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Stability of the separable solutions for a nonlinear boundary diffusion problem

Tianling Jin, Jingang Xiong, Xuzhou Yang

Abstract

In this paper, we study a nonlinear boundary diffusion equation of porous medium type arising from a boundary control problem. We give a complete and sharp characterization of the asymptotic behavior of its solutions, and prove the stability of its separable solutions.

Stability of the separable solutions for a nonlinear boundary diffusion problem

Abstract

In this paper, we study a nonlinear boundary diffusion equation of porous medium type arising from a boundary control problem. We give a complete and sharp characterization of the asymptotic behavior of its solutions, and prove the stability of its separable solutions.
Paper Structure (8 sections, 26 theorems, 278 equations)

This paper contains 8 sections, 26 theorems, 278 equations.

Table of Contents

  1. Introduction
  2. Steady states and some quantitative elliptic estimates
  3. Existence, uniqueness, and infinite speed of propagation
  4. Extinction or blow up in finite time, and integral bounds
  5. Uniform upper and lower bounds
  6. Convergence
  7. Convergence to a steady state with rates
  8. Sharp rates and higher order asymptotics

Key Result

Theorem 1.1

Let $n\ge 2$, $\Omega \subset \mathbb{R}^n$ be a bounded smooth domain, $p$ satisfy 2 and $a\in C^\infty(\partial \Omega)$. Suppose that $u_0\in C^\infty(\partial \Omega)$ and $u_0$ is positive. Then 1--eq:initial admits a unique positive smooth solution $u$ on $\overline\Omega\times[0,T^*)$, where

Theorems & Definitions (50)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Proposition 2.1
  • Lemma 2.2: Local maximum principle
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4: Weak Harnack inequality
  • ...and 40 more