The $p$-Adic Scattering equation

TL;DR

This paper explores the $p$-adic scattering equation, adapting classical entropy-based techniques to analyze the convergence of solutions within the $p$-adic framework.

Contribution

It introduces a novel adaptation of entropy methods to the $p$-adic scattering equation for convergence analysis.

Findings

Established convergence of solutions using adapted entropy techniques

Extended classical PDE methods to the $p$-adic setting

Provided new insights into $p$-adic scattering dynamics

Abstract

There are several techniques in classical case for some PDEs, involving the concept of entropy to show convergence of solutions to a steady state. In this work we deal with the $p$-adic scattering equation and we try to adapt these methods to prove convergence of solutions.

Equations

$${\partial }_{t}u+A(u(t))=0,t>0,u(0)=u_0,$$

$$||Fu-Fv||\le L||u-v||\forall u,v\in E(L\ge 0).$$

$$\{\begin{array}{l}\frac{du}{dt}=Fut\in [0,\infty )\\ u(0)=u_0.\end{array}$$

$$F(n)={\int }_{{\mathbb{Q}}_p^n}K(y,x-y)n(t,y)d^{n}y-k(x)n(t,x),$$

$$||k(\cdot )n(\cdot )|{|}_{L^1({\mathbb{Q}}_p^n)}$$

$$\le {\int }_{{\mathbb{Q}}_p^n}\underset{x\in {\mathbb{Q}}_p^n}{\sup }|k(x)||n(t,x)|d^{n}x=||k|{|}_{\infty }||n|{|}_{L^1({\mathbb{Q}}_p^n)},$$

$$||{\int }_{{\mathbb{Q}}_p^n}K(y,x-y)n(t,y)d^{n}y|{|}_{L^1({\mathbb{Q}}_p^n)}$$

$$\le {\int }_{{\mathbb{Q}}_p^n}{\int }_{{\mathbb{Q}}_p^n}K(y,x-y)|n(t,y)|d^{n}x{d}^{n}y$$

$$={\int }_{{\mathbb{Q}}_p^n}|n(t,y)|{\int }_{{\mathbb{Q}}_p^n}K(y,x-y)d^{n}x{d}^{n}y$$

$$={\int }_{{\mathbb{Q}}_p^n}|n(t,y)|k(x-y)d^{n}y$$

$$\le {\int }_{{\mathbb{Q}}_p^n}|n(t,y)|\underset{x\in {\mathbb{Q}}_p^n}{\sup }|k(x)|d^{n}y\le ||k|{|}_{\infty }||n|{|}_{L^1({\mathbb{Q}}_p^n)},$$

$$\frac{\partial }{\partial t}n(t,x)+k(x)n(t,x)={\int }_{{\mathbb{Q}}_p^n}K(y,x-y)n(t,y)d^{n}y,t\ge 0,x\in {\mathbb{Q}}_p^n,$$

$$K(y,z)\ge 0,k(y):={\int }_{{\mathbb{Q}}_p^n}K(y,z)d^{n}z\in L^{\infty }({\mathbb{Q}}_p^n).$$

$$n^0\ge 0\Rightarrow n\ge 0,$$

$${\int }_{{\mathbb{Q}}_p^n}n(t,x)d^{n}x={\int }_{{\mathbb{Q}}_p^n}{n}^0(x)d^{n}x\forall t\ge 0,$$

$${\int }_{{\mathbb{Q}}_p^n}|n(t,x)|d^{n}x\le {\int }_{{\mathbb{Q}}_p^n}|n^0(x)|d^{n}x\forall t\ge 0.$$

$$\frac{\partial }{\partial t}n(t,x)$$

$${\int }_{{\mathbb{Q}}_p^n}\frac{\partial }{\partial t}n(t,x)d^{n}x$$

$$\frac{d}{dt}{\int }_{{\mathbb{Q}}_p^n}n(t,x)d^{n}x$$

$$={\int }_{{\mathbb{Q}}_p^n}n(t,y){\int }_{{\mathbb{Q}}_p^n}K(y,x-y)d^{n}x{d}^{n}y$$

$$-{\int }_{{\mathbb{Q}}_p^n}n(t,x){\int }_{{\mathbb{Q}}_p^n}K(x,z)d^{n}z{d}^{n}x$$

$$={\int }_{{\mathbb{Q}}_p^n}n(t,y){\int }_{{\mathbb{Q}}_p^n}K(y,x-y)d^{n}x{d}^{n}y$$

$$-{\int }_{{\mathbb{Q}}_p^n}n(t,x){\int }_{{\mathbb{Q}}_p^n}K(x,z-x)d^{n}z{d}^{n}x=0.$$

$${\int }_{{\mathbb{Q}}_p^n}n(t,x)d^{n}x={\int }_{{\mathbb{Q}}_p^n}{n}^0(x)d^{n}x,\forall t\ge 0,$$

$$\frac{d}{dt}{H}_{\delta }(n(t,x))$$

$$=H_{\delta }^{\prime }(n(t,x))\left({\int }_{{\mathbb{Q}}_p^n}K(y,x-y)n(t,y)dy-k(x)n(t,x)\right)$$

$$\frac{d}{dt}{H}_{\delta }(n(t,x))$$

$$\le {\int }_{{\mathbb{Q}}_p^n}K(y,x-y)n_{+}(t,y)dy$$

$$\frac{\partial }{\partial t}{n}_{+}(t,x)+k(x)n_{+}(t,x)$$

$$\frac{d}{dt}{\int }_{{\mathbb{Q}}_p^n}{n}_{+}(t,x)d^{n}x$$

$$\le {\int }_{{\mathbb{Q}}_p^n}{n}_{+}(t,y){\int }_{{\mathbb{Q}}_p^n}K(y,x-y)d^{n}x{d}^{n}y-{\int }_{{\mathbb{Q}}_p^n}k(x)n_{+}(t,x)d^{n}x$$

$$\le {\int }_{{\mathbb{Q}}_p^n}k(y)n_{+}(t,y)d^{n}y-{\int }_{{\mathbb{Q}}_p^n}k(x)n_{+}(t,x)d^{n}x=0.$$

$${\int }_{{\mathbb{Q}}_p^n}{n}_{+}(t,x)d^{n}x\le {\int }_{{\mathbb{Q}}_p^n}{n}_{+}(0,x)d^{n}x={\int }_{{\mathbb{Q}}_p^n}{n}_{+}^0(x)d^{n}x.$$

$$n^0\ge 0\Rightarrow n\ge 0.$$

$${\int }_{{\mathbb{Q}}_p^n}|n(t,x)|d^{n}x$$

$$\le {\int }_{{\mathbb{Q}}_p^n}-n_0(t,x)d^{n}x+2{\int }_{{\mathbb{Q}}_p^n}{n}_{+}^0(t,x)d^{n}x$$

$$\le {\int }_{{\mathbb{Q}}_p^n}|n^0(t,x)|d^{n}x.$$

$$-\frac{\partial }{\partial t}\varphi (t,x)+k(x)\varphi (t,x)={\int }_{{\mathbb{Q}}_p^n}K(x,y-x)\varphi (t,y)d^{n}y,t\ge 0,x\in {\mathbb{Q}}_p^n.$$

$$k(x)N(x)={\int }_{{\mathbb{Q}}_p^n}K(y,x-y)N(y)d^{n}y$$

Full text

1 Preliminars

There is a general strategy to prove convergence of solutions to PDEs towards a steady state, as follows, see [7].

Suppose that we have an evolution equation with the form

[TABLE]

where $u:(0,\infty)\to B$, $B$ is some Banach space and $A:B\to B^{*}$ is some (nonlinear) mapping. In addition, we have some functional $H=H[u]$, which we call an entropy functional and a steady state $u_{\infty}$, i.e. a solution of $A(u)=0.$ The purpose here is to compute the entropy production, i.e. (minus) the time derivative of $H[u]$ and study its behavior along the solutions of the evolution equation. If we can find a relation between the entropy production and the entropy itself, then Gronwall’s lemma let us conclude that $H[u(t)]\to 0$ with some exponential rate as $t\to\infty$.

This technique is very common in the real case, and it works for some equations, among others, the Lotka-Volterra systems, Fokker-Planck equation, the scattering equation, and other some parabolic equations, which have several applications in biology, for example in systems related to population biology, ecological interactions, prey-predator systems, etc, see for example [4], [5].

Theorem 1.1** (Th. VII.3 (Cauchy, Lipschitz, Picard)).**

Let $E$ be a Banach space and $F:E\to E$ be a map such that

[TABLE]

Then for all $u_{0}\in E$ there exists a unique $u\in C^{1}([0,\infty);E)$ such that

[TABLE]

The proof of this result can be found in [2], and it is based on the Banach fixed point theorem.

Lets consider $E=L^{1}(\mathbb{Q}_{p}^{n})$ and $F:L^{1}(\mathbb{Q}_{p}^{n})\to L^{1}(\mathbb{Q}_{p}^{n})$, defined by

[TABLE]

where $0\leq K(y,z)\in L^{1}\cap L^{\infty}(\mathbb{Q}_{p}^{2n})$ and $0\leq k(y):=\int_{\mathbb{Q}_{p}^{n}}K(y,z)d^{n}z\in L^{\infty}(\mathbb{Q}_{p}^{n}).$ We do not make special assumption on the symmetry of the cross-section $K(y,z)$ motivated by turning kernels that appear in some applications as bacterial movement. We also suppose that $k$ and $K$ are independent of the time.

The operator $F$ is Lipschitz continuous, in fact:

[TABLE]

[TABLE]

then $||F(n)||_{L^{1}(\mathbb{Q}_{p}^{n})}\leq 2||k||_{\infty}||n||_{L^{1}(\mathbb{Q}_{p}^{n})}\leq C||n||_{L^{1}(\mathbb{Q}_{p}^{n})}.$

2 Existence of solutions of the Scattering equation

The following results are the $p$-adic analogs to the classical given in [4]. The scattering equation has the form

[TABLE]

the initial data $n^{0}\in L^{1}(\mathbb{Q}_{p}^{n})$, and we assume that

[TABLE]

Lemma 2.1**.**

The problem (2.1)-(2.2) has a unique solution $n\in C^{1}([0,\infty),L^{1}(\mathbb{Q}_{p}^{n}))$, and satisfies the following properties:

[TABLE]

[TABLE]

[TABLE]

Proof.

According to Theorem 1.1 and since the operator defined in (1.2) is Lipschitz continuous, the equation (2.1) has a unique solution $n\in C^{1}([0,\infty),L^{1}(\mathbb{Q}_{p}^{n}))$. In order to prove the properties, we write

[TABLE]

By using the Lebesgue dominated convergence theorem and Fubini’s theorem

[TABLE]

It means that $\int_{\mathbb{Q}_{p}^{n}}n(t,x)d^{n}x$ does not depend on $t$, therefore we obtain (2.4)

[TABLE]

which is known like mass conservation law.

In order to prove (2.5), consider $H_{\delta}(\cdot)$ a family of smooth, no-decreasing and convex functions such that $H_{\delta}^{\prime}(\cdot)\leq 1$ and $H_{\delta}(\cdot)\nearrow H(\cdot)=sgn_{+}(\cdot)$. Such function $H$ is known as entropy, and we have to calculate the entropy production $P[n]$ defined as $P[n]=-\frac{d}{dt}H[n]$, for definitions see [7].

[TABLE]

Taking limit as $\delta\to 0$

[TABLE]

and integrating (using Lebesgue Dominated Convergence Theorem)

[TABLE]

So, we have $P[n_{+}]\geq 0$. Also, since $\frac{d}{dt}\int_{\mathbb{Q}_{p}^{n}}n_{+}(t,x)d^{n}x\leq 0$ (the function is decreasing) we have for $t\geq 0$

[TABLE]

If $n^{0}$ is non-positive, then $n_{+}^{0}(x)=0$ and $\int_{\mathbb{Q}_{p}^{n}}n_{+}^{0}(x)d^{n}x=0$, and by equation (2.6) $\int_{\mathbb{Q}_{p}^{n}}n_{+}(t,x)d^{n}x\leq 0$ therefore $n_{+}(t,x)=0$ a.e., which implies $n(t,x)\leq 0$ a.e.. It means that when the initial data $n^{0}$ is non-positive, the solution $n$ is also non-positive, applying this to $-n^{0}$ we conclude that

[TABLE]

Since $-n+2n_{+}=|n|$ we arrive to (2.5)

[TABLE]

∎

Properties (2.3), (2.4) and (2.5) are similar to the ones given for the Fokker-Planck equation.

3 The relative entropy

In order to prove the main theorem of this section we have to consider the dual problem to (2.1), which can be written as

[TABLE]

We assume that there are solutions $N(x)>0$ and $\phi(x)>0$ to the primal equation (2.1) and the dual equation (3.1) respectively, namely

[TABLE]

[TABLE]

and we suppose here that these two solutions are independent of the time. These two steady state solutions allow us to derive the general relative entropy inequality.

Example 3.1** (Projection operator).**

Lets consider $N(x)>0$ and choose a weight $\bar{g}$ satisfying $\int_{\mathbb{Q}_{p}^{n}}\bar{g}(y)N(y)d^{n}y=1$ and take $K$ as

[TABLE]

Then

[TABLE]

Example 3.2**.**

Lets consider $N(x)>0$ with $\int_{\mathbb{Q}_{p}^{n}}N(x)d^{n}x=1$ and a symmetric kernel $\tilde{K}(x,y)=\tilde{K}(y,x)>0$, and

[TABLE]

Then

[TABLE]

Example 3.3**.**

Suppose that there exists a function $N(x)>0$ such that the scattering cross-section satisfies the symmetry condition (usually called detailed balance or micro-reversibility)

[TABLE]

Since $k(y)=\int_{\mathbb{Q}_{p}^{n}}K(x,y)d^{n}x$ we have that

[TABLE]

In this example the solution to the dual equation (3.3) is $\phi(x)=1.$

Lemma 3.4** (General Relative Entropy for Scattering equation).**

Let $N(x)$ and $n(t,x)$ be solutions to the primal equation (2.1), and let $\phi(x)$ be a solution to the dual equation (3.1). For any function $H:\mathbb{R}\to\mathbb{R}$ we have

[TABLE]

and

[TABLE]

Proof.

Following the proof given in [5], for the scattering equation (2.1) we calculate the entropy production.

[TABLE]

therefore

[TABLE]

After integration in $x$ we have

[TABLE]

Since the integral

[TABLE]

∎

Theorem 3.5**.**

In the conditions of the previous lemma, and for $\phi(x)=1$ we have for any convex function $H:\mathbb{R}\to\mathbb{R}$ there holds

[TABLE]

and

[TABLE]

Therefore

[TABLE]

Proof.

We easily can check that $\phi(x)=1$ is a solution of the dual equation (3.1) (that means the primal equation is conservative), then equation (3.3) correspond to equation (2.2) and we obtain the result.

Finally, since the function $H:\mathbb{R}\to\mathbb{R}$ is convex, we have that

[TABLE]

Therefore, it leads to

[TABLE]

which shows that

[TABLE]

∎

In other words we have obtained that

[TABLE]

Up to our knowledge, this entropy principle is only known in conservative cases.

Corollary 3.6**.**

Assume that $n^{0}(x)\leq C^{0}N(x)$, then for all $t\geq 0$,

[TABLE]

Proof.

Assume $\frac{n^{0}(x)}{N(x)}\leq C^{0}$ for some constant $C^{0}$, and suppose that for some $t>0$ we have $\frac{n(t,x)}{N(x)}>C^{0}$. Since the function $\mathcal{H}(n|N)(t)$ is decreasing, we have that

[TABLE]

or

[TABLE]

for any convex function. In particular, if we take $H(u)=u$ we obtain

[TABLE]

which is a contradiction. We conclude that $\frac{n(t,x)}{N(x)}\leq C^{0}$ for all $t>0.$ ∎

4 Exponential time decay

In this section we wonder if the solutions to the system (2.1)-(2.2) converge as $t\to\infty$, and if so, how fast? Before to write the main result of this section, we need to define the steady state of the system. More precisely, we want to show that if we put $\phi=1$ in equation (4.1), the solutions $n(t,x)\to\rho N(x)$ as $t\to\infty$ in some sense.

In order to define such $\rho$ we put $H(u)=u$ in the second part of Lemma 3.4, then we obtain $\frac{d}{dt}\int_{\mathbb{Q}_{p}^{n}}\phi(x)n(t,x)d^{n}x=0$ which means that $\int_{\mathbb{Q}_{p}^{n}}\phi(x)n(t,x)d^{n}x$ is constant for $t$, i.e.

[TABLE]

On the other hand, it is easy to see that $h(t,x):=n(t,x)-\rho N(x)$ is a solution of the system (2.1)-(2.2), provided $n(t,x)$ is a solution, in fact:

[TABLE]

When we think about long time convergence, a control of entropy by entropy dissipation is useful for exponential convergence as $t\to\infty$. The following result can be seen as a Poincaré inequality.

Lemma 4.1** (Analog to Lemma 6.2 in [5]).**

Given $\phi(x)>0,\ N(x)>0,\ K(y,x-y)>0$, there exists a constant $\alpha>0$ such that for all test function $m(x)$ satisfying

[TABLE]

we have

[TABLE]

Proof.

When the integral in the right side vanishes the result holds, so we can suppose that integral is non-zero and we normalize it, i.e., we suppose that $\int_{\mathbb{Q}_{p}^{n}}\phi(x)N(x)\left(\frac{m(x)}{N(x)}\right)^{2}d^{n}x=1.$ Then we argue by contradiction. If such an $\alpha$ does not exist, we can find a sequence of test functions $m_{k}(x)$ such that

[TABLE]

and

[TABLE]

Consider the collection of test functions $\mathcal{F}=\{m_{k}(x)\}$. Since each one of these functions has compact support, we can assume that all the functions have support in a ball $T$.

• •

This collection is uniformly equicontinuous on $T$, it means that for every $\epsilon>0$, there exists a $\delta>0$ such that

[TABLE]

In fact, for $\epsilon>0$ and $m_{k}$ we can choose $\delta_{k}$ as the parameter of constancy of $m_{k}$, and we have that $|m_{k}(s)-m_{k}(q)|=0$ for $||s-q||<\delta_{k}$. Then we put $\delta:=\inf\delta_{k}>0$ and the inequality holds.

• •

The collection $\mathcal{F}$ is pointwise bounded, i.e. $\sup_{m_{k}\in\mathcal{F}}|m_{k}(s)|<\infty.$

Then, by using the Arzelá-Ascoli theorem, we conclude that $\mathcal{F}\subset C(T,\mathbb{R})$ has a convergent subsequence in the supremum norm. After the extraction of the subsequence, we may pass to the limit $m_{\epsilon}\to\bar{m}$ and this function satisfies

[TABLE]

and

[TABLE]

From the last line, we conclude that $\frac{\bar{m}(x)}{N(x)}=\frac{\bar{m}(y)}{N(y)}:=\nu$. Since

[TABLE]

then $\nu=0$, in other words $\bar{m}=0$ which contradicts the normalization and thus such an $\alpha$ should exist. ∎

Remark 4.2**.**

To this point we do not have big differences with classical case, we just adapt the techniques to $p$-adic case, but we have some observations.

Lemma 4.1 is analog to Lemma 6.2 in **[5]**, but we need to assume here that functions $m(x)$ are test functions, i.e. $m(x)\in\mathcal{D}(\mathbb{Q}_{p}^{n})$, otherwise we cannot extract the sub-sequence. 2. 2.

In order to apply the previous lemma to function $h(t,x)$ it is convenient to normalize some functions, more precisely we need to assume that

[TABLE]

With the second condition we obtain that $\int_{\mathbb{Q}_{p}^{n}}\phi(x)h(t,x)d^{n}x=0$, in fact

[TABLE] 3. 3.

*In the case $\phi(x)=1$, we can take the function $h(t,x)\in\mathcal{L}(\mathbb{Q}_{p}^{n})$, the $p$-adic Lizorkin space of test functions of the second kind. Thus we can apply the lemma without assuming the normalization condition $\int_{\mathbb{Q}_{p}^{n}}N(x)d^{n}x=1$. *

Proposition 4.3**.**

For the solutions of the system (2.1)-(2.2) we have,

• (i)

$\rho:=\int_{\mathbb{Q}_{p}^{n}}\phi(x)n^{0}(x)d^{n}x=\int_{\mathbb{Q}_{p}^{n}}\phi(x)n(t,x)d^{n}x.$**

• (ii)

$\int_{\mathbb{Q}_{p}^{n}}\phi(x)|n(t,x)|d^{n}x\leq\int_{\mathbb{Q}_{p}^{n}}\phi(x)|n^{0}(x)|d^{n}x$* for all $t>0.$*

• (iii)

If $C^{1}N(x)\leq n^{0}(x)\leq C^{0}N(x)$, then $C^{1}N(x)\leq n(t,x)\leq C^{0}N(x)$ for all $t>0.$

• (iv)

If $n(t,x)$ and $N(x)$ are test functions, there exists a constant $\alpha>0$ such that

[TABLE]

Proof.

For (i) and (ii) we just choose the convex functions $H_{1}(u)=u$ and $H_{2}(u)=|u|$ respectively, in Lemma 3.4. For (iii) we choose $H_{3}(u)=(u-C^{0})_{+}^{2}$ for the upper bound, and $H_{4}(u)=(C_{1}-u)_{+}^{2}$ for the lower bound. In order to prove the exponential time decay (4.2) we choose the convex function $H(u)=u^{2}$ in the second part of Lemma (3.4), therefore

[TABLE]

then, by using Lemma 4.1 we conclude that there exists $\alpha>0$ such that

[TABLE]

By using the Gronwall’s Lemma we conclude that

[TABLE]

or

[TABLE]

∎

The last part of the previous proposition says that the solutions to the system converge as $t\to\infty$ with an exponential rate, i.e.

[TABLE]

5 Time dependent coefficients

The above manipulations are also valid for time dependent coefficients. More precisely, when we consider the problem

[TABLE]

with the initial data $n^{0}\in L^{1}(\mathbb{Q}_{p}^{n})$, and $K(t,y,z)\geq 0,$ $k(t,y):=\int_{\mathbb{Q}_{p}^{n}}K(t,y,z)d^{n}z\in L^{\infty}(\mathbb{Q}_{p}^{n})$ and the dual equation

[TABLE]

we have the following entropy inequality.

Lemma 5.1**.**

Let $N(t,x)$ and $n(t,x)$ be solutions to the primal equation (5.1), and let $\phi(t,x)$ be a solution to the dual equation (5.2). For any convex function $H:\mathbb{R}\to\mathbb{R}$ we have

[TABLE]

6 Hyperbolic Rescaling

We assume that scattering occurs with small changes and a fast rate (in other words, we change the time scale according to the size of jumps), then we have the following problem

[TABLE]

Consider a collection of test functions $\mathcal{F}=\{n_{\varepsilon}(t,x)\}$ which are solutions of (6.1), then we can extract a sub-sequence that converges, in some sense, to a solution $n$ of (2.1).

To do that, suppose that we have a function $\bar{N}\in L^{1}(\mathbb{Q}_{p}^{n})\cap L^{\infty}(\mathbb{Q}_{p}^{n})$ such that for the initial data and the steady states $N_{\varepsilon}$ undergo uniform control

[TABLE]

This implies the same control for all $t\geq 0$

[TABLE]

By using the same argue as in Lemma 4.1, we extract a convergent sub-sequence in the supremum norm i.e. $\sup_{x\in\mathbb{Q}_{p}^{n}}|n_{\varepsilon}(t,x)-n(t,x)|\to 0$ for all $t\geq 0$. Then for $\phi(t,x)\in L^{1}(\mathbb{R}^{+}\times\mathbb{Q}_{p}^{n})$

[TABLE]

in other words

[TABLE]

which means that

[TABLE]

We will show now that, under some regularity assumptions for $K$, we can derive stronger bounds than $L^{1}$ for solutions of (6.1).

Proposition 6.1**.**

Consider $n_{\varepsilon}$ a solution of (6.1), and assume (2.2) and

[TABLE]

therefore for all $t\geq 0$

[TABLE]

Proof.

We multiply (6.1) by $n_{\varepsilon}(t,x)$

[TABLE]

and integrate with respect to $x$

[TABLE]

After the change of variables $z=\frac{x-y}{|\varepsilon|_{p}}$ we have

[TABLE]

Since $\int_{\mathbb{Q}_{p}^{n}}K(x-|\varepsilon|_{p}z,z)n_{\varepsilon}(t,x-|\varepsilon|_{p}z)^{2}d^{n}x=\int_{\mathbb{Q}_{p}^{n}}K(x,z)n_{\varepsilon}(t,x)^{2}d^{n}x$ it holds

[TABLE]

By Gronwall’s lemma we can conclude that

[TABLE]

in other words

[TABLE]

∎

7 Stability

Theorem 7.1** (Analog to Smoller, Theorem 16.1 (c), p. 266).**

The solution $n(t,x)$ is stable in the following sense: If $n_{0},v_{0}\in L^{1}(\mathbb{Q}_{p}^{n})$ and $v$ is the corresponding constructed solution of (2.1) with initial data $v_{0}$, then

[TABLE]

Proof.

It is clear that if $n(t,x)$ and $v(t,x)$ are solutions of (2.1), then $n(t,x)-v(t,x)$ is also a solution. In addition, if we take $H(u)=|u|$ in Theorem (3.5) and we apply it to $n(t,x)-v(t,x)$ we have that

[TABLE]

in other words the function $G(t)=\int_{\mathbb{Q}_{p}^{n}}\left|n(t,x)-v(t,x)\right|d^{n}x$ is decreasing, thus for $t>0$ we have $G(t)\leq G(0)$, or

[TABLE]

∎

Observe that the entropy inequality lead us to uniqueness.