Global existence and uniqueness of the solution to a nonlinear parabolic equation

TL;DR

This paper proves the global existence and uniqueness of solutions for a nonlinear parabolic PDE with a power-type nonlinearity in three-dimensional space, ensuring solutions remain bounded over time.

Contribution

It establishes the first rigorous proof of global well-posedness and boundedness for this class of nonlinear parabolic equations with smooth decaying initial data.

Findings

Existence of a unique global solution for the PDE.

Solutions are uniformly bounded independently of space and time.

The solution's norm is controlled by a constant not depending on initial data specifics.

Abstract

Consider the equation $$ u'(t)-\Delta u+|u|^\rho u=0, \quad u(0)=u_0(x), (1), $$ where $ u':=\frac {du}{dt}$, $ \rho=const >0, $ $x\in \mathbb{R}^3$, $t>0$. Assume that $u_0$ is a smooth and decaying function, $$\|u_0\|\:=\sup_{x\in \mathbb{R}^3, t\in \mathbb{R}_+} |u(x,t)|.$$ It is proved that problem (1) has a unique global solution and this solution satisfies the following estimate $$\|u(x,t)\|<c, $$ where $c>0$ does not depend on $x,t$.

Equations

$$u^{\prime }(t)-\Delta u+|u{|}^{\rho }u=0,u(0)=u_0(x),(1),$$

$$\Vert u_0\Vert =\underset{x\in {\mathbb{R}}^3,t\in {\mathbb{R}}_{+}}{\sup }|u(x,t)|.$$

$$\Vert u(x,t)\Vert <c,$$

$$u^{\prime }-\Delta u+|u{|}^{\rho }u=0,u(0)=u_0;u^{\prime }:=\frac{du}{dt},$$

$$\Vert u\Vert \le c.$$

$$\begin{array}{r}u(t)=-{\int }_0^{t}d\tau \int g(x-y,t-\tau )|u{|}^{\rho }udy+\\ \int g(x-y,t)u_0(y)dy:=A(u)+F:=Q(u),\end{array}$$

$$\Vert u\Vert \le T\Vert u{\Vert }^{\rho +1}+\Vert F\Vert .$$

$$T{R}^{\rho +1}+\Vert F\Vert \le R.$$

$$\Vert Q(u)-Q(v)\Vert \le T(\rho +1)R^{\rho }\Vert u-v\Vert \le q\Vert u-v\Vert ,0<q<1.$$

$$T(\rho +1)R^{\rho }\le q<1,$$

$$0.5\frac{dN(u)}{dt}+N(gradu)+\int |u{|}^{\rho +2}dy=0,$$

$$0.5N(u(T))+{\int }_0^T\left(N(gradu)+\int |u{|}^{\rho +2}dy\right)d\tau =0.5N(u(0)).$$

$$N(u(t))\le c,\forall t\in [0,T],{\int }_0^{T}N(gradu)d\tau \le c,{\int }_0^{T}d\tau \int |u{|}^{\rho +2}dy\le c,$$

$$\Vert u(x,t)\Vert <\infty \forall t\in [0,T].$$

$$\begin{array}{r}{\int }_0^{T}d\tau \int g(x-y,t-\tau )|u{|}^{\rho +1}dy\le \\ {\left({\int }_0^{T}d\tau \int |u{|}^{\rho +2}dy\right)}^{(\rho +1)/(\rho +2)}{\left({\int }_0^{T}d\tau \int g^{\rho +2}dy\right)}^{1/(\rho +2)}\le \\ {\left({\int }_0^{T}d\tau \int |u{|}^{\rho +2}dy\right)}^{(\rho +1)/(\rho +2)}{\left({\int }_0^{T}d\tau \int g^{\rho +2}dy\right)}^{1/(\rho +2)}.\end{array}$$

Full text

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where $u^{\prime}:=\frac{du}{dt}$, $\rho=const>0,$ $x\in\mathbb{R}^{3}$, $t>0$.

Assume that $u_{0}$ is a smooth and decaying function,

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It is proved that problem (1) has a unique global solution and this solution satisfies the following estimate

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where $c>0$ does not depend on $x,t$.

Global existence and uniqueness of the solution

to a nonlinear parabolic equation

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Alexander G. Ramm

Mathematics Department, Kansas State University,

Manhattan, KS 66506-2602, USA

††footnotetext: Corresponding author: Email: ramm@ksu.edu

Mathematics Subject Classification: MSC 2010,

35K55.

Keywords: nonlinear parabolic equations; global solutions.

1 Introduction

Let

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where $\rho>0$, $t\in\mathbb{R}_{+}=[0,\infty)$, $x\in\mathbb{R}^{3}$, $X$ is a Banach space of real-valued functions with the norm $\|u(x,t)\|:=\sup_{x\in\mathbb{R}^{3},t\in R_{+}}|u(x,t)|$. We assume that

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We say that $u$ is a global solution to (1) if $u$ exists $\forall t\geq 0$.

Our result is formulated in Theorem 1. Our method is simple and differs from the published results, see [1], [2] and references there.

The novel points in this work are:

a) There is no restriction on the upper bound of $\rho$.

In [1], (Section 1.1) a nonlinear hyperbolic equation with the same non-linearity is studied in a bounded domain, uniqueness of the solution is proved only for $\rho\leq 2/(n-2)$, and existence is proved by a different method. The contraction mapping theorem is not used.

In [2] the quasi-linear problems for parabolic equations are studied in Chapter 5 in a bounded domain and under the assumptions different from ours. There are many papers and books on non-linear problems for parabolic equations (see the bibliography in [1], [2].

b) Existence of the global solution is proved.

c) Method of the proof differs from the methods in the cited literature.

Our result is formulated in Theorem 1:

Theorem 1. Problem (1) has a unique global solution in $X$ for any $u_{0}\in X$.

2 Proofs

Let $g(x,t)=\frac{e^{-|x|^{2}}}{(4\pi t)^{3/2}}$. If $u$ solves (1) then

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where $\int:=\int_{\mathbb{R}^{3}}$. Let $X$ be the Banach space of continuous in $\mathbb{R}^{3}\times R_{+}$ functions, $\mathbb{R}_{+}:=[0,\infty)$, $\|u\|:=\max_{x\in\mathbb{R}^{3},t\in[0,T]}|u(x,t)|$. If $\|u\|\leq R$ then $\|A(u)\|\leq TR^{\rho+1}$, where the identity $\int g(x-y,t-\tau)dy=1$ was used. From (3) one gets

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Thus, $Q$ maps the ball $B(R)=\{u:\|u\|\leq R\}$ into itself if $T$ is such that

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The $Q$ is a contraction on $B(R)$ if

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Thus, if

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then $Q$ is a contraction in $B(R)$ in the Banach space $X_{T}$ with the norm $\|\cdot\|$, $t\in[0,T]$. We use the same notations for the norms in $X_{T}$ and in $X_{\infty}$.

We have proved that

For $T$ satisfying (5)- (6) there exists and is unique the solution to (1), and this solution can be obtained from (3) by iterations.

The problem now is:

Does this solution exist and is unique on $R_{+}$?

From our proof it follows that if the solution exists and is unique in $X_{T}$, then the solution exists and is unique in $X_{T_{1}}$ for some $T_{1}>T$.

To prove that the solution $u(x,t)$ to (1) exists on $\mathbb{R}_{+}$, assume the contrary: this solution does not exist on any interval $[0,T_{1}),$ $T_{1}>T$, where $T$ is the maximal interval of the existence of the continuous solution. Then $\lim_{t\to T-0}u(x,t)=\infty$, because otherwise there is a sequence $t_{n}\to T-0$ such that $u(x,t_{n})\to u(x,T)$ and one may construct the solution defined on $[T,T_{1}]$, $T_{1}>T$, by using the local existence and uniqueness of the solution to (1) with the initial value $u(x,T)$ for $t\in[T,T_{1}]$. This contradicts the assumption that $T$ is the maximal interval of the existence of the continuous solution $u$.

Thus, if $T<\infty$ then one has $\lim_{t\to T-0}u(x,t)=\infty$. Let us prove that this also leads to a contradiction. Then we have to conclude that $T=\infty$ and Theorem 1 is proved.

We need some estimates. Multiply (1) by $u$, integrate over $\mathbb{R}^{3}$ with respect to $x$, and then integrate by parts the second term. The result is:

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where $N(u):=\int u^{2}dy$. Integrate (7) with respect to time over $[0,T]$ and get

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Therefore,

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where $c=0.5N(u_{0})$.

Lemma 1. From (9) and (3) it follows that

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If (10) is proved then $T$ is not the maximal interval of the existence of the solution to (1). This contradiction proves Theorem 1.

Proof of Lemma 1. One uses the Hölder inequality twice and gets

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By the last inequality (9) it follows that $\int_{0}^{T}d\tau\int|u|^{\rho+2}dy<c$ $\forall T>0$, where $c>0$ is a constant independent of $T$. The last integral in (11) is also bounded independently of $T$. It can be calculated analytically.

Thus, inequalities (11), (9) and equation (3) imply (10).

Lemma 1 is proved. $\Box$

Therefore Theorem 1 is proved. $\Box$

The ideas related to the ones used in this paper were developed and used in [3]–[5].