# Sharp large time behaviour in $N$-dimensional reaction-diffusion equations
of bistable type

Jean-Michel Roquejoffre
Institut de Mathématiques de Toulouse; UMR 5219
Université de Toulouse; CNRS
Université Toulouse III, 118 route de Narbonne, 31062 Toulouse, France
<EMAIL_ADDRESS>

Violaine Roussier-Michon
Institut de Mathématiques de Toulouse; UMR 5219
Université de Toulouse; CNRS
INSA Toulouse, 135 av. Rangueil, 31077 Toulouse, France
<EMAIL_ADDRESS>
###### Abstract

We study the large time behaviour of the reaction-diffsuion equation
$\partial_{t}u=\Delta u+f(u)$ in spatial dimension $N$, when the nonlinear
term is bistable and the initial datum is compactly supported. We prove the
existence of a Lipschitz function $s^{\infty}$ of the unit sphere, such that
$u(t,x)$ converges uniformly in $\mathbb{R}^{N}$, as $t$ goes to infinity, to
$U_{c_{*}}\bigg{(}|x|-c_{*}t+\displaystyle\frac{N-1}{c_{*}}\mathrm{ln}t+s^{\infty}\Big{(}\displaystyle\frac{x}{|x|}\Big{)}\bigg{)}$,
where $U_{c*}$ is the unique 1D travelling profile. This extends earlier
results that identified the locations of the level sets of the solutions with
$o_{t\to+\infty}(t)$ precision, or identified precisely the level sets
locations for almost radial initial data.

##  1 Introduction

###  1.1 Question under study

The paper is devoted to the large time behaviour of the solution of the
reaction-diffusion equation

$\displaystyle\partial_{t}u=\Delta u+f(u),$ $\displaystyle\quad
t>0\,,\,x\in\mathbb{R}^{N}$ (1) $\displaystyle u(0,x)=u_{0}(x),$
$\displaystyle\quad\quad\quad\quad x\in\mathbb{R}^{N}$ (2)

where $f\in{\cal C}^{\infty}([0,1],\mathbb{R})$. We will assume the existence
of $\theta\in(0,1)$ such that

$f<0\ \hbox{on $(0,\theta)$},\ f>0\ \hbox{on $(\theta,1)$},\quad
f^{\prime}(0)<0,\ f^{\prime}(1)<0,\quad\int_{0}^{1}f>0.$

Thus $f$ is said, in reference to the equation $\dot{u}=f(u)$, of the bistable
type. A typical example is

$f(u)=u(u-\theta)(1-u),\ \ \ 0<\theta<\frac{1}{2}.$

We consider compactly supported initial datum $u_{0}$ of the form

$\displaystyle\exists R_{2}>R_{1}>0\,,\,\forall
x\in\mathbb{R}^{N}\,,\quad{\mathbf{1}}_{B_{R_{1}}}(x)\leq
u_{0}(x)\leq{\mathbf{1}}_{B_{R_{2}}}(x),$ (3)

where ${\mathbf{1}}_{A}$ is the indicator of the set $A$ and $B_{R}$ is the
ball of $\mathbb{R}^{N}$ of radius $R$ centered at the origin. Equation (1)
has a unique classical solution $u(t,x)$ in ${\cal
C}^{\infty}([0,+\infty[\times\mathbb{R}^{N},[0,1])$ emanating from $u_{0}$,
see [11] for instance. From Aronson and Weinberger [1], as soon as $R_{1}>0$
is large enough, the solution $u$ spreads at a fixed speed $c_{*}>0$, in the
following sense:

$\min_{|x|\leq ct}u(t,x)\to 1\mbox{ as }t\to+\infty\,,\mbox{ for all }0\leq
c<c_{*}$

and

$\sup_{|x|\geq ct}u(t,x)\to 0\mbox{ as }t\to+\infty\,,\mbox{ for all
}c>c_{*}.$

The goal of this work is to sharpen this result. The goal of this paper is to
prove the following

###### Theorem 1.1

Let $u_{0}$ satisfy assumption (3). There is a Lipschitz function
$s^{\infty}$, defined on the unit sphere of $\mathbb{R}^{N}$, such that the
solution $u$ of (1) emanating from $u_{0}$ satisfies

$\lim\limits_{t\to+\infty}\sup_{x\in\mathbb{R}^{N}}\left|u(t,x)-U_{*}\biggl{(}|x|-c_{*}t+\frac{N-1}{c_{*}}{\mathrm{ln}}t+s^{\infty}\Big{(}\frac{x}{|x|}\Big{)}\biggl{)}\right|=0.$

###  1.2 Relation to existing works

In the case $N=1$, equation (1) with $N=1$ reads

$\partial_{t}u=\partial_{xx}u+f(u),\quad t>0\,,\,x\in\mathbb{R}.$ (4)

It admits one-dimensional travelling fronts $U(x-ct)$ if and only if
$c=c_{*}$, the just mentionned spreading speed. The profile $U$, satisfies

$U^{\prime\prime}+c_{*}\,U^{\prime}+f(U)=0,\quad x\in\mathbb{R},$ (5)

together with the conditions at infinity

$\lim\limits_{x\to-\infty}U(x)=1\quad\mbox{ and
}\quad\lim\limits_{x\to+\infty}U(x)=0.$ (6)

Any solution $U$ to (5)-(6) is a shift of a fixed profile $U_{*}$:
$U(x)=U_{*}(x+s)$ with some fixed $s\in\mathbb{R}$. The large time behaviour
of (4) has a history of important contributions, the most fundamental being
perhaps that of Fife and McLeod [8]. They proved that the solution of (4)
starting from an initial datum $u_{0}(x)$ that is roughly front-like at
infinity, namely

$\limsup_{x\to-\infty}u_{0}(x)>\theta,\quad\liminf_{x\to+\infty}u_{0}(x)<\theta$

gives rise to a solution $u(t,x)$ that converges to a travelling wave
exponentially in time. Precisely, there exists $x_{0}\in\mathbb{R}$ (depending
on $u_{0}$ in a way that is not explicit in general) such that

$\sup_{x\in\mathbb{R}}|u(t,x)-U_{*}(x+c_{*}t+x_{0})|\leq Ce^{-\omega t},$

where $\omega>0$ is essentially the first nonzero eigenvalue of the linear
operator

$-\partial_{xx}-f^{\prime}(U_{*}(.+x_{0})).$

The large time behaviour of the solutions to (1) has not been described at
that level of precision, to the exception of a former paper by the second
author [20] tackling the case of almost spherically symmetric initial data,
and which will be the starting point of our work. This contribution proves the
convergence to travelling waves, shifted by the logarithmic delay
$\displaystyle\frac{N-1}{c_{*}}\mathrm{ln}t$ plus an additional, possibly
angle dependent constant. While the result takes advantage of the nearly
spherically symmetric, it emphasises the fact that the part of the shift that
is constant in time is, in general, angle dependent. We also refer to [21], a
work that also identifies the fact that alsmost spherically symmetric, but
nonsymmetric initial data will remain so for all later times.

In several space dimensions $N\geq 2$, a line of results, in a spirit
different from that of Theorem 1.1, is the convergence in profile of the
solutions. Namely, $u(t,x)$ is followed in the reference frame where it is
bounded away from 0 or 1, and its asymptotic shape is characterised. We
mention a very interesting contribution of Jones [12], stating that the level
sets of the solution of (1), whatever the nonlinearity is, will have
oscillations only of the size $O_{t\to+\infty}(1)$. This is a consequence of
the following fact: if $\lambda$ is a regular value of $u$, the normal to the
$\lambda$-level set of $u$ meets the convex hull of the support of the initial
datum. A simple proof of this fact is given by Berestycki in [2]. This work
has been revisited in [19].

Instead of a bistable nonlinearity, we may consider (1) with a nonlinearity
$f>0$ and concave between 0 and 1 (so-called Fisher-KPP nonlinearity, in
reference to the seminal paper [13]). It is a well-known fact that one-
dimensional waves exist for all speeds $c\geq c_{*}=2\sqrt{f^{\prime}(0)}$. If
$U_{*}$ is a wave with bottom speed, we recently proved in [17], in
collaboration with L. Rossi, that the dynamics of $u$ is

$\lim\limits_{t\to+\infty}\sup_{x\in\mathbb{R}^{N}}\left|u(t,x)-U_{*}\biggl{(}|x|-c_{*}t+\frac{N+2}{c_{*}}{\mathrm{ln}}t+s^{\infty}\Big{(}\frac{x}{|x|}\Big{)}\biggl{)}\right|=0.$
(7)

Thus, in both cases, there is a logarithmic delay. However they are of
different nature. In the bistable case, the delay is purely due to curvature
terms, as will be clear from Section 3, and as had already been elucidated in
[20]. In the Fisher-KPP case, there is an additional shift
$\displaystyle\frac{3}{c_{*}}\mathrm{ln}t$, which is already present in one
space dimension, and that is called the Bramson shift [4], [5]. It comes from
the fact that, as 0 is the most unstable value in the range of $u$ \- that is,
the growth for the linearised equation $\dot{v}=f^{\prime}(u)v$ is maximal
when $u=0$ -, the dynamics of $u(t,x)$ is driven by its tail, which implies a
different behaviour that is very much related to the one-dimensional Dirichlet
heat equation. Bramson’s proof is probabilistic, and a new interpretation of
this result is proposed in [14]. Before the complete proof of [17], the
position of the level sets had been identified with $O_{t\to+\infty}(1)$
precision by Gärtner [9], that is, they expand like
$c_{*}t-\displaystyle\frac{N+2}{2}\mathrm{ln}t$. Estimate (7) is proved on the
basis of the ideas of [17].

As an illustration to our explanation, we mention the recent contribution [6],
which treats the porous medium equation with Fisher-KPP nonlinearity. It
identifies the position of the level sets with $O_{t\to+\infty}(1)$ precision,
that is, they expand like $c_{*}t-\displaystyle\frac{N-1}{c_{*}}\mathrm{ln}t$.
It may look surprising, as the nonlinearity is the Fisher-KPP one. However,
this can be explained by the fact that the porous medium equation is really a
free boundary problem, so that the solution has no tail. This entails a
behaviour that is more closely related to what is observed in the bistable
case.

###  1.3 Strategy of the proof of Theorem 1.1, organisation of the paper

Let us explain how the proof of Theorem 1.1 proceeds. The first step is to
identify the reference frame in which $u(t,x)$ is nontrivial, for this we
apply the existing analysis of the second author [20]. Once this is done, we
write, as in [17], equation (1) in polar coordinates, shifted in the correct
reference frame. This has the inconvenience of cancelling out, at large times,
the angular diffusion, which deprives us of an important source of
compactness. To recover it we estimate the angular derivative, something that
was quite useful in the fisher-KPP case [17]. However, while one could use the
maximum principle in [17] in a relatively easy fashion - the asymptotic
equation was the linear heat equation in the tail of the solution - one cannot
do it here. Indeed, what drives the propagation is the body of the solution,
not its tail. As a result, there is no obvious application of the maximum
principle, and the estimate proceeds by applying a Fife-McLeod type idea to
the angular derivative of $u$, by comparing it to its radial derivative. This
is done in three successive steps detailed in Section 3. Once this is under
control, a stability result, once again in the Fife-Mc Leod type, but
complicated by the presence of angular terms, helps concluding the proof.

The organisation of the paper follows the main steps of this strategy. In
Section 2, we trap the solution between two 1D travelling waves moving like
$c_{*}t-\displaystyle\frac{N-1}{c_{*}}\mathrm{ln}t$, thus characterising the
reference frame in which the solution is nontrivial; we also prepare the
equations. Section 3 is devoted to the main estimate, namely an estimate on
the angular variable of $u$. The proof of Theorem 1.1 is concluded in Section
4. We make some final remarks in Section 5.

##  2 Radial bounds and preparation of the equations

The main result of this section, that we will deduce from Theorem 1 of [20],
is the following.

###### Proposition 2.1

Let $u$ solve (1) with initial datum $u_{0}$ satisfying (3). There are four
real numbers $t_{0}>0$, $C>0$ and $s_{-}<s_{+}$ such that, for all $t\geq
t_{0}$ and $x\in\mathbb{R}^{N}$, we get

$U_{*}(|x|-c_{*}t+\frac{N-1}{c_{*}}\mathrm{ln}t-s_{-})-C\,\frac{{\mathrm{ln}}t}{t}\leq
u(t,x)\leq
U_{*}(|x|-c_{*}t+\frac{N-1}{c_{*}}\mathrm{ln}t-s_{+})+C\,\frac{{\mathrm{ln}}t}{t}.$
(8)

Proof. Let $u_{0}$ satisfy assumption (3) and $u$ be the unique solution to
(1) emanating from $u_{0}$. Define $R_{0}>0$ and $\delta_{0}>0$, depending on
the non-linearity $f$, as in theorem 1 of [20].

We first build a super-solution named $\bar{u}$ as follows. Choose
$\varepsilon\in(0,\displaystyle\frac{\delta_{0}}{\sqrt{R_{2}+1}})$ and
$\bar{R}>R_{0}$ such that

$\forall x\in\mathbb{R}^{N}\,,\quad
u_{0}(x)\leq{\mathbf{1}}_{B_{R_{2}}}(x)\leq
U_{*}(|x|-\bar{R})+\varepsilon{\mathbf{1}}_{B_{R_{2}+1}}(x)$

Let $\bar{u}$ be the solution to (1) emanating from
$U_{*}(|x|-\bar{R})+\varepsilon{\mathbf{1}}_{B_{R_{2}+1}}(x)$. By the maximum
principle, we get that for all $t>0$ and all $x\in\mathbb{R}^{N}$,
$u(t,x)\leq\bar{u}(t,x)$ and we just have to compare $\bar{u}$ with a front.
This is done by theorem 1 in [20]. Indeed, defining

$X=\\{u:\mathbb{R}^{N}\to\mathbb{R}\,|\,\exists\tilde{u}\in
H^{1}(\mathbb{R}^{+})\mbox{ such that }u(x)=\tilde{u}(|x|)\mbox{ for
}x\in\mathbb{R}^{N}\\}$

we get

$\|\bar{u}(0,x)-U_{*}(|x|-\bar{R})\|_{X}\leq\varepsilon\sqrt{R_{2}+1}\leq\delta_{0}$

and therefore, by theorem 1 and the remarks below in [20], there exist
$L\in\mathbb{R}$ and $C>0$ such that for all $t>0$ and $x\in\mathbb{R}^{N}$,

$|\bar{u}(t,x)-U_{*}(|x|-c_{*}t+\frac{N-1}{c_{*}}\ln t+L)|\leq C\,\frac{\ln
t}{t}$

Defining $s^{+}=-L$ leads to the right hand side of (8) for any $t\geq 0$.

Dealing with a sub-solution is not so simple because a small perturbation as
$max(U_{*}(|x|+\underline{R})+\varepsilon,0)$ may not developp a front. We
therefore use Aronson and Weinberger’s result [1] to wait until the solution
$u$ has propagated enough. Fix $\varepsilon>0$ and $\underline{R}>R_{0}$ such
that $U_{*}(-\underline{R})\leq 1-\varepsilon$. Then, for all
$x\in\mathbb{R}^{N}$, we have

$U_{*}(|x|-\underline{R})\leq U_{*}(-\underline{R})\leq 1-\varepsilon.$

On the other hand, by Aronson and Weinberger’s result [1] with $c=c_{*}/2$,
there exists $t_{\varepsilon}>0$ such that for any $t\geq t_{\varepsilon}$ and
$|x|\leq ct$, $1-\varepsilon\leq u(t,x)\leq 1$. Choose $t_{0}\geq
t_{\varepsilon}$ such that

$\|U_{*}(\cdot-\underline{R})\|_{H^{1}(ct_{0},\infty)}\leq\delta_{0}$

and define
$\underline{u}(t_{0},x)=U_{*}(|x|-\underline{R}){\mathbf{1}}_{B_{ct_{0}}}(x)$.
Then, $\underline{u}(t_{0},x)\leq u(t_{0},x)$ for any $x\in\mathbb{R}^{N}$.

Let $\underline{u}$ be the solution to (1) emanating from
$\underline{u}(t_{0},x)$ at $t=t_{0}$. The maximum principle ensures that
$\underline{u}(t,x)\leq u(t,x)$ for any $t\geq t_{0}$ and $x\in\mathbb{R}^{N}$
and we just have to compare $\underline{u}$ with a front. Since
$\|\underline{u}(t_{0},x)-U_{*}(|x|-\underline{R})\|_{X}=\|U_{*}(\cdot-\underline{R})\|_{H^{1}(ct_{0},\infty)}\leq\delta_{0}$,
theorem 1 in [20] applies and there exists $L\in\mathbb{R}$ and $C>0$ such
that forall $t\geq t_{0}$ and $x\in\mathbb{R}^{N}$,

$|\underline{u}(t,x)-U_{*}(|x|-c_{*}t+\frac{N-1}{c_{*}}\ln t+L)|\leq
C\,\frac{\ln t}{t}$

Defining $s_{-}=-L$ leads to the left hand side of (8) for any $t\geq t_{0}$.
$\Box$

This proposition makes it clear that the transition zone, where $u$ is neither
close to $1$ nor $0$, is located around
$R(t)=c_{*}t-\displaystyle\frac{N-1}{c_{*}}\ln t$. We therefore choose to
handle the initial equation (1) in a frame, moving at speed $\dot{R}(t)$ in
any radial direction. Let us explain those transformations on the equations.

From now on, we take $t=1$ as initial time and (2) is replaced by
$u(1,x)=u_{0}(x)$. This will be handier in view of the following
transformations and, since equation (1) is invariant by translation in time,
there is no loss of generality. We first use the polar coordinates

$x\mapsto(r=|x|>0,\Theta=\frac{x}{|x|}\in\mathbb{S}^{N-1})$

then (1) becomes

$\partial_{t}u=\partial_{rr}u+\frac{N-1}{r}\partial_{r}u+\frac{\Delta_{\Theta}u}{r^{2}}+f(u),\quad{t>1,\
r>0,\ \Theta\in\mathbb{S}^{N-1}}.$

Here, $\Delta_{\Theta}$ is the Laplace-Beltrami operator on the unit sphere of
$\mathbb{R}^{N}$. Its precise expression will not be needed in the sequel. The
initial condition reads $u(1,r,\Theta)=u_{0}(r,\Theta)$.

Since we mentionned that the transition zone is located around
$R(t)=c_{*}t-k\ln t$ with $k=(N-1)/c_{*}$, we choose the change of variables
$r^{\prime}=r-R(t)$ and $u(t,r,\Theta)=u_{1}(t,r-R(t),\Theta)$. We drop the
primes and indexes, and (1) becomes

$\partial_{t}u=\partial_{rr}u+c_{*}\partial_{r}u+\biggl{(}\frac{N-1}{r+c_{*}t-k{\mathrm{ln}}t}-\frac{k}{t}\biggl{)}\partial_{r}u+\frac{\Delta_{\Theta}u}{(r+c_{*}t-k{\mathrm{ln}}t)^{2}}+f(u).$
(9)

The equation is valid for $t>1$, $r>-2t+k{\mathrm{ln}}t$, and
$\Theta\in\mathbb{S}^{N-1}$ and the initial condition becomes
$u(1,r,\Theta)=u_{0}(r+c_{*},\Theta)$.

To unravel the mechanisms at work, our first guess is that the term in
$\Delta_{\Theta}v$ will not matter too much, because it decays like $t^{-2}$
(an integrable power of $t$), except in the zone $r\sim-c_{*}t$, where we know
(for instance [1]) that $u(t,r,\Theta)$ goes to 1 as $t\to+\infty$. This
confirms the information given by proposition 2.1 that the dynamics is like
that of the one-dimensional equation. On the other hand, in the advection
term, we have

$\displaystyle\frac{N-1}{r+c_{*}t-k{\mathrm{ln}}t}\sim_{t\to+\infty}\frac{N-1}{c_{*}t},$

except for extremely large $r$. This is nonintegrable in $t$, but we balance
it with the $\displaystyle\frac{k}{t}$ term since we chose

$k=\frac{N-1}{c_{*}}.$ (10)

This heuristics confirms that $R(t)=c_{*}t-\displaystyle\frac{N-1}{c_{*}}\ln
t$ is the right moving frame to observe the large time dynamics of (1). In the
sequel, we will keep the notation $k$, keeping in mind that $k$ is given by
formula (10). Also, from now on, we will only consider solutions of (9).

##  3 Boundedness of the angular derivative

This section is devoted to the following estimate

###### Theorem 3.1

Let $u$ solve (9) with initial datum $u_{0}(\cdot+c_{*},\cdot)$. Then, there
is $C>0$ such that

$\forall t\geq
1\,,\quad\|\nabla_{\Theta}u(t,.,.)\|_{L^{\infty}((-c_{*}t/2,+\infty)\times\mathbb{S}^{N-1})}\leq
C.$ (11)

The proof of this theorem 3.1 is based on a bootstrap argument. We will first
prove that the quantity on the left handside is an $o(t)$, which will allow us
to prove that it is an $O(t^{\varepsilon})$ for all $\varepsilon>0$, which
will in turn leads us to $O(1)$. The main idea is to adapt the construction by
Fife and McLeod [8] of sub and super-solutions, but at the level of the linear
equation. The main ingredient is that $\partial_{r}u$ becomes bounded away
from 0 on every compact set, so that it may serve as a comparison function.
And so, the main step (namely section 3.2) will consist in comparing
$|\nabla_{\Theta}u|$ to a suitable multiple of $\partial_{r}u$, as it almost
satisfies the same equation. The main body of the work will consist in
quantifying what this innocent ”almost” means. This idea of using the
longitudinal derivative of the solution as a comparison tool (as oppose to
that of the wave, which has a long history dating back to Fife-McLeod) was
first used in [16], to prove the convergence to travelling waves in
cylindrical geometry.

Proof of theorem 3.1. So, let $u$ solve equation (9) with datum
$u_{0}(\cdot+c_{*},\cdot)$.

###  3.1 The $o(t)$ estimate

Let us perform the revert change of variables explained in the previous
section to come back to $u$ solution to equation (1). Pick any direction
$\Theta$ on the unit sphere. We may, even if it means rotating, assume that
$\Theta=0$, so that we are looking in the direction $Ox_{1}$. Let
$x^{\prime}=(x_{2},...,x_{N})$ be the coordinates orthogonal to the direction
$Ox_{1}$. Consider the sector

$\Sigma_{t}=\\{x\in\mathbb{R}^{N}\,|\,x_{1}>0,\
\frac{|x^{\prime}|}{x_{1}}\leq\frac{1}{t^{3/4}}\\},$

Notice that for $x\in\Sigma_{t}$, when $x_{1}\sim t$, we have
$|x^{\prime}|\leq t^{1/4}$. Write (1) in $\Sigma_{t}$, in the reference frame
moving like $c_{*}t-k{\mathrm{ln}}t$ in the direction $Ox_{1}$, it reads
$X_{1}=x_{1}-c_{*}t+k\ln t$,
$u(t,x)=u_{1}(t,X_{1},x^{\prime})=u_{1}(t,x_{1}-c_{*}t+k\ln t,x^{\prime})$ so
that dropping indexes,

$\partial_{t}u=\Delta u+\left(c_{*}-\frac{k}{t}\right)\partial_{1}u+f(u).$

We also have, because $r=|(X_{1}+c_{*}t-k{\mathrm{ln}}t,x^{\prime})|$ in
$\Sigma_{t}-(c_{*}-k{\mathrm{ln}}t)e_{1}$:

$r-c_{*}t+k{\mathrm{ln}}t=X_{1}+o_{t\to+\infty}(1),\ \hbox{uniformly in
$\Sigma_{t}-(c_{*}-k{\mathrm{ln}}t)e_{1}$}$

Proposition 2.1 and parabolic regularity implies that the trajectories
$(u(T+t,X_{1},x^{\prime}))_{T>0}$ are relatively compact in
$C^{2}([-\tau,\tau]\times\mathbb{R}\times[-M,M]^{N-1})$ for all $\tau>0$ and
$M>0$. If $u_{\infty}(t,X_{1},x^{\prime})$ is a limiting trajectory we have
for $(t,X_{1},x^{\prime})\in\mathbb{R}^{N+1}$

$\displaystyle\partial_{t}u_{\infty}=\Delta
u_{\infty}+\left(c_{*}-\displaystyle\frac{k}{t}\right)\partial_{1}u_{\infty}+f(u_{\infty})$
(12) $\displaystyle U_{*}(X_{1}-s_{-})\leq u_{\infty}(t,X_{1},x^{\prime})\leq
U_{*}(X_{1}-s_{+}).$ (13)

From Theorem 1.1 of [18] there is $s_{\infty}\in\mathbb{R}$ such that for
$(t,X_{1},x^{\prime})\in\mathbb{R}^{N+1}$

$u_{\infty}(t,X_{1},x^{\prime})=U_{*}(X_{1}-s_{\infty}).$

Parabolic regularity implies

$\lim_{t\to+\infty}|\nabla_{x^{\prime}}u(t,X_{1},x^{\prime})|=0,\ \hbox{
uniformly in }(t,X_{1})\in\mathbb{R}_{+}\times\mathbb{R}\mbox{ and
}x^{\prime}\mbox{ on every compact set.}$

Let us translate this result in the variables of equation (9). Because

$\nabla_{\Theta}u(t,r,0)=(r+c_{*}t-k{\mathrm{ln}}t)\nabla_{x^{\prime}}u(t,X_{1},0),$
(14)

we have the expected estimate on $\nabla_{\Theta}u$ for $\Theta=0$. Note that
our argument is uniform in the direction considered, so that we have in the
end, for $u$ solution to (9)

$\lim_{t\to+\infty}\frac{\|\nabla_{\Theta}u(t,.,.)\|_{L^{\infty}((-c_{*}t/2,+\infty)\times\mathbb{S}^{N-1})}}{(r+c_{*}t-k{\mathrm{ln}}t)}=0.$

Parabolic regularity again implies the following corollary.

###### Corollary 3.2

We have

$\lim_{t\to+\infty}\frac{\|\Delta_{\Theta}u(t,.,.)\|_{L^{\infty}((-c_{*}t/2,+\infty)\times\mathbb{S}^{N-1})}}{(r+c_{*}t-k{\mathrm{ln}}t)^{2}}=0.$

We also extract from the preceding argument an additional corollary.

###### Corollary 3.3

For every $M>0$, there is $T_{M}>0$ and $\delta_{M}>0$, the function
$M\mapsto\delta_{M}$ decreasing, such that

$-\partial_{r}u(t,r,\Theta)\geq\delta_{M}\ \hbox{for $t\geq T_{M}$, $-M\leq
r\leq M$, $\Theta\in{\mathbb{S}}^{N-1}$}.$

###  3.2 The $O(t^{\varepsilon})$ estimate

For $u$ solution to (9), denote

$V(t,r,\Theta)=-\partial_{r}u(t,r,\Theta).$

The equation for $V$ is

$\biggl{(}\partial_{t}+L(t)-f^{\prime}(u)\biggl{)}V=-\frac{N-1}{(r+c_{*}t-k{\mathrm{ln}}t)^{2}}V+\frac{2\Delta_{\Theta}u}{(r+c_{*}t-k{\mathrm{ln}}t)^{3}},$
(15)

the expression of $L(t)$ being

$L(t)=-\partial_{rr}-c_{*}\partial_{r}-\biggl{(}\frac{N-1}{r+c_{*}t-k{\mathrm{ln}}t}-\frac{k}{t}\biggl{)}\partial_{r}-\frac{\Delta_{\Theta}}{(r+c_{*}t-k{\mathrm{ln}}t)^{2}}.$
(16)

If $\Theta=(\theta_{1},...,\theta_{N-1})$, we set

$u_{i}=\partial_{\theta_{i}}u,$ (17)

we have

$\biggl{(}\partial_{t}+L(t)-f^{\prime}(u)\biggl{)}u_{i}=0.$ (18)

A super-solution for (18) is looked for under the form

$\overline{v}(t,r,\Theta)=\xi(t)V(t,r,\Theta)+q(t),$ (19)

that is, almost as in Fife-McLeod [8], up to the fact that we work at the
level of the linearised equation. In the sequel, we will always study
$\overline{v}$ in the range $r\geq-c_{*}t/2$, the range
$[-c_{*}t+k{\mathrm{ln}}t,-c_{*}t/2)$ being taken care of by Proposition 2.1.
Also, it is enough to construct $\overline{v}$ for $t$ large enough.

We shall now introduce new notations to explain how we will choose the
functions $\xi(t)$ and $q(t)$. Denote $F=\|f^{\prime}\|_{\infty}>0$ and

$\varepsilon(t)=\big{\|}\frac{N-1}{r+c_{*}t-k{\mathrm{ln}}t}V-\frac{2\Delta_{\Theta}u}{(r+c_{*}t-k{\mathrm{ln}}t)^{2}}\big{\|}_{\infty}\,.$

Then, $\varepsilon(t)$ is a nonnegative function tending to $0$ at infinity,
this last estimate comes from Corollary 3.2. Pick $M>0$ and $\delta>0$ such
that

$f^{\prime}(u(t,r,\Theta))\leq-\delta\ \hbox{if $|r|\geq M.$}$

By corollary 3.3, there exists $T_{M}>0$ and $\delta_{M}>0$ such that
$V\geq\delta_{M}$ for $t\geq T_{M}$, $|r|\leq M$ and
$\Theta\in{\mathbb{S}}^{N-1}$. Moreover, if $V^{-}$ stands for the negative
part of $V$, we also get from corollary 3.3 that

$\lim_{t\to+\infty}\lim_{N\to\infty}\sup_{r\in(-c_{*}t/2,+\infty)\backslash(-N,N)}V^{-}(t,r,\Theta)=0.$

Then, for any $\eta>0$, there exist $T>0$ and $N>0$ such that
$V(t,r,\Theta)\geq 0$ for $r\in[-N,N]$ and $|V^{-}(t,r,\Theta)|\leq\eta$ for
$r\in(-c_{*}t/2,+\infty)\backslash(-N,N)$. Choose now $\eta>0$ small enough so
that $0<\eta<\min(\delta_{M},\displaystyle\frac{\delta\delta_{M}}{2F})$ and
$N>M$.

Now that all those preliminaries are given, let define $\xi$ and $q$ as the
unique solutions to the ODE system for $t>T$

$\left\\{\begin{array}[]{rll}\dot{q}+\displaystyle\displaystyle\frac{\delta}{4}q=&\displaystyle\frac{\varepsilon(t)\xi}{c_{*}t/2-k{\mathrm{ln}}t}\,,\\\
\dot{\xi}=&\displaystyle\frac{\delta+F}{\delta_{M}+\eta}q\,,\end{array}\right.$
(20)

the initial data at $(q(T),\xi(T))$ being nonnegative and sufficiently large
so that $\overline{v}(T,r,\Theta)\geq|u_{i}(T,r,\Theta)|$. Then, for any
$t\geq T$, $q(t)\geq 0$ and $\xi(t)\geq 0$ and we shall prove that
$\overline{v}$ defined by (19) with $\xi$ and $q$ verifying (20) is a
supersolution to (18). We have

$\begin{array}[]{rll}&\biggl{(}\partial_{t}+L(t)-f^{\prime}(u)\biggl{)}\overline{v}\\\
=&\dot{\xi}(t)V+\xi(t)\biggl{(}\partial_{t}V+L(t)V-f^{\prime}(u)V\biggl{)}+\dot{q}-f^{\prime}(u)q\\\
\geq&\dot{\xi}(t)V+\dot{q}-f^{\prime}(u)q-\displaystyle\frac{\varepsilon(t)\xi}{r+c_{*}t-k{\mathrm{ln}}t},\end{array}$

Consider first the set where $|r|\geq M$. Since $\dot{\xi}\geq 0$, $q\geq 0$,
$|V^{-}|\leq\eta$ and
$\eta<\min(\delta_{M},\displaystyle\frac{\delta\delta_{M}}{2F})$,

$\begin{array}[]{rll}\biggl{(}\partial_{t}+L(t)-f^{\prime}(u)\biggl{)}\overline{v}\geq&-\dot{\xi}\eta+\dot{q}+\delta
q-\displaystyle\frac{\varepsilon(t)\xi}{c_{*}t/2-k{\mathrm{ln}}t},\\\
=&-\eta\displaystyle\frac{\delta+F}{\delta_{M}+\eta}q+\dot{q}+\delta
q-\displaystyle\frac{\varepsilon(t)\xi}{c_{*}t/2-k{\mathrm{ln}}t}\\\
\geq&\dot{q}+\displaystyle\frac{\delta}{4}q-\displaystyle\frac{\varepsilon(t)\xi}{c_{*}t/2-k{\mathrm{ln}}t}=0,\end{array}$

In the range $|r|\leq M$, the super-solution condition is true since

$\begin{array}[]{rll}\biggl{(}\partial_{t}+L(t)-f^{\prime}(u)\biggl{)}\overline{v}\geq&\dot{\xi}\delta_{M}+\dot{q}-Fq-\displaystyle\frac{\varepsilon(t)\xi}{c_{*}t/2-k{\mathrm{ln}}t},\\\
=&\delta_{M}\displaystyle\frac{\delta+F}{\delta_{M}+\eta}q+\dot{q}-Fq-\displaystyle\frac{\varepsilon(t)\xi}{c_{*}t/2-k{\mathrm{ln}}t}\\\
\geq&\dot{q}+\displaystyle\frac{\delta}{4}q-\displaystyle\frac{\varepsilon(t)\xi}{c_{*}t/2-k{\mathrm{ln}}t}=0,\end{array}$

Finally, for any $r\geq-c_{*}t/2$, $\overline{v}$ is a supersolution to (18).

Set $\varepsilon>0$. In order to prove $|\nabla_{\Theta}u|$ is a
$O(t^{\varepsilon})$, it suffices to study $(q(t),\xi(t))$ as time goes to
infinity. The equation being linear, it is enough to study it with
$(q(T),\xi(T))=(1,1)$. The first equation in (20) gives

$\displaystyle q(t)\leq$ $\displaystyle
e^{-\delta(t-T)/4}+\int_{T}^{t}e^{-\delta(t-s)/4}\displaystyle\frac{\varepsilon(s)\xi(s)}{c_{*}s/2-k\ln
s}ds\,,$ $\displaystyle\leq$ $\displaystyle
e^{-\delta(t-T)/4}+C\varepsilon\xi(t)\int_{T}^{t}\displaystyle\frac{e^{-\delta(t-s)/4}}{1+s}ds\leq
C\varepsilon\displaystyle\frac{\xi(t)}{1+t}$

with $C$ a universal constant. Indeed, $\dot{\xi}\geq 0$ and $T$ can be chosen
large enough so that for any $s\geq T$, $0\leq\varepsilon(s)\leq\varepsilon$.
We also have estimated $c_{*}t-k{\mathrm{ln}}t$ by a suitably small multiple
of $1+t$. Plugging this result in the second equation of (20), we get for any
$t\geq T$

$\dot{\xi}(t)\leq C\varepsilon\frac{\xi(t)}{1+t},$

with a universal constant $C$. Therefore, $\xi(t)\leq(1+t)^{C\varepsilon}$,
which is precisely the desired estimate. The same will hold for $q$.

###  3.3 Conclusion

We start from

$|\nabla_{\Theta}u(t,r,\Theta)|\leq Ct^{1/20}.$

Set $u_{ij}=\partial_{\theta_{i}\theta_{j}}u$, we have

$\biggl{(}\partial_{t}+L(t)-f^{\prime}(u)\biggl{)}u_{ij}=f^{\prime\prime}(u)u_{i}u_{j}=O(t^{1/10}).$

Parabolic regularity implies (this involves a rescaling of $\Theta$ by $t$,
then scaling back):

$|u_{ij}(t,r,\Theta)|\leq Ct^{21/20}.$

This allows in turn a more precise estimate in the equation for
$V=-\partial_{r}u$, because we have now

$\biggl{(}\partial_{t}+L(t)-f^{\prime}(u)\biggl{)}V=O(t^{-39/20}),$

in the range $r\in(-c_{*}t/2,+\infty)$. Clearly, the right handside is an
integrable power of $t$, we may therefore redo the previous step, replacing
$\displaystyle\frac{\varepsilon(t)\xi(t)}{1+t}$
by$\displaystyle\frac{\xi(t)}{(1+t)^{39/20}}$. This leads to the desired
estimate on $\overline{v}$ and then on $\nabla_{\Theta}u$, ending the proof of
theorem 3.1. $\Box$.

##  4 Convergence

Theorem 1.1 will result from the following stability result, once again close
in spirit to Fife and McLeod [8].

###### Theorem 4.1

Let $u$ be a solution to (9) and $s$ a Lipschitz function of
${\mathbb{S}}^{N-1}$. For every $\varepsilon>0$, there is $T_{\varepsilon}>0$
and $\eta_{\varepsilon}>0$ (depending possibly on $\|s\|_{\infty}$ and
$\|\nabla s\|_{\infty}$) such that, if we have

$|u(T_{\varepsilon},r,\Theta)-U_{*}(r+s(\Theta))|\leq\eta_{\varepsilon},\quad
r>-c_{*}T_{\varepsilon}+{\mathrm{ln}}T_{\varepsilon}\,,\,\Theta\in{\mathbb{S}}^{N-1},$
(21)

then there is $T_{\varepsilon}^{\prime}\geq T_{\varepsilon}$ such that, for
all $t\geq T_{\varepsilon}^{\prime}$ we have:

$|u(t,r,\Theta)-U_{*}(r+s(\Theta))|\leq\varepsilon,\quad
r>-c_{*}T_{\varepsilon}^{\prime}+{\mathrm{ln}}T_{\varepsilon}^{\prime}\,,\,\Theta\in{\mathbb{S}}^{N-1}.$

Let us postpone the proof of theorem 4.1 to prove theorem 1.1.

Proof of Theorem 1.1: Let $u$ solve (1) with initial datum $u_{0}$ satisfying
(3). Perform transformations listed in section 2 to deal with polar
coordinates in the radial moving frame. We still denote $u$ the solution of
(9). Define, for all $\tau>0$,

$\Omega_{\tau}=\\{t\in[-\tau,\tau],r\in[-c_{*}t+k{\mathrm{ln}}t,+\infty),\Theta\in{\mathbb{S}}^{N-1}\\}$

Then, by parabolic regularization and Ascoli’s theorem, the family
$(u(T+.,.,.))_{T>0}$ is relatively compact in $C(\Omega_{\tau})$ for every
$\tau>0$. Therefore, there is a sequence $(t_{n})_{n}$ going to infinity such
that $(u(t_{n}+.,.,.))_{n}$ converges, uniformly in every $\Omega_{\tau}$, to
a uniformly continuous function $u_{\infty}(t,r,\Theta)$ satisfying for
$t\in\mathbb{R}$

$\begin{array}[]{rll}\partial_{t}u_{\infty}=\partial_{rr}u_{\infty}+c_{*}\partial_{r}u+f(u_{\infty})&\hbox{in
${\mathcal{D}}^{\prime}(\mathbb{R}^{2}\times{\mathbb{S}}^{N-1})$}\\\
U_{*}(r-s_{-})\leq u_{\infty}(t,r,\Theta)\leq U_{*}(r-s_{+}).&\end{array}$

thanks to the radial barriers obtained in section 2. Moreover,
$u_{\infty}(.,.,\Theta)$ is $C^{1}$ in $t$ and $C^{2}$ in $r$ due to parabolic
regularity, for every $\Theta\in{\mathbb{S}}^{N-1}$. From [8], for every
$\Theta\in{\mathbb{S}}^{N-1}$, there is $s(\Theta)\in\mathbb{R}$ such that

$u_{\infty}(t,r,\Theta)=U_{*}(r+s(\Theta)).$

From Theorem 3.1, the function $s$ is Lipschitz. Moreover, from Theorem 4.1,
the whole family $(u(t,.,.))_{t>0}$ converges uniformly on
$[0,+\infty)\times\mathbb{S}^{N-1}$ to the function $(r,\Theta)\mapsto
U_{*}(r+s(\Theta))$. Reverting to the original variables proves theorem 1.1.
$\Box$

As said above, Theorem 4.1 is a Fife and Mc Leod’s type result, and so will be
obtained from the construction of sub and super solutions very much inspired
from [8]. However, while those in [8] were explicit, the ones constructed here
solve a nonlinear differential system, and so must be studied with a little
care. This is the object of the following intermediate lemma.

For any $\varepsilon>0$, $C>0$ and $T_{0}>0$ such that
$c_{*}T_{0}-k{\mathrm{ln}}T_{0}\geq 100$, we define

$g_{\varepsilon}(t,\xi)=\frac{C}{\varepsilon}\big{|}\frac{N-1}{c_{*}t-k{\mathrm{ln}}t+\xi}-\frac{k}{t}\big{|}\,,\,t>T_{0}\,,\,\xi\in\mathbb{R}$

###### Lemma 4.2

Let $\delta$, $\gamma$ and $\eta$ be strictly positive constants. Consider the
differential system

$\left\\{\begin{array}[]{rlll}\dot{q}+\delta q=&g_{\varepsilon}(t,\xi)&t>T\\\
\gamma\dot{\xi}=&Cq+g_{\varepsilon}(t,\xi)&t>T\\\ q(T)=&\eta,\
\xi(T)=0.&\end{array}\right.$ (22)

where $\varepsilon$, $C$ and $g_{\varepsilon}$ are defined above. There is
$T\geq T_{0}$ and $K>0$ depending on all the constants involved in (22),
except $\eta$, such that (22) has a solution defined on $[T,+\infty)$ which
satisfies for any $t\geq T$,

$0\leq q(t)\leq K(\eta+\frac{1}{\varepsilon
T^{1/2}})e^{-\delta/2(t-T)}+\frac{K}{t^{3/2}},\quad\xi(t)\leq
K(\eta+\frac{1}{\varepsilon T^{1/2}}).$ (23)

Proof. In what follows, $K$ denotes a generic positive constant that may
differ from lines to lines. We first derive a logarithmic bound on $\xi$, then
the desired bound. By definition, there exists $K>0$ such that for any $t>T$,

$0\leq g_{\varepsilon}(t,\xi(t))\leq\frac{K}{\varepsilon(t+\xi(t))},$

so that we have

$q(t)\leq\eta
e^{-\delta(t-T)}+\frac{K}{\varepsilon}\int_{T}^{t}\frac{e^{-\delta(t-s)}}{s+\xi(s)}ds.$

Let $T^{*}\geq T$ be the largest $\tau\geq T$ such that

$\hbox{for all $t\in[T,\tau]$,}\ \xi(t)\leq({\mathrm{ln}}t)^{2}.$ (24)

So, for any $s\in[T,T^{*}]$, we may write

$\frac{1}{\varepsilon(s+\xi(s))}\leq\frac{K}{\varepsilon s},$

so that for any $t\in[T,T^{*}]$, cutting the integral at
$s=\displaystyle\frac{t}{2}$, we get

$q(t)\leq\frac{K}{\varepsilon t}+(\frac{K}{\varepsilon
T}+\eta)e^{-\frac{\delta}{2}(t-T)}.$

This implies in turn

$\dot{\xi}(t)\leq\frac{K}{\varepsilon t},$

hence $\xi(t)\leq\displaystyle\frac{K}{\varepsilon}{\mathrm{ln}}t$, a
contradiction with the definition of $T^{*}$, as soon as $T$ is large enough,
say, of the order $\varepsilon^{-2}$. So for any $t\geq T$,
$\xi(t)\leq({\mathrm{ln}}t)^{2}$. But then, we have a more precise estimate of
$g_{\varepsilon}(t,\xi(t))$. Using the actual value of $k$ we have

$g_{\varepsilon}(t,\xi(t))\leq\frac{C(N-1)}{\varepsilon
c_{*}t}\biggl{(}\frac{1}{1-\ln
t/t+({\mathrm{ln}}t)^{2}/(c_{*}t)}-1\biggl{)}\leq\frac{K{\mathrm{ln}}t}{\varepsilon
t^{2}}\leq\frac{K}{\varepsilon t^{3/2}},$

thus an integrable power of $t$. Plugging this new estimate into (22) yields
(23), hence the lemma. $\Box$

Proof of Theorem 4.1. Let $u$ be a solution to (9) and $s$ as in the statment
of theorem 4.1. Let $\varepsilon>0$ and choose
$T_{\varepsilon}=O(\frac{1}{\varepsilon^{2}})$ and
$\eta_{\varepsilon}=O(\varepsilon^{2})$. We first regularise $s$: set

$s^{\pm}_{\varepsilon}(\Theta)=(\rho_{\varepsilon}*s)(\Theta)\pm
C\varepsilon,$

where $\rho_{\varepsilon}$ is an $\varepsilon$-approximation of the identity
and $C>0$ large enough (say, a multiple of $\|\nabla s\|_{\infty}$). Then, by
assumption (21), we have

$U_{*}(r+s^{-}_{\varepsilon}(\Theta))-C\varepsilon\leq
u(T_{\varepsilon},r,\Theta)\leq
U_{*}(r+s^{+}_{\varepsilon}(\Theta))+C\varepsilon.$

Also notice that

$\|\nabla_{\Theta}s^{+}_{\varepsilon}\|_{\infty}\leq\|\nabla_{\Theta}s\|_{\infty},\
\|\Delta_{\Theta}s^{+}_{\varepsilon}\|_{\infty}\leq\frac{\|\nabla_{\Theta}s\|_{\infty}}{\varepsilon}.$

Write equation (9) for $u$ as

$N\\!L[u]=0.$

We are going to construct sub and super-solutions to (9), starting from
$t=T_{\varepsilon}$, at small distance of $u(t,r,\Theta)$. We display details
for the super-solution’s case, as it goes in the same way for sub-solutions.
We try the form, in the spirit of Fife and Mc Leod [8],

$\overline{u}(t,r,\Theta)=U_{*}(r+s^{+}_{\varepsilon}(\Theta)-\xi^{+}(t))+q^{+}(t).$
(25)

where $\xi^{+}$ and $q^{+}$ are chosen in the following way.

We first define $\mu_{0}\in(0,\min(\theta,1-\theta)))$ and $\delta>0$ such
that

$f^{\prime}(u)\leq-\delta\ \hbox{on }[0,\mu_{0}]\cup[1-\mu_{0},1].$

Choose $M>0$ such that $U_{*}(\rho)\in[0,\mu_{0}]\cup[1-\mu_{0},1]$ if
$\rho\notin[-M,M]$ and $\delta_{M}>0$ such that
$U^{\prime}_{*}(\rho)\leq-\delta_{M}$ if $\rho\in[-M,M]$.

Now define $\xi^{+}$ and $q^{+}$ as the unique solution to (22) with
$\gamma=\delta_{M}$ and $\eta=C\varepsilon$. Then, choosing $T_{\varepsilon}$
possibly larger, we know by lemma 4.2 that there exists $K>0$ such that for
any $t\geq T_{\varepsilon}$,

$0\leq q(t)\leq K(C\varepsilon+\frac{1}{\varepsilon
T^{3/2}})e^{-\delta/2(t-T)}+\frac{K}{t^{3/2}},\quad\xi(t)\leq
K(C\varepsilon+\frac{1}{\varepsilon T^{1/2}}).$

Adjusting $\varepsilon>0$ such that
$2K(C\varepsilon+\frac{1}{\sqrt{T_{\varepsilon}}})<\mu_{0}$, it implies that

$0\leq q^{+}(t)\leq\mu_{0}\mbox{ and }\dot{\xi}^{+}(t)\geq 0$

Once these preliminaries are done with, we may compute $NL[\bar{u}]$. Set
$\rho:=r+s^{+}_{\varepsilon}(\Theta)-\xi^{+}(t)$. Then,

$\begin{array}[]{rll}N\\!L[\overline{u}]=&-\dot{\xi}^{+}(t)U_{*}^{\prime}(\rho)+\dot{q}^{+}(t)-f(U_{*}(\rho)+q^{+})+f(U_{*}(\rho))\\\
&-\biggl{(}\displaystyle\frac{N-1}{\rho+c_{*}t-k{\mathrm{ln}}t+\xi^{+}(t)-s^{+}_{\varepsilon}(\Theta)}-\displaystyle\frac{k}{t}\biggl{)}U_{*}^{\prime}(\rho)-\displaystyle\frac{U_{*}^{\prime}(\rho)\Delta_{\Theta}s^{+}_{\varepsilon}+U_{*}^{\prime\prime}(\rho)|\nabla_{\Theta}s^{+}_{\varepsilon}|^{2}}{(\rho+c_{*}t-k{\mathrm{ln}}t+\xi^{+}(t)-s^{+}_{\varepsilon}(\Theta))^{2}}.\end{array}$

Consider first the range $|\rho|\geq M$. Then, plugging $\overline{u}$ in the
expression of $N\\!L$ reveals that a sufficient condition for
$N\\!L[\overline{u}]\geq 0$ is

$\dot{q}^{+}+\delta
q^{+}\geq\frac{\|\nabla_{\Theta}s\|_{\infty}}{\varepsilon\biggl{(}\rho+c_{*}t-k{\mathrm{ln}}t+\xi^{+}(t)\biggl{)}^{2}}+\biggl{|}\frac{N-1}{\rho+c_{*}t-k{\mathrm{ln}}t+\xi^{+}(t)}-\frac{k}{t}\biggl{|}.$
(26)

In the range $\rho\leq M$, a sufficient condition is

$\delta_{M}\dot{\xi}^{+}-Cq^{+}\geq\frac{\|\nabla_{\Theta}s\|_{\infty}}{\varepsilon\biggl{(}\rho+c_{*}t-k{\mathrm{ln}}t+\xi^{+}(t)\biggl{)}^{2}}+\biggl{|}\frac{N-1}{\rho+c_{*}t-k{\mathrm{ln}}t+\xi^{+}(t)}-\frac{k}{t}\biggl{|},$
(27)

Both conditions are satisfied as $\xi^{+}$ and $q^{+}$ satisfy (22). Note that
we always may absorb the first term in the right handside of (26) or (27) in a
function of the type $g_{\varepsilon}(t,\xi^{+}(t))$. Thus $\bar{u}$ is a
supersolution to (9) and for any $t\geq T_{\varepsilon}$

$u(t,r,\theta)\leq U_{*}(r+s(\theta))+\varepsilon$

Dealing in the same way with a subsolution finishes the proof of the theorem.
$\Box$.

##  5 Further questions and final remarks

Instead of working with a bistable nonlinearity, one could think of dealing
with a nonlinearity $f$ for which there is $\theta\in(0,1)$ such that

$f\equiv 0\ \hbox{on $[0,\theta]$},\ \ \ f>0\ \hbox{on $(\theta,1)$},\ \ \
f(1)=0.$

We believe that Theorem 1.1 holds in this case. However, the degeneracy of $f$
near 0 would certainly impose the construction of sub and super solutions with
exponential weights. The main step would once again be the gradient estimate,
which already carries more technicalities than the basic Fife-Mc Leod
construction. Some rather tedious computations should therefore be expected.

A more challenging question is to assess when the shift $s^{\infty}(\Theta)$
is trivial, that is, constant with respect to $\Theta$. The analysis of the
second author in [20] shows that the set of (almost spherically symmetric)
initial data giving raise to a nontrivial shift is quite big (open and dense).
The issue is therefore whether any non spherically symmetric initial datum
will generate a nontrivial shift. We note that [20] does not preclude
codimension 1 sets of initial data generating trivial shifts. We hope to
address this question in the future.

Another, possibly easier question, concerns further regularity for
$s^{\infty}$. In [20] it is shown to be $L^{2}$, in this work we upgrade its
regularity to Lipschitz. Whether it is $C^{2}$, or whether the derivative may
develop discontinuities even if the initial datum is smooth, is something that
does not immediately stem from our analysis. We simply note that, given the
goal that is ours in this paper, additional regularity is only anecdotical, in
the sense that it would have slightly simplified the convergence proof.
However the question is interesting in its own right and we note that, in [17]
we could not go further than Lipschitz regularity either. Whether more
involved considerations would have allowed us to reach further regularity, or,
as opposed to this, a new phenomenon occurs, is something that we do not know.

Finally, we believe that it would be interesting to understand the rate of
convergence of $u(t,x)$ to the shifted wave. In he Fisher-KPP case, it is a
very interesting problem that was first raised by Ebert and Van Saarloos [7],
in one spatial dimension. They proposed a full expansion in terms of powers of
$t^{-1/2}$ and found universal (that is, not depending on the initial datum
and not even on the nonlinearity $f$) terms. The analysis of [7] was carried
out in the formal style, and a first mathematically rigorous proof of the
expansion up to the order $t^{-1+\delta}$ ($\delta>0$ is any small positive
number) was given in [15]. An analysis, still in the formal style, of [3],
finds an expansion that is different from that of [7] at higher orders, in the
sense that $t^{-1}\mathrm{ln}t$ terms pop up. The analysis of [3] is confirmed
in a mathematically rigorous way by Graham in [10]. So, coming back to our
model (1), we believe that pushing the expansion, perhaps to exponentially
small terms, would be of interest. Making a more extensive use of the concept
of radial waves, as defined in [20], could be a starting point.

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# Effective Field Theory Analysis on $\mu$ Problem in Low-Scale Gauge
Mediation

Sibo Zheng
Department of Physics, Chongqing University, Chongqing 401331, P.R. China

Abstract

Supersymmetric models based on the scenario of gague mediation often suffer
from the well-known $\mu$ problem. In this paper, we reconsider this problem
in low-scale gauge mediation in terms of effective field theory analysis. In
this paradigm, all high energy input soft mass can be expressed via loop
expansions. If the corrections coming from messenger thresholds are small, as
we assume in this letter, then all RG evaluations can be taken as linearly
approximation for low-scale supersymmetric breaking. Due to these
observations, the parameter space can be systematically classified and studied
after constraints coming from electro-weak symmetry breaking are imposed. We
find that some old proposals in the literature are reproduced, and two new
classes are uncovered. We refer to a microscopic model, where the specific
relations among coefficients in one of the new classes are well motivated.
Also, we discuss some primary phenomenologies.

May 2011

##  1 Introduction

###  1.1 Motivations

Among scenarios that solve the stringent constraints on experiments of flavor
violations in supersymmetric (SUSY) models, gauge mediation [1, 2, 3, 4, 5] is
an appealing candidate. In particular, SUSY models with low-scale SUSY-
breaking can be directly examined at the LHC. However, gauge mediation suffers
from a well-known fine tuning problem related to electro-weak symmetry
breaking (EWSB), i.e, the $\mu$ and B$\mu$ problem [6]. On general grounds
there is a small hierarchy between these two input scales at mediation scale,
$B_{\mu}\sim~{}(16\pi^{2})\mu^{2}$. There have been a lot of efforts on
solving this problem in the literature [7]. However, except all important
subtleties are taken into account, it is not concrete to argue whether a SUSY
model suffers such a fine tuning problem.

What is the most important essentials related to $\mu$ problem are all the
input soft masses that are determined both by the SUSY-breaking and the
messenger sectors as well as the renormalization group (RG) evaluations for
these soft masses. The specific relations in the mediation scale can be
substantially modified by the RG corrections. A typical example is SUSY models
of gaugino mediation [8], in which the high energy input parameters are three
gaugino masses (together with $L^{-1}$, $\tan\beta$ and $sign{\mu}$). It
obviously does not realize EWSB and satisfy other considerations. But the
correct and large RG corrections turns it into a viable SUSY model at electro-
weak scale.

###  1.2 Strategy

In this paper, we systematically analyze the $\mu$ problem from viewpoint of
effective field theory, where the SUSY-breaking sector is referred by spurion
superfields. We introduce two separate spurion superfields, which generalize
the minimal setup of gauge mediation where only one spurion field needs to be
considered. These spurion superfields introduce relevant dynamical scales,

$\displaystyle{}X_{G}=M_{G}\left(1+\theta^{2}\Lambda_{G}\right),~{}~{}~{}~{}~{}~{}X_{D}=M_{D}\left(1+\theta^{2}\Lambda_{D}\right)$
(1.1)

$X_{G}$ and $X_{D}$ are responsible for SUSY-breaking related to generations
of scalar scalars, gaugino masses as well as $\mu$ and $B_{\mu}$ terms of
Higgs superfields, respectively. In this note, for simplicity we assume
$M_{D}/M_{G}\sim 0.1-1$ so that the corrections coming from multiplet
messenger thresholds are small.

Using $X_{G}$ and $X_{D}$, we can express all the high energy input soft mass
terms through loop expansions 111We take the effective number of messengers as
$N_{eff,G}=1$ for the squark and gaugino masses. This value can vary when the
models derive from the minimal setup (see e.g, [9]). It thus changes the
relative ratio between squark and gaugino masses. . Note that the specific
relations $\Lambda_{G}/M_{G}\lesssim 10^{-1},\Lambda_{D}/M_{D}\lesssim
10^{-1}$ have to be imposed such that we can express

$\displaystyle{}m^{2}_{\tilde{f}_{i}}$ $\displaystyle=$
$\displaystyle\sum^{3}_{i=1}\left(C_{2}(\tilde{f},i)\frac{g^{4}_{i}}{(16\pi^{2})^{2}}\right)\Lambda^{2}_{G},$
$\displaystyle M_{i}$ $\displaystyle=$
$\displaystyle\frac{g^{2}_{i}}{(16\pi^{2})}\Lambda_{G}$ (1.2)

for squark and gaugino masses and

$\displaystyle{}m^{2}_{{H_{\mu},d}}=\sum^{2}_{i=1}\left(C_{2}(H_{\mu,d},i)\frac{g^{4}_{i}}{(16\pi^{2})^{2}}\right)\Lambda^{2}_{G}+\Lambda^{2}_{D}\left[\frac{C^{(1)}_{H_{\mu,d}}}{(16\pi^{2})}+\frac{C^{(2)}_{H_{\mu,d}}}{(16\pi^{2})^{2}}+\cdots\right],$
(1.3)

for Higgs masses squared $m^{2}_{H_{\mu}}$ and $m^{2}_{H_{d}}$. The
contributions in (1.3) arise from the ordinary gauge mediation and the
superpotential for hidden sector $X_{D}$. The $\mu$ and B$\mu$ terms, on the
other hands, only receive the contributions coming from $X_{D}$-sector 222Note
that a tree-level $\mu$ term can exist in the SUSY limit $\Lambda_{D}=0$.,

$\displaystyle{}B_{\mu}$ $\displaystyle=$
$\displaystyle\Lambda^{2}_{D}\left[\frac{C^{(1)}_{B\mu}}{(16\pi^{2})}+\frac{C^{(2)}_{B\mu}}{(16\pi^{2})^{2}}+\cdots\right]$
$\displaystyle\mu$ $\displaystyle=$ $\displaystyle
C^{(0)}_{\mu}M_{D}+\Lambda_{D}\left[\frac{C^{(1)}_{\mu}}{(16\pi^{2})}+\frac{C^{(2)}_{\mu}}{(16\pi^{2})^{2}}+\cdots\right],$
(1.4)

Note that when a particular loop coefficient $C^{(i)}$ is determined to be
non-zero, all higher order corrections can be neglected because of the
perturbative nature. Furthermore, a coefficient $C^{(i)}$ smaller than loop
factor $1/(16\pi^{2})$ is actually an $(i+1)$-loop effect. Following this fact
we assume all $C^{(i)}$ are bounded as,

$\displaystyle{}\frac{1}{16\pi^{2}}<C^{(i)}\lesssim 1$ (1.5)

The upper bound in (1.5) is due to the perturbativity of new Yukawa couplings
in superpotential for hidden sector $X_{D}$. Otherwise, large Yukawa couplings
will suffer the problem of Landau poles.

Having systematically analyzed the high energy input soft masses, we can
obtain their values at EW scale by taking the RG corrections into account. The
RG equations [11] for these soft masses in minimal supersymmetric standard
model (MSSM) are quite involved. In this note we discuss gauge mediation with
low-scale SUSY breaking,

$\displaystyle{}10^{2}TeV\lesssim\Lambda_{G,D}\lesssim 10^{6}TeV$ (1.6)

In this region the NLSP particle are mainly prompt decays or can be long
lived, which is of highly interest at searches of SUSY signals at colliders.
Also in the region (1.6) the RG evaluations for soft masses mainly receive
their corrections coming from $\ln(M/M_{Z})$ factor, which is of order unity .
This contribution can be taken as linear approximation [12]. Following the RG
equations at one-loop, one obtains the soft masses at EW scale via replacing
(1.3) and (1.2) with

$\displaystyle{}\mu\rightarrow\mu$ $\displaystyle=$ $\displaystyle\hat{\mu},$
$\displaystyle B_{\mu}\rightarrow~{}B_{\mu}-\frac{\delta
C_{B_{\mu}}}{(16\pi^{2})^{2}}\mu\Lambda_{G}$ $\displaystyle=$
$\displaystyle\hat{B_{\mu}},$ $\displaystyle
m^{2}_{H_{\mu}}\rightarrow~{}m^{2}_{H_{\mu}}-\frac{\delta
C_{H_{\mu}}}{(16\pi^{2})^{2}}\Lambda^{2}_{G}$ $\displaystyle=$
$\displaystyle\hat{m}^{2}_{H_{\mu}},$ (1.7) $\displaystyle
m^{2}_{H_{d}}\rightarrow~{}m^{2}_{\tilde{H}_{d}}+\frac{\delta
C_{H_{d}}}{(16\pi^{2})^{2}}\Lambda^{2}_{G}$ $\displaystyle=$
$\displaystyle\hat{m}^{2}_{H_{d}}$

where $\delta~{}C_{i}$ are both positive real numbers of order one 333The
$\ln(M)$ terms are all of order one in low-scale gauge mediation, which are
absorbed into the $\delta~{}C_{i}$..

Then, as a primary analysis about the physical parameter space, we impose the
necessary condition for EWSB,

$\displaystyle{}\left(\hat{m}^{2}_{H_{\mu}}+\mid\hat{\mu}\mid^{2}\right)\left(\hat{m}^{2}_{H_{d}}+\mid\hat{\mu}\mid^{2}\right)\thickapprox\hat{B}^{2}_{\mu}$
(1.8)

In terms of (1.2) and expressions for input soft masses through loop
expansions, we can analyze the possible parameter space composed of
$(C^{(i)}_{\mu},C^{(i)}_{B_{\mu}},C^{(i)}_{H_{\mu,d}},M_{G,D},\Lambda_{G,D})$
under constraints (1.5) order by order. An important observation which can be
used to simplify the classifications is that if $C^{(i)}_{\mu}=0$ at $i$th-
loop, then we must also have $C^{(i)}_{B_{\mu}}=0$. Reversely, the statement
is not true. This conclusion can be proved by arguments of effective field
theory (see [13] for primary analysis via SUSY algebra).

Other relevant experimental constraints on parameters in Higgs sector include
negative mass squared for $H_{\mu}$ scalar,

$\displaystyle{}\hat{m}^{2}_{H_{\mu}}<0$ (1.9)

all masses of Higgs scalars are bigger than $\mathcal{O}(115)$GeV, and the
mass of the lightest chargino has to be larger than $\mathcal{O}(100)$GeV. We
will also discuss possible implications arising from these constraints on
allowed parameter space.

###  1.3 Outlines

The outline of this paper is organized as follows. In section 2, we divide the
discussions into three classes,

$\displaystyle{}Case~{}(1)$ $\displaystyle:$
$\displaystyle~{}~{}~{}~{}\hat{\mu}^{2}>>\mid\hat{m}^{2}_{H_{\mu}}\mid,~{}~{}\hat{\mu}^{2}>>\hat{m}^{2}_{H_{d}}$
$\displaystyle Case~{}(2)$ $\displaystyle:$
$\displaystyle~{}~{}~{}~{}\hat{\mu}^{2}>\mid\hat{m}^{2}_{H_{\mu}}\mid,~{}~{}\hat{\mu}^{2}<<\hat{m}^{2}_{H_{d}}$
$\displaystyle Case~{}(3)$ $\displaystyle:$
$\displaystyle~{}~{}~{}~{}\hat{\mu}^{2}>\mid\hat{m}^{2}_{H_{\mu}}\mid,~{}~{}\hat{\mu}^{2}\sim\hat{m}^{2}_{H_{d}}$
(1.10) $\displaystyle Case~{}(4)$ $\displaystyle:$
$\displaystyle~{}~{}~{}~{}\hat{\mu}^{2}<<\mid
m^{2}_{H_{\mu}}\mid,~{}~{}\hat{\mu}^{2}\sim\hat{m}^{2}_{H_{d}}$

First, we find that there is parameter space allowed for the case $(1)$, which
is not extensively studied before as far as we know. There are parameter
spaces allowed for case $(2)$ to case $(4)$. Second, the typical parameter
space for case $(3)$ and case $(4)$ corresponds to the one-loop $\mu$/two-loop
B$\mu$ as well as “lopsided gauge mediation ” [14] respectively, which both
have been discussed in the literature. Finally, the second case is totally
new. Although the spectra for soft masses in Higgs sector in this case is
similar to those in [10] , it is not covered by that approach. Here only one
substantial hierarchy among soft masses in Higgs sector is needed, in
comparison with at least two proposed in [10].

In section 3, we discuss the microscopic models of hidden sector $X_{D}$ where
the typical values in parameter space for case $(2)$ is well motivated. The
new Yukawa couplings appearing in the hidden superpotential
$W(X_{D},H_{\mu,d})$ are all around unity or smaller. No severe fine tunings
are allowed in the hidden sector. This is also different from the model
buildings for the approach in [10].

In section 4, we discuss the phenomenologies for models belonging to case
$(2)$. Following the typical parameter values, we find that Higgs scalar
expect $h^{0}$ are nearly degenerate at large $\hat{m}_{H_{d}}$ and small
$\tan\beta\thickapprox 0.1$ is favored. The direct consequence for this class
of SUSY models is that the next-to-lightest supersymmetric particle (NLSP) is
mostly Bino-like and its prompt two-body decay modes are dominated by final
state $\gamma$ and $Z^{0}$ plus missing energy, with branching ratio
$cos^{2}\theta_{W}$ and $sin^{2}\theta_{W}$ respectively. The decay channel to
$h^{0}$ plus missing energy is negligible as a result of dramatically
suppression.

In section 5, we summarize the main results in this note.

##  2 Classifications

Now we discuss the allowed parameter space in the four classes in (1.3) in
turn. First,we introduce such three dimensional parameters for later use,

$\displaystyle{}x=\Lambda_{G}/M_{G}\lesssim
10^{-1},~{}~{}~{}~{}y=\Lambda_{D}/M_{D}\lesssim
10^{-1},~{}~{}~{}~{}~{}z=\Lambda_{G}/\Lambda_{D}.$ (2.1)

The first two are small positive, real numbers as mentioned in the
introduction, while the last parameter $z$ is also real and positive, in the
range of $10^{-2}$ to $10^{2}$ in general. When $z=1$, we reproduce the
physics of gauge mediation with one spurion superfield. Since the mass
parameters in Higgs sector receive contributions both from the $X_{G}$ and
$X_{D}$ sectors either directly or via RG evaluation, it will be of use to
discuss which one is the dominant contribution.

###  2.1 Case $(1)$: Large $\hat{\mu}$ Term

In this case, the most stringent constraint comes from (1.8) , from which we
have when $\mu$ is generated at tree-level

$\displaystyle{}\left(C^{(0)}_{\mu}\right)^{2}$ $\displaystyle\thickapprox$
$\displaystyle\left[\frac{C^{(1)}_{B\mu}}{(16\pi^{2})}y^{2}+\frac{C^{(2)}_{B\mu}}{(16\pi^{2})^{2}}y^{3}+\cdots\right]-\frac{C^{(0)}_{\mu}\delta
C_{B_{\mu}}}{(16\pi^{2})^{2}}\frac{\Lambda_{G}}{M_{D}}$ (2.2)

First, consider the case for $C^{(1)}_{B_{\mu}}\neq 0$. Eq.(2.2) implies that,

$\displaystyle{}\left(C^{(0)}_{\mu}\right)^{2}\thickapprox\frac{C^{(1)}_{B_{\mu}}}{16\pi^{2}}y^{2},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}C^{(1)}_{B_{\mu}}>\frac{C^{(0)}_{\mu}\delta
C_{B_{\mu}}}{16\pi^{2}}\frac{M_{D}}{\Lambda_{G}}$ (2.3)

Second, if $C^{(1)}_{B_{\mu}}=0$ and $C^{(2)}_{B_{\mu}}\neq 0$, then one
obtains,

$\displaystyle{}\left(C^{(0)}_{\mu}\right)^{2}\thickapprox\frac{C^{(1)}_{B_{\mu}}}{(16\pi^{2})^{2}}y^{3}$
(2.4)

which is not allowed according to (1.5). Note that $\delta C_{B_{\mu}}$ is a
positive real coefficient of order one. As the tree-level $\mu$ term respects
the supersymmetry, it is less of interest in comparison with the mechanism
that all mass scales are originated from softly broken SUSY, unless a
dynamical generation for tree-level $\mu$ term is realized so that it is a
natural consequence.

If $\mu$ is generated at one-loop, (2.2) is replaced by,

$\displaystyle{}\left(C^{(1)}_{\mu}\right)^{2}$ $\displaystyle\thickapprox$
$\displaystyle
16\pi^{2}\left[C^{(1)}_{B_{\mu}}+\frac{C^{(2)}_{B_{\mu}}}{(16\pi^{2})}y+\cdots\right]-\frac{C^{(1)}_{\mu}\delta
C_{B_{\mu}}}{(16\pi^{2})}\frac{\Lambda_{G}}{\Lambda_{D}}$ (2.5)

A non-zero $C^{(1)}_{B_{\mu}}$ will cause the tension for its permitted value,
as stated in (1.5). What is more promising is that the $B_{\mu}$ term is
generated at two-loop. In this case, we obtain,

$\displaystyle{}C^{(1)}_{\mu}>z,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left(C^{(1)}_{\mu}\right)^{2}\thickapprox
yC^{(2)}_{B_{\mu}}$ (2.6)

Take the constraints for case $(1)$ in (1.3) into account, we find that no
matter the mainly contribution to $\hat{m}_{H_{\mu,d}}$ comes from $X_{D}$ or
$X_{G}$ sector, there is only a set of critical values,

$\displaystyle{}(z_{*})^{2}=\frac{1}{16\pi^{2}},~{}~{}~{}~{}~{}~{}and~{}~{}~{}~{}~{}~{}C^{(1)}_{\mu*}=1$
(2.7)

if we relax the large differences in (1.3) for case $(1)$ to some moderate
values.

###  2.2 Case $(2)$: Large $\hat{m}_{H_{d}}$

We obtain after imposing the constraint in (1.3) for this case ,

$\displaystyle{}\begin{array}[]{c}\mid C_{H_{\mu,d}}^{(2)}\mid\lesssim
z^{2}\\\ C^{(0)}_{\mu}>\frac{x}{(16\pi^{2})}\\\
C^{(0)}_{\mu}<<\frac{x}{(16\pi^{2})}\\\ \cdots\\\
\end{array}~{}~{}~{}~{}~{}~{}or~{}~{}\begin{array}[]{c}\mid
C_{H_{\mu}}^{(2)}\mid<z^{2}\\\ C_{H_{d}}^{(2)}>>z^{2}\\\
C^{(0)}_{\mu}>\frac{x}{(16\pi^{2})}\\\
\left(C^{(0)}_{\mu}\right)^{2}<<\frac{C^{(2)}_{H_{d}}}{(16\pi^{2})}y^{2}\\\
\cdots\\\ \end{array}~{}~{}~{}~{}~{}~{}or~{}~{}~{}\begin{array}[]{c}\mid
C_{H_{\mu}}^{(2)}\mid>z^{2}\\\ \mid C_{H_{d}}^{(2)}\mid<<z^{2}\\\
C^{(0)}_{\mu}<<\frac{x}{(16\pi^{2})}\\\
\left(C^{(0)}_{\mu}\right)^{2}>\frac{C^{(2)}_{H_{d}}}{(16\pi^{2})}y^{2}\\\
\cdots\\\ \end{array}$ (2.22)

if $C^{(0)}_{\mu}\neq 0$. Here the constraints neglected denote those coming
from (1.8). We see that the first and third sets of choices is obviously not
consistent. The second one implies that $\ C^{(2)}_{H_{d}}y>>1$, which is not
permitted as a result of the fact that $C^{(2)}_{H_{d}}$ is of order unity and
$y$ is smaller than one.

if $C^{(0)}_{\mu}=0$ and $C^{(1)}_{\mu}\neq 0$, we get

$\displaystyle{}\begin{array}[]{c}\mid C_{H_{d}}^{(2)}\mid\lesssim z^{2}\\\
\left(C^{(1)}_{\mu}\right)^{2}>x^{2}\\\
\left(C^{(1)}_{\mu}\right)^{2}<<x^{2}\\\ \cdots\\\
\end{array}~{}~{}~{}~{}or~{}~{}\begin{array}[]{c}\mid
C_{H_{\mu}}^{(2)}\mid\lesssim z^{2}\\\ C_{H_{d}}^{(2)}>>z^{2}\\\
\left(C^{(1)}_{\mu}\right)^{2}\gtrsim z^{2}\\\
\left(C^{(1)}_{\mu}\right)^{2}<<C_{H_{d}}^{(2)}\\\ \cdots\\\
\end{array}~{}~{}~{}~{}~{}~{}or~{}~{}~{}\begin{array}[]{c}\mid
C_{H_{\mu}}^{(2)}\mid>z^{2}\\\ C_{H_{d}}^{(2)}<<z^{2}\\\
\left(C^{(1)}_{\mu}\right)^{2}>C_{H_{d}}^{(2)}\\\
\left(C^{(1)}_{\mu}\right)^{2}<<z^{2}\\\ \cdots\\\ \end{array}$ (2.37)

The first and third classes are obviously not consistent, while the second one
is allowed. We impose the constraint in (1.8), which implies that when
$C^{(1)}_{B\mu}=0$, $\sqrt{C^{(2)}_{H_{d}}}\thickapprox z/(16\pi^{2})>>z$.
Thus this choice is not allowed. On the other hand, if $C^{(1)}_{B\mu}\neq 0$,
we obtain an additional constraint,

$\displaystyle{}C^{(1)}_{\mu}\sqrt{C^{(2)}_{H_{d}}}\thickapprox
C^{(1)}_{B\mu}>>C^{(1)}_{\mu}z$ (2.38)

which is actually consistent with the second class of choices in (2.37). For
example, we take such typical values for these parameters,

$\displaystyle{}C^{(1)}_{\mu}\sim C^{(1)}_{B\mu}\sim z\sim
0.1,~{}~{}~{}C^{(2)}_{H_{\mu}}\sim
0.01~{}~{}~{}and~{}~{}~{}~{}~{}C^{(2)}_{H_{d}}\sim 1$ (2.39)

In the next section, we will construct a class of models to motivate choices
of parameters in (2.39).

###  2.3 Case $(3)$: Standard Proposal

First, when $C^{(0)}_{\mu}\neq 0$, the only possible parameter space
corresponds to the circumstance under which the contributions to soft masses
of $H_{\mu,d}$ are dominated by the $X_{G}$ hidden sector. Otherwise, we will
always get extremely small $C^{(0)}_{\mu}$, i.e,
$C^{(0)}_{\mu}<<1/(16\pi^{2})$, which indeed is high-order effects according
to our understandings. Follow this observation, we obtain the constraints,

$\displaystyle{}\begin{array}[]{c}\mid C_{H_{\mu}}^{(2)}\mid\lesssim z^{2},\\\
\mid C_{H_{d}}^{(2)}\mid\lesssim z^{2},\\\
C^{(0)}_{\mu}\thickapprox\left(\frac{x}{16\pi^{2}}\right),\\\
\left(C^{(0)}_{\mu}\right)^{2}\thickapprox\frac{C^{(1)}_{B\mu}}{(16\pi^{2})}y^{2}\\\
\end{array}$ (2.44)

Second, if $\mu$ term is generated at one-loop, there is only one possibility
needed to be considered,

$\displaystyle{}\begin{array}[]{c}\mid C_{H_{\mu}}^{(2)}\mid<z^{2},\\\ \mid
C_{H_{d}}^{(2)}\mid<z^{2},\\\ C^{(1)}_{\mu}\thickapprox z^{2},\\\
\left(C^{(1)}_{\mu}\right)^{2}\thickapprox
16\pi^{2}C^{(1)}_{B\mu}~{}~{}~{}or~{}~{}~{}\left(C^{(1)}_{\mu}\right)^{2}\thickapprox
yC^{(2)}_{B\mu},~{}~{}~{}(C^{(1)}_{B\mu}=0)\\\ \end{array}$ (2.49)

for either one-loop or two-loop $B_{\mu}$. All other choices are directly
excluded. These two choices have been well known. As shown in (2.49), for the
one-loop generation of $B_{\mu}$ , there will be no parameter space allowed
except $C^{(2)}_{B_{\mu}}$ close to its lower bounded value, which actually a
two-loop effect. In this sense, the second class in (2.49) is often referred
as the standard proposal to solve the problem.

###  2.4 Case $(4)$: Small $\hat{\mu}$ and Large $\hat{B_{\mu}}$

Similarly to the previous discussions, the fist class of choices for
$C^{(0)}_{\mu}\neq 0$ is given by,

$\displaystyle{}C_{\mu}^{(0)}<<\frac{1}{16\pi^{2}}\frac{\Lambda_{G}}{M_{D}},~{}~{}~{}~{}~{}~{}~{}C_{\mu}^{(0)}<<\frac{1}{16\pi^{2}}y$
(2.50)

which contradicts with the statement in (1.5). Second, if $C^{(0)}_{\mu}=0$
and $C^{(1)}_{\mu}\neq 0$, we obtain,

$\displaystyle{}C_{\mu}^{(1)}<<z,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}C_{\mu}^{(1)}<<C^{(2)}_{H_{d}}$
(2.51)

These choices are argued to be consistent in a class of hidden model [14].

In summary, we reproduce two old proposals in the literature as well as
discover two new approaches. In particular , the proposal discussed in [10] is
a specific choice of the case $(2)$. This statement will be more apparent in
the next section, in which the model buildings and the parameter space is
obviously different from the discussions shown in [10].

##  3 A Hidden Model

In this section we consider the hidden superpotential for SUSY-breaking sector
$X_{D}$ associated with the Higgs sector. We refer to the hidden
superpotential as 444This superpotential is the one in [14] that has been used
to analyze the models of large $B_{\mu}$ and small $\mu$. Here we use it for
different purpose and as an illustration for constructing mass spectrum in
(2.39). It is also interesting to discuss other possibilities such as models
of MSSM singlets.

$\displaystyle{}W_{hid}=\lambda_{\mu}H_{\mu}DS+\lambda_{d}H_{d}\bar{D}\bar{S}+X_{D}D\bar{D}+\frac{1}{2}X_{D}\left(a_{S}S^{2}+a_{\bar{S}}\bar{S}^{2}+a_{S\bar{S}}S\bar{S}\right)$
(3.1)

in terms of which the typical parameter space in (2.39) for the case $(2)$ can
be realized. In (3.1), $D$ and $\bar{D}$ are bi-fundamental chiral superfileds
under the representation of the EW groups, $S$ and $\bar{S}$ are MSSM
singlets, and $X_{D}$ and $X_{S}$ refer to the relevant hidden SUSY-breaking
sector.

Integrate out the massive component fields in $S,\bar{S},D$ and $\bar{D}$, we
obtain the effective Kahler potential [15],

$\displaystyle{}K_{eff}=-\frac{1}{32\pi^{2}}Tr~{}\left[\mathcal{M}\mathcal{M}^{{\dagger}}\ln\left(\frac{\mathcal{M}\mathcal{M}^{{\dagger}}}{\Lambda^{2}}\right)\right]$
(3.2)

where

$\displaystyle{}\mathcal{M}\mathcal{M}^{{\dagger}}=\left(\begin{array}[]{cc}\mid
X_{D}\mid^{2}+\mid\lambda_{d}\mid^{2}\mid~{}H_{d}\mid^{2}&\lambda_{\mu}H_{\mu}X_{D}^{*}+\lambda_{d}H_{d}^{*}X_{S}\\\
(\lambda_{\mu}H_{\mu}X_{D}^{*}+\lambda_{d}H_{d}^{*}X_{S})^{*}&\mid
X_{S}\mid^{2}+\mid\lambda_{\mu}\mid^{2}\mid~{}H_{\mu}\mid^{2}\\\
\end{array}\right)$ (3.5)

The soft masses can be obtained via evaluating the eigenvalues of matrix
(3.5), which are given by 555For the explicit expressions for functions
$P(v,w),Q(v,w)$ and $R(v,w)$ we refer the readers to [14].,

$\displaystyle{}m^{2}_{H_{\mu}}$ $\displaystyle=$
$\displaystyle\frac{\lambda_{\mu}^{2}}{16\pi^{2}}\Lambda_{D}^{2}\left[c_{\theta}^{2}P(v,w)+s_{\theta}^{2}P(\lambda~{}v,w)\right],$
$\displaystyle m^{2}_{H_{d}}$ $\displaystyle=$
$\displaystyle\frac{\lambda_{d}^{2}}{16\pi^{2}}\Lambda_{D}^{2}\left[s_{\theta}^{2}P(v,w)+c_{\theta}^{2}P(\lambda~{}v,w)\right],$
(3.6) $\displaystyle\mu$ $\displaystyle=$
$\displaystyle\frac{\lambda_{\mu}\lambda_{d}}{16\pi^{2}}\Lambda_{D}s_{\theta}c_{\theta}\left[Q(\lambda~{}v,w)-Q(v,w)\right],$
$\displaystyle B_{\mu}$ $\displaystyle=$
$\displaystyle\frac{\lambda_{\mu}\lambda_{d}}{16\pi^{2}}\Lambda^{2}_{D}s_{\theta}c_{\theta}\left[R(\lambda~{}v,w)-R(v,w)\right]$

where the dimensionless coefficients are defined as,

$\displaystyle{}\lambda=\frac{a_{\bar{S}}-a_{S}\tan^{2}\theta}{a_{S}-a_{\bar{S}}\tan^{2}\theta},~{}~{}~{}~{}\tan
2\theta=\frac{2a_{S\bar{S}}}{a_{\bar{S}}-a_{S}},~{}~{}~{}~{}~{}v=M_{S}/M_{D},~{}~{}~{}~{}w=\Lambda_{S}/\Lambda_{D},$
(3.7)

Compare (1.3) and (1.2) with (3), we obtain the explicit expressions,

$\displaystyle{}C^{(1)}_{\mu}$ $\displaystyle=$
$\displaystyle\lambda_{\mu}\lambda_{d}s_{\theta}c_{\theta}\left[Q(\lambda~{}v,w)-Q(v,w)\right],$
$\displaystyle C^{(1)}_{B_{\mu}}$ $\displaystyle=$
$\displaystyle\lambda_{\mu}\lambda_{d}s_{\theta}c_{\theta}\left[Q(\lambda~{}v,w)-Q(v,w)\right],$
$\displaystyle C^{(2)}_{H_{\mu}}$ $\displaystyle=$
$\displaystyle(16\pi^{2})\lambda_{\mu}^{2}\left[c_{\theta}^{2}P(v,w)+s_{\theta}^{2}P(\lambda~{}v,w)\right],$
(3.8) $\displaystyle C^{(2)}_{H_{d}}$ $\displaystyle=$
$\displaystyle(16\pi^{2})\lambda_{d}^{2}\left[s_{\theta}^{2}P(v,w)+c_{\theta}^{2}P(\lambda~{}v,w)\right]$

Figure 1:  The values of $w$s vary as $v$s in the region where $\sin\theta\sim
0.02$, with $\lambda_{\mu}=4.8$, $\lambda_{d}=0.4$ and $\lambda=0.1$. The
dashing, solid and the bottom lines represent the curves for $\mu$, $B_{\mu}$
and $H_{\mu}$ respectively. The favored region is near $v\sim 0.2-0.3$. In
this region $C^{(2)}_{H_{d}}\sim 0.4-0.6$.

Now we discuss the parameter space that is composed of
$\lambda_{\mu,d},\lambda,v$, $\theta$ and $w$ ( with an overall scale
$\Lambda_{D}$ ), by impose the concrete choices in (2.39) on (3). In
particular, the perturbativity of Yukawa couplings in (3.1) and the absence of
obvious fine tunings between them require,

$\displaystyle{}0.1\lesssim\lambda_{\mu,d}\lesssim
5,~{}~{}~{}~{}~{}~{}~{}0.1\lesssim\lambda\lesssim
10,~{}~{}~{}~{}~{}~{}~{}~{}~{}0.1\lesssim a_{S,\bar{S}}\lesssim 5,$ (3.9)

Also the value of $v$ is restricted to be around $0.1-10$ according to the
assumption we follow.

First, we take the curves for $C^{(1)}_{\mu}$ and $C^{(2)}_{B_{\mu}}$ for
consideration. In the region of small $\sin\theta$ value where substantial
simplifications happen, the large ratio of $\lambda_{\mu}/\lambda_{d}$ is
favored. If $\lambda$ is set to be around $0.8$, these two curves overlap from
$v\sim 0.1$ to $v\sim 0.3$, during which $C^{(2)}_{H_{\mu}}$ is close to its
typical value, as shown in fig. $(1)$.

When we move to the region of parameter space where $\cos\theta$ closes to 1,
similar results can be obtained, with $a_{S}\sim 0.1a_{\bar{S}}$ and
$a_{S\bar{S}}\sim 0.1a_{\bar{S}}$ as shown in fig $(2)$. In this region, it is
more easier to obtain large $m_{H_{d}}$ term. The typical value for
$C^{(1)}_{H_{d}}$ is around $1.0-3.0$.

The main goal of this section is to construct a concrete model in which the
typical parameter space is allowed. We want to mention that the parameter
space we discover is part of the total physical one.

Figure 2:  The values of $w$s vary as $v$s in the region where $\cos\theta\sim
0.02$, with $\lambda_{\mu}=4$, $\lambda_{d}=0.4$ and $\lambda=0.1$. The
dashing, solid and the bottom lines represent the curves for $\mu$, $B_{\mu}$
and $H_{\mu}$ respectively. The favored region is near $v\sim 0.3-0.4$.

##  4 Phenomenology

Now we discuss the phenomenological implications for the mass spectra
represented by the second class of models in (1.3). In the typical parameter
space, we find,

$\displaystyle{}\mid\hat{m}^{2}_{H_{\mu}}\mid<\hat{\mu}^{2}\sim
M^{2}_{r}\sim\hat{m}^{2}_{\tilde{f}}\sim 10^{-1}\hat{B}_{\mu}\sim
10^{-2}\hat{m}^{2}_{H_{d}}$ (4.1)

###  4.1 Higgs Mass Spectrum

The tree-level Higgs mass spectra are given by,

$\displaystyle{}\hat{m}^{2}_{A^{0}}=\frac{2\hat{B}_{\mu}}{\sin
2\beta}=2\mid\mu\mid^{2}+\hat{m}^{2}_{H_{\mu}}+\hat{m}^{2}_{H_{d}}\thickapprox\hat{m}^{2}_{H_{d}}$
(4.2)

The last expression is obtained in terms of specific relations in (4.1).
Eq.(4.2) also implies that $\tan\beta\thickapprox 0.1$. The masses of the
remaining neutral Higgs scalars are,

$\displaystyle{}\hat{m}^{2}_{h^{0},H^{0}}=\frac{1}{2}\left(\hat{m}^{2}_{A^{0}}+m_{Z}^{2}\mp\sqrt{(\hat{m}^{2}_{A^{0}}-m_{Z}^{2})^{2}+4m_{Z}^{2}\hat{m}^{2}_{A^{0}}\sin^{2}2\beta}\right)$
(4.3)

from which we obtain up to leading order of
$O(m^{2}_{Z}/\hat{m}^{2}_{H_{d}})$,

$\displaystyle{}\hat{m}_{h^{0}}\thickapprox\frac{1}{\sqrt{2}}m_{Z}+\mathcal{O}\left(\frac{m^{2}_{Z}}{\hat{m}^{2}_{H_{d}}}\right),~{}~{}~{}~{}~{}~{}~{}~{}~{}\hat{m}_{H^{0}}\thickapprox\hat{m}_{H_{d}}+\mathcal{O}\left(\frac{m^{2}_{Z}}{\hat{m}^{2}_{H_{d}}}\right)$
(4.4)

The masses for charged Higgs scalars are,

$\displaystyle{}\hat{m}_{H^{\pm}}\thickapprox\hat{m}_{H_{d}}+\mathcal{O}\left(\frac{m^{2}_{Z}}{\hat{m}^{2}_{H_{d}}}\right)$
(4.5)

These mass spectra suggest that $\Lambda_{G}\gtrsim 10^{2}$TeV.

###  4.2 Bino-like NLSP

The NLSP is either the neutrilino $\tilde{\chi}^{0}_{1}$ or chargino
$\tilde{\chi}^{+}$ for the class of models in (4.2). Using the fact that
$m_{Z}<\mid\hat{\mu}\pm M_{1}\mid$ and $m_{Z}<\mid\hat{\mu}\pm M_{2}\mid$ for
$\Lambda_{G}\gtrsim 10^{2}$TeV, we have

$\displaystyle{}\hat{m}_{\tilde{\chi}^{0}_{1}}$ $\displaystyle\thickapprox$
$\displaystyle M_{1}+\mathcal{O}\left(\frac{m^{2}_{W}}{\hat{\mu}^{2}}\right),$
$\displaystyle\hat{m}^{2}_{\tilde{\chi}_{+}}$ $\displaystyle\thickapprox$
$\displaystyle
M^{2}_{2}+\frac{1}{2}m^{2}_{W}+\mathcal{O}\left(\frac{m^{2}_{W}}{\hat{\mu}^{2}}\right)$
(4.6)

So neutrilino is the NLSP. Under the same limit, the matrix $N$ [16] that is
used to dialogize the neurtilino mass matrix is given by ,

$\displaystyle{}N_{11}$ $\displaystyle\thickapprox$ $\displaystyle 1,$
$\displaystyle N_{12}$ $\displaystyle\thickapprox$ $\displaystyle 0,$
$\displaystyle N_{13}$ $\displaystyle\thickapprox$ $\displaystyle
s_{W}s_{\beta}\frac{m_{Z}(\hat{\mu}+M_{1}\cot\beta)}{\mid\hat{\mu}\mid^{2}-M_{1}^{2}}$
(4.7) $\displaystyle N_{14}$ $\displaystyle\thickapprox$ $\displaystyle-
s_{W}c_{\beta}\frac{m_{Z}(\hat{\mu}+M_{1}\tan\beta)}{\mid\hat{\mu}\mid^{2}-M_{1}^{2}}$

which leas to the conclusion that the NLSP neutrilino is mostly Bino-like. It
promptly decays into final states of $\gamma$, $Z$ and $h$ plus missing energy
when $\Lambda_{G}\lesssim 10^{3}$TeV, which can be directly produced at
Tevatron and LHC ( see [20] for recent discussions on neutrilino NLSP at
colliders). The decay widths for these final states are given by,

$\displaystyle{}\Gamma(\tilde{\chi}^{0}_{1}\rightarrow\gamma+\tilde{G})$
$\displaystyle\thickapprox$ $\displaystyle c^{2}_{W}\mathcal{A},$
$\displaystyle\Gamma(\tilde{\chi}^{0}_{1}\rightarrow Z+\tilde{G})$
$\displaystyle\thickapprox$
$\displaystyle\left[s_{W}^{2}+\frac{1}{8}\sin^{2}2\beta\left(\frac{m_{Z}}{\hat{\mu}}\right)^{2}\left(1-\frac{m^{2}_{Z}}{\hat{m}^{2}_{\tilde{\chi}^{0}_{1}}}\right)^{4}\right]\mathcal{A}\thickapprox
s^{2}_{W}\mathcal{A},$ $\displaystyle\Gamma(\tilde{\chi}^{0}_{1}\rightarrow
h+\tilde{G})$ $\displaystyle\thickapprox$ $\displaystyle 0$ (4.8)

with

$\displaystyle{}\mathcal{A}=\frac{\hat{m}^{5}_{\tilde{\chi}_{0}}}{16\pi
M^{2}_{G}\Lambda^{2}_{G}}=\left(\frac{\hat{m}_{\tilde{\chi}_{0}}}{100GeV}\right)^{5}\left(\frac{100TeV}{\sqrt{M_{G}\Lambda_{G}}}\right)^{4}\frac{1}{0.1mm}$
(4.9)

To derive the final results in (4.2) we use the definite value
$\tan\beta\thickapprox 0.1$, as imposed by (4.2). This small value
dramatically suppresses the decay width of channel
$\tilde{\chi}^{0}_{1}\rightarrow h+\tilde{G}$. The branching ratios for the
channels involved are found to be,

$\displaystyle{}Br(\tilde{\chi}^{0}_{1}$ $\displaystyle\rightarrow$
$\displaystyle\gamma+\tilde{G})\thickapprox 74\%$ $\displaystyle
Br(\tilde{\chi}^{0}_{1}$ $\displaystyle\rightarrow$ $\displaystyle
Z+\tilde{G})\thickapprox 26\%$ (4.10)

The spectra (4.1) lead to a large $m_{A}$ (relative to $m_{Z}$) and small
$\tan\beta$. No matter the value of $\tan\beta$, large $m_{A}$ is sufficient
to reduce the Higgs search of MSSM to the SM Higgs search. In this sense, in
the model we study here phenomenologies related to the rates of Higgs
prodcutions and decays are similar with those under the decoupling limit [17].
However, as reviewed in [18], it is still possible to discover or exclude
heavy $m_{H^{\pm}}$ up to 1 TeV via $\tau$ $\mu$ decays at LHC with 300 fb-1.

To descriminate between our model and the others described by the decoupling
limit, one should turn to analyze the neutralino and chargino sectors. Due to
the smaller $\mu$ term as well as smaller gaugino masses than $m_{A}$ as shown
in (4.1), lighter neutrilinos and charginos are allowed, in comparison with
what one expects in the ordinary models that solve the $\mu$ problem, with
$\mu\sim m_{H}\sim m_{A}$. Typically, if the masses of neutrilinos and
charginos are of order $\mathcal{O}(100)$ GeV, the decays with leptonic final
states such as $\tilde{C}^{\pm}_{2}\rightarrow\tilde{N}_{1}l^{\pm}\nu$ and
$\tilde{N}_{2}\rightarrow\tilde{N}_{1}l^{+}l^{-}$ are very interesting [19].
Because these decays can be prompt and searched directly at the LHC . However,
in ordinary models with $\mu\sim m_{H}\sim m_{A}$ , one expects the masses of
neutralinos and charginos are heavier than $\mathcal{O}(200)$ GeV if we take
the decoupling limit. A double increase in $m_{\tilde{\chi}_{0}}$ will results
in the decay widths
$\Gamma(\tilde{\chi}^{0}_{1}\rightarrow\gamma/Z+\tilde{G})$ dramatically
enhanced, as shown from (4.9), which is beyond the reach of LHC.

##  5 Conclusions

In this paper $\mu$ problem in gauge mediation is studied in terms of
effective field theory analysis. As the subtleties that determine the SUSY
soft mass parameters at EW scale are the high energy input boundary values and
their RG evaluations, we use the loop expansions to capture the mainly
property of the former contribution, and restrict our discussion to negligible
multiple messenger threshold corrections as well as low-scale SUSY-breaking to
simplify the later contribution. Following these facts, we classify the
problem into four classes, and explore the parameter space allowed by imposing
primary constraints coming from EWSB. We reproduce two old proposals in the
literature as well as discover two new classes.

As illustration for model buildings, we refer to a hidden theory proposed in
[14] for different purpose. We find in some regions where no fine tunings
happen these typical values of parameters in one of the two new classes can be
realized. Also, we discuss the phenomenological implications predicted by this
new class of models. The Higgs scalars expect $h^{0}$ are heavy and of order
$m_{H_{d}}$. The NLSP is Bino-like neutrilino, whose two-body decays are
prompt and mainly composed of $\gamma$ and $Z$ plus missing energy. The
channel for neutrilno decaying into $h^{0}$ is kinetically permitted, however,
dramatically suppressed.

$\bf{Acknowledgement}$
This work is supported in part by the Fundamental Research Funds for the
Central Universities with project number CDJRC10300002.

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