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Global existence and blow-up of solutions to the double nonlinear porous medium equation

Bolys Sabitbek, Berikbol Torebek

Abstract

In this study, we examine a double nonlinear porous medium equation subject to a novel nonlinearity condition within a bounded domain. First, we introduce the blow-up solution for the problem under consideration for the negative initial energy. By introducing a set of potential wells, we construct invariant sets of solutions for the double nonlinear porous medium equation. For subcritical and critical initial energy scenarios, we derive the global existence and asymptotic behavior of weak solutions, as well as blow-up phenomena occurring within a finite time for the positive solution to the double nonlinear porous medium equation.

Global existence and blow-up of solutions to the double nonlinear porous medium equation

Abstract

In this study, we examine a double nonlinear porous medium equation subject to a novel nonlinearity condition within a bounded domain. First, we introduce the blow-up solution for the problem under consideration for the negative initial energy. By introducing a set of potential wells, we construct invariant sets of solutions for the double nonlinear porous medium equation. For subcritical and critical initial energy scenarios, we derive the global existence and asymptotic behavior of weak solutions, as well as blow-up phenomena occurring within a finite time for the positive solution to the double nonlinear porous medium equation.

Paper Structure

This paper contains 15 sections, 18 theorems, 156 equations.

Table of Contents

  1. Introduction
  2. Setting problem
  3. Literature overview
  4. Main objectives of the paper
  5. Blow-up results for $J(u_0)<0$
  6. Family of Potential Wells
  7. Low initial energy $J(u_0)<d$
  8. Invariant sets of solutions
  9. Global existence of weak solutions
  10. Asymptotic behavior of weak solutions
  11. Finite time blow-up for $J(u_0)<d$
  12. Critical initial energy $J(u_0)=d$
  13. Global existence for $J(u_0)=d$
  14. Asymptotic behavior of solutions
  15. Finite time blow-up for $J(u_0)=d$

Key Result

Theorem 2.1

Let $\Omega$ be a bounded domain of $\mathbb{R}^n$ with smooth boundary $\partial \Omega$. Let a function $f$ satisfy where $m\geq 1$ and $\sigma >0$, and $0<\beta\leq \frac{\lambda_{1,p}(\alpha - p)}{p}$. If $u_0^m \in L^{\infty}(\Omega)\cap W_0^{1,p}(\Omega)$ satisfies the inequality then there cannot exist a positive solution $u$ of (main_eqn_p>2) existing for all times $T^*$ and a positive

Theorems & Definitions (34)

  • Definition 1.1
  • Theorem 2.1
  • Lemma 2.2: Theorem 1.1, KL06
  • proof : Proof of Theorem \ref{['thm_p>2']}
  • Lemma 3.1
  • proof : Proof of Lemma \ref{['lem-1']}
  • Lemma 3.2
  • proof : Proof of Lemma \ref{['lem-2']}
  • Lemma 3.3
  • proof : Proof of Lemma \ref{['lem-3']}
  • ...and 24 more