mathds{T}_{L})$ so that

$\mathcal{V}\subset\mathcal{H}\subset\mathcal{V}^{\prime}.$

We follow the framework presented in Section 1.2 of Chapter 1.

###### Analyticity

We prove convergence in compact spatial domain where $L<\infty$. Let
$H^{1}_{+}(\mathds{T}_{L})$ denote the positive cone in
$H^{1}(\mathds{T}_{L})$, i.e.,

$H^{1}_{+}(\mathds{T}_{L}):=\\{u\in H^{1}(\mathds{T}_{L}):u(x)>0,\;\;\forall
x\in H^{1}(\mathds{T}_{L})\\}.$

Note that $H^{1}(\mathds{T}_{L})$ is a Banach algebra. So the Lyapunov
functional,

$\displaystyle F(u)$ $\displaystyle=\frac{1}{2p}e^{-\frac{2p}{p+1}\int
u^{p+1}dx}\int u^{p}K*u^{p}-\epsilon[(u^{p})_{x}]^{2}dx,$
$\displaystyle=\frac{1}{2p}e^{-\frac{2p}{p+1}\int u^{p+1}dx}\int
u^{p}K*u^{p}-\epsilon p^{2}u^{2p-2}u_{x}^{2}dx.$

is analytic in the neighborhood of $H^{1}_{+}(\mathds{T}_{L}).$ In order to
show that $F$ satisfies the Łojaisewicz inequality, let’s analyze the
derivatives of $F.$

###### Hessian of F

Recall that

$\displaystyle DF(u)(v)=e^{-\frac{2p}{p+1}\int u^{p+1}dx}\int
u^{p-1}\Big{[}\epsilon(u^{p})_{xx}+K*u^{p}-c(t)u\Big{]}vdx.$

The Hessian at an equilibrium $\tilde{u}$ is the following,

$\displaystyle D^{2}F(\tilde{u})(v,w)$
$\displaystyle=D(e^{-\frac{2p}{p+1}\int\tilde{u}^{p+1}dx})(w)\int\tilde{u}^{p-1}\cancelto{0}{\Big{[}\epsilon(\tilde{u}^{p})_{xx}+K*\tilde{u}^{p}-c(t)u\Big{]}}vdx$
$\displaystyle+e^{-\frac{2p}{p+1}\int\tilde{u}^{p+1}dx}D(\int\tilde{u}^{p-1}\Big{[}\epsilon(\tilde{u}^{p})_{xx}+K*\tilde{u}^{p}-c(t)\tilde{u}\Big{]}vdx)(w)$
$\displaystyle=e^{-\frac{2p}{p+1}\int\tilde{u}^{p+1}dx}\int(p-1)\tilde{u}^{p-2}w\cancelto{0}{\Big{[}\epsilon(\tilde{u}^{p})_{xx}+K*\tilde{u}^{p}-c(t)\tilde{u}\Big{]}}vdx$
$\displaystyle\quad+e^{-\frac{2p}{p+1}\int\tilde{u}^{p+1}dx}\int\tilde{u}^{p-1}\bigg{[}\epsilon
p(\tilde{u}^{p-1}w)_{xx}+pK*(\tilde{u}^{p-1}w)$
$\displaystyle\quad-2p\Big{\\{}\int\tilde{u}^{p-1}(\epsilon(\tilde{u}^{p})_{xx}+K*\tilde{u}^{p})wdx\Big{\\}}\tilde{u}-2pE(\tilde{u})w\bigg{]}vdx$
$\displaystyle=\langle v,D^{2}F(\tilde{u})w\rangle_{L^{2}}.$

Note that $\tilde{c}=\lim_{t\to\infty}c(t)=2pE(\tilde{u})$ so that,

$\epsilon(\tilde{u}^{p})_{xx}+K*\tilde{u}^{p}=\tilde{c}\tilde{u}.$

We can identify
$D^{2}F(\tilde{u}):\mathcal{V}\longrightarrow\mathcal{V}^{\prime}$ as follows,

$D^{2}F(\tilde{u}):w\mapsto e^{-\frac{2p}{p+1}}\tilde{u}^{p-1}\left(\epsilon
p(\tilde{u}^{p-1}w)_{xx}+{pK*(\tilde{u}^{p-1}w)}-{2p\tilde{c}(\int\tilde{u}^{p}wdx)\tilde{u}}-\tilde{c}w\right).$

Since $\tilde{u}\in C^{\infty}$ is bounded above, and below away from $0$, the
multiplication with $\tilde{u}$, $v\mapsto\tilde{u}v$ is an isomorphism from
$\mathcal{V}$ to $\mathcal{V}$, and from $\mathcal{V}^{\prime}$ to
$\mathcal{V}^{\prime}$. As we discussed in Proposition 3.1,
$\Delta:\mathcal{V}\to\mathcal{V}^{\prime}$ with periodic boundary condition
is semi-Fredholm. Therefore,

$w\mapsto\tilde{u}^{p-1}(\tilde{u}^{p-1}w)_{xx}$

is semi-Fredholm from $\mathcal{V}$ to $\mathcal{V}^{\prime}$. So by Theorem
1.4 $D^{2}F(\tilde{u})\in\mathcal{L}(\mathcal{V};\mathcal{V}^{\prime})$ is
semi-Fredholm.

###### Convergence result

We use Theorem 1.6 and Theorem 1.7 to deduce the convergence result as
follows.

######  Theorem 7.2 (Convergence result).

Let $u(t)\in\mathcal{V}$ a solution of (5.2). If $\\{u(t)\\}_{t\geq 0}$ is
precompact in $\mathcal{V}$, e.g., as in Theorem 7.1, then there exists
$\tilde{u}\in\mathcal{E}:=\\{u\in\mathcal{V}:\nabla F(u)=0\\}$ such that

$\lim_{t\rightarrow\infty}\|u(t)-\tilde{u}\|_{\mathcal{V}}=0$

Moreover, let $\theta$ be any Łojasiewicz exponent of $F$ at $\tilde{u}$. Then
we have

$\|u(t)-\tilde{u}\|_{\mathcal{H}}=\left\\{\begin{array}[]{ll}O(e^{-\delta
t})&\text{if}\;\;\theta=\frac{1}{2},\;\;\;\;\text{for some}\;\;\delta>0\\\
O(t^{-\theta/(1-2\theta)})&\text{if}\;\;0<\theta<\frac{1}{2}\end{array}\right.$

###### Proof.

We use the uniform bound on $u$ (Proposition 7.3 and Corollary 7.1) to check
the angle condition

$\displaystyle\langle\nabla{F(u)},u^{\prime}\rangle$
$\displaystyle=\alpha(t)^{-1}\int
u^{p-1}\Big{[}\epsilon(u^{p})_{xx}+K*u^{p}-c(t)u\Big{]}^{2}dx$
$\displaystyle\geq\sigma\|\nabla
F(u)\|_{\mathcal{H}}\|u^{\prime}\|_{\mathcal{H}},$

and the rate condition

$\|u^{\prime}\|_{\mathcal{H}}\geq\gamma\|\nabla F(u)\|_{\mathcal{H}}.$

Suppose $\tilde{u}\in\omega(u)$ and let
$\bar{F}=\lim_{t\to\infty}F(u(t))=F(\tilde{u})$ and $\theta$ be the
Łojasiewicz exponent at $\tilde{u}$. Since the argument is the same with the
proof of Theorem 1.7, we conclude the proof by showing the following
calculation,

$\displaystyle-\frac{d}{dt}(\bar{F}-F(u(t)))^{\theta}$
$\displaystyle=(\bar{F}-F(u(t)))^{1-\theta}\langle\nabla{F(u)},u^{\prime}\rangle$
$\displaystyle\geq\sigma(\bar{F}-F(u(t)))^{1-\theta}\|\nabla
F(u)\|_{\mathcal{H}}\|u^{\prime}\|_{\mathcal{H}}$ $\displaystyle\geq\sigma
C\|u^{\prime}\|_{\mathcal{H}}.\qed$

###### Instability of constant solutions

Note that $\bar{u}$ is a constant satisfying $L\bar{u}^{p+1}=1$, $\bar{u}$ is
a fixed point of (7.1) as well as (5.2). So as we did in Theorem 6.5, we can
find a perturbation $\phi$ at $\bar{u}$ such that

$F(\bar{u}+\phi)>F(\bar{u}).$

Indeed, at $\bar{u}$

$\displaystyle D^{2}F(\bar{u})(v)=e^{-\frac{2p}{p+1}}\bar{u}^{2p-2}(\epsilon
pv_{xx}+pK*v-\frac{2p}{L}\int_{\mathds{T}_{L}}vdy-v).$ (7.12)

Note that, the constant function $\mathds{1}$ is an eigenvector with negative
eigenvalue,

$D^{2}F(\bar{u})(\mathds{1})=-e^{-\frac{2p}{p+1}}\bar{u}^{2p-2}(p+1)\mathds{1}.$

It is easy to see that the invariant manifold $\mathcal{M}$ attracts the
solutions.

For the perturbations in the tangent manifold at $\bar{u}$, i.e.,
$T_{\bar{u}}\mathcal{M}=\\{\phi\in\mathcal{V}:\int_{\mathds{T}_{L}}\phi
dx=0\\}$,

$\displaystyle
D^{2}F(\bar{u})(\phi)=e^{-\frac{2p}{p+1}}\bar{u}^{2p-2}(\epsilon
p\phi_{xx}+pK*\phi-\phi).$ (7.13)

Let $\phi_{n}(x)=\cos(\lambda_{n}x)$ where $\lambda_{n}=\frac{2n\pi}{L}$ then

$D^{2}F(\bar{u})(\phi_{n})=e^{-\frac{2p}{p+1}}\bar{u}^{2p-2}(-\epsilon
p\lambda_{n}^{2}+p\widehat{K}(\lambda_{n})-1)\phi_{n}.$

So for small enough $\epsilon$, $D^{2}F(\bar{u})$ have positive eigenvalues,
which shows the instability of the constant solution $\bar{u}.$

In conclusion, we have the following convergence result to a nontrivial
equilibrium.

######  Theorem 7.3.

For large enough $L>0$ there exist small enough $\epsilon>0$ and $\delta>0$
such that the solution $u(t)$ of (7.1) with the initial
$u_{0}=\bar{u}+\delta\cos(\frac{2\pi}{L}x)$, with $L^{p+1}$-normalization if
needed, converges to a nontrivial equilibrium.

### Bibliography

  * [1] P.-A. Absil, R. Mahony, and B. Andrews, Convergence of the iterates of descent methods for analytic cost functions, SIAM Journal on Optimization, 16 (2005), pp. 531–547.
  * [2] E. Akin and J. Hofbauer, Recurrence of the unfit, Mathematical Biosciences, 61 (1982), pp. 51–62.
  * [3] S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta metallurgica, 27 (1979), pp. 1085–1095.
  * [4] F. J. A. Artacho, R. Campoy, and P. T. Vuong, The boosted dc algorithm for linearly constrained dc programming, arXiv preprint arXiv:1908.01138, (2019).
  * [5] F. J. A. Artacho, R. M. Fleming, and P. T. Vuong, Accelerating the dc algorithm for smooth functions, Mathematical Programming, 169 (2018), pp. 95–118.
  * [6] H. Attouch and M. Teboulle, Regularized lotka-volterra dynamical system as continuous proximal-like method in optimization, Journal of optimization theory and applications, 121 (2004), pp. 541–570.
  * [7] A. Barakat and P. Bianchi, Convergence rates of a momentum algorithm with bounded adaptive step size for nonconvex optimization, in Asian Conference on Machine Learning, PMLR, 2020, pp. 225–240.
  * [8] T. Bárta, R. Chill, and E. Fašangová, Every ordinary differential equation with a strict lyapunov function is a gradient system, Monatshefte für Mathematik, 166 (2012), pp. 57–72.
  * [9] A. Blanchet and J. Bolte, A family of functional inequalities: Łojasiewicz inequalities and displacement convex functions, Journal of Functional Analysis, 275 (2018), pp. 1650 – 1673.
  * [10] J. Bolte, A. Daniilidis, and A. Lewis, The łojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems, SIAM Journal on Optimization, 17 (2007), pp. 1205–1223.
  * [11] M. Broom, C. Cannings, and G. T. Vickers, On the number of local maxima of a constrained quadratic form, Proceedings: Mathematical and Physical Sciences, 443 (1993), pp. 573–584.
  * [12] J. W. Cahn, On spinodal decomposition, Acta metallurgica, 9 (1961), pp. 795–801.
  * [13] S. J. Chapman, S. D. Howison, and J. R. Ockendon, Macroscopic models for superconductivity, Siam Review, 34 (1992), pp. 529–560.
  * [14] A. Christlieb, J. Jones, K. Promislow, B. Wetton, and M. Willoughby, High accuracy solutions to energy gradient flows from material science models, Journal of Computational Physics, 257 (2014), pp. 193–215.
  * [15] T. H. Colding and W. P. Minicozzi II, Lojasiewicz inequalities and applications, arXiv preprint arXiv:1402.5087, (2014).
  * [16] , Arnold-thom gradient conjecture for the arrival time, arXiv preprint arXiv:1712.05381, (2017).
  * [17] J. Diakonikolas and M. I. Jordan, Generalized momentum-based methods: A hamiltonian perspective, SIAM Journal on Optimization, 31 (2021), pp. 915–944.
  * [18] T. P. Dinh and H. A. Le Thi, Recent advances in dc programming and dca, Transactions on computational intelligence XIII, (2014), pp. 1–37.
  * [19] C. M. Elliott and A. Stuart, The global dynamics of discrete semilinear parabolic equations, SIAM journal on numerical analysis, 30 (1993), pp. 1622–1663.
  * [20] D. J. Eyre, An unconditionally stable one-step scheme for gradient systems, Unpublished article, 6 (1998).
  * [21] G. Friesecke and R. L. Pego, Solitary waves on FPU lattices: I. qualitative properties, renormalization and continuum limit, Nonlinearity, 12 (1999), p. 1601.
  * [22] , Solitary waves on FPU lattices: II. linear implies nonlinear stability, Nonlinearity, 15 (2002), p. 1343.
  * [23] , Solitary waves on Fermi–Pasta–Ulam lattices: III. Howland-type Floquet theory, Nonlinearity, 17 (2003), p. 207.
  * [24] , Solitary waves on Fermi–Pasta–Ulam lattices: IV. proof of stability at low energy, Nonlinearity, 17 (2004), p. 229.
  * [25] G. Friesecke and J. A. Wattis, Existence theorem for solitary waves on lattices, Communications in mathematical physics, 161 (1994), pp. 391–418.
  * [26] K. Glasner and S. Orizaga, Improving the accuracy of convexity splitting methods for gradient flow equations, Journal of Computational Physics, 315 (2016), pp. 52–64.
  * [27] A. Haraux and M. A. Jendoubi, The convergence problem for dissipative autonomous systems: classical methods and recent advances, Springer, 2015.
  * [28] D. Henry, Geometric theory of semilinear parabolic equations, vol. 840, Springer, 2006.
  * [29] M. Herrmann, Unimodal wavetrains and solitons in convex Fermi–Pasta–Ulam chains, Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences, 140 (2010), pp. 753–785.
  * [30] J. Hofbauer and K. Sigmund, Evolutionary game dynamics, Bulletin of the American mathematical society, 40 (2003), pp. 479–519.
  * [31] A. Hoffman and C. E. Wayne, A simple proof of the stability of solitary waves in the Fermi-Pasta-Ulam model near the KdV limit, arXiv preprint arXiv:0811.2406, (2008).
  * [32] K. Hovsepian, P. Anselmo, and S. Mazumdar, Supervised inductive learning with lotka–volterra derived models, Knowledge and information systems, 26 (2011), pp. 195–223.
  * [33] P.-E. Jabin and H. Liu, On a non-local selection–mutation model with a gradient flow structure, Nonlinearity, 30 (2017), p. 4220.
  * [34] R. Jordan, D. Kinderlehrer, and F. Otto, The variational formulation of the Fokker–Planck equation, SIAM journal on mathematical analysis, 29 (1998), pp. 1–17.
  * [35] K. Kurdyka and A. Parusinski, wf-stratification of subanalytic functions and the lojasiewicz inequality, Comptes rendus de l’Académie des sciences. Série 1, Mathématique, 318 (1994), pp. 129–133.
  * [36] T. I. Lakoba and J. Yang, A generalized petviashvili iteration method for scalar and vector hamiltonian equations with arbitrary form of nonlinearity, Journal of Computational Physics, 226 (2007), pp. 1668–1692.
  * [37] S. law Lojasiewicz, Ensembles semi-analytiques, IHES notes, (1965).
  * [38] U. Le and D. E. Pelinovsky, Convergence of Petviashvili’s method near periodic waves in the fractional Korteweg–de Vries equation, SIAM Journal on Mathematical Analysis, 51 (2019), pp. 2850–2883.
  * [39] H. A. Le Thi and T. P. Dinh, Dc programming and dca: thirty years of developments, Mathematical Programming, 169 (2018), pp. 5–68.
  * [40] H. A. Le Thi, T. P. Dinh, et al., Convergence analysis of difference-of-convex algorithm with subanalytic data, Journal of Optimization Theory and Applications, 179 (2018), pp. 103–126.
  * [41] H. A. Le Thi, V. Huynh, and T. Pham Dinh, Convergence analysis of dc algorithm for dc programming with subanalytic data, Ann. Oper. Res. Technical Report, LMI, INSA-Rouen, (2009).
  * [42] G. Li and T. K. Pong, Calculus of the exponent of kurdyka–łojasiewicz inequality and its applications to linear convergence of first-order methods, Foundations of computational mathematics, 18 (2018), pp. 1199–1232.
  * [43] H. Liu, W. Cai, and N. Su, Entropy satisfying schemes for computing selection dynamics in competitive interactions, SIAM Journal on Numerical Analysis, 53 (2015), pp. 1393–1417.
  * [44] S. Lojasiewicz, Une propriété topologique des sous-ensembles analytiques réels, Les équations aux dérivées partielles, 117 (1963), pp. 87–89.
  * [45] S. Lojasiewicz and M. Zurro, On the gradient inequality, Bulletin of the Polish Academy of Sciences-Mathematics, 47 (1999), pp. 143–146.
  * [46] A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Springer Science & Business Media, 2012.
  * [47] T. Mizumachi and R. L. Pego, Asymptotic stability of Toda lattice solitons, Nonlinearity, 21 (2008), p. 2099.
  * [48] P. Ochs, Local convergence of the heavy-ball method and ipiano for non-convex optimization, Journal of Optimization Theory and Applications, 177 (2018), pp. 153–180.
  * [49] P. Ochs, Y. Chen, T. Brox, and T. Pock, ipiano: Inertial proximal algorithm for nonconvex optimization, SIAM Journal on Imaging Sciences, 7 (2014), pp. 1388–1419.
  * [50] J. J. Palis and W. De Melo, Geometric theory of dynamical systems: an introduction, Springer Science & Business Media, 2012.
  * [51] R. L. Pego and T.-S. Van, Existence of solitary waves in one dimensional peridynamics, Journal of Elasticity, (2018), pp. 1–30.
  * [52] D. Pelinovsky and Y. Stepanyants, Convergence of Petviashvili’s iteration method for numerical approximation of stationary solutions of nonlinear wave equations, SIAM J. Numerical Analysis, 42 (2004), pp. 1110–1127.
  * [53] D. E. Pelinovsky and Y. A. Stepanyants, Convergence of petviashvili’s iteration method for numerical approximation of stationary solutions of nonlinear wave equations, SIAM Journal on Numerical Analysis, 42 (2004), pp. 1110–1127.
  * [54] V. I. Petviashvili, Equation of an extraordinary soliton, Fizika plazmy, 2 (1976), pp. 469–472.
  * [55] J. Shin, H. G. Lee, and J.-Y. Lee, Unconditionally stable methods for gradient flow using convex splitting runge–kutta scheme, Journal of Computational Physics, 347 (2017), pp. 367–381.
  * [56] L. Simon, Asymptotics for a class of non-linear evolution equations, with applications to geometric problems, Annals of Mathematics, (1983), pp. 525–571.
  * [57] P. D. Tao and L. T. H. An, Convex analysis approach to dc programming: theory, algorithms and applications, Acta mathematica vietnamica, 22 (1997), pp. 289–355.
  * [58] P. D. Tao et al., Algorithms for solving a class of nonconvex optimization problems. methods of subgradients, in North-Holland Mathematics Studies, vol. 129, Elsevier, 1986, pp. 249–271.
  * [59] S. M. Wise, C. Wang, and J. S. Lowengrub, An energy-stable and convergent finite-difference scheme for the phase field crystal equation, SIAM Journal on Numerical Analysis, 47 (2009), pp. 2269–2288.
  * [60] Z. Wu and M. Li, General inertial proximal gradient method for a class of nonconvex nonsmooth optimization problems, Computational Optimization and Applications, 73 (2019), pp. 129–158.