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Global existence and blow-up of solutions to porous medium equation and pseudo-parabolic equation, I. Stratified Groups

Michael Ruzhansky, Bolys Sabitbek, Berikbol Torebek

Abstract

In this paper, we prove a global existence and blow-up of the positive solutions to the initial-boundary value problem of the nonlinear porous medium equation and the nonlinear pseudo-parabolic equation on the stratified Lie groups. Our proof is based on the concavity argument and the Poincaré inequality, established in [38] for stratified groups.

Global existence and blow-up of solutions to porous medium equation and pseudo-parabolic equation, I. Stratified Groups

Abstract

In this paper, we prove a global existence and blow-up of the positive solutions to the initial-boundary value problem of the nonlinear porous medium equation and the nonlinear pseudo-parabolic equation on the stratified Lie groups. Our proof is based on the concavity argument and the Poincaré inequality, established in [38] for stratified groups.

Paper Structure

This paper contains 7 sections, 4 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Nonlinear porous medium equation
  3. Blow-up solutions of the nonlinear porous medium equation
  4. Global existence for the nonlinear porous medium equation
  5. Nonlinear pseudo-parabolic equation
  6. Blow-up phenomena for the pseudo-parabolic equation
  7. Global solution for the pseudo-parabolic equation

Key Result

Theorem 2.1

Let $\mathbb{G}$ be a stratified group with $N_1$ being the dimension of the first stratum. Let $D \subset \mathbb{G}$ be an admissible domain. Let $2\leq p<\infty$ with $p\neq N_1$. Assume that function $f$ satisfies where for some where $R = \sup_{x \in D} |x'|$ and $x=(x',x")$ with $x'$ being in the first stratum. Let $u_0 \in L^{\infty}(D)\cap\mathring{S}^{1,p}(D)$ satisfy the inequality

Theorems & Definitions (10)

  • Definition 1.1
  • Theorem 2.1
  • Remark 2.2
  • proof : Proof of Theorem \ref{['thm_p>2']}
  • Theorem 2.3
  • proof : Proof of Theorem \ref{['thm_GEp']}
  • Theorem 3.1
  • proof : Proof of Theorem \ref{['thm_p>21']}
  • Theorem 3.2
  • proof : Proof of Theorem \ref{['thm_GEp1']}