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Convergence of Solutions of the Porous Medium Equation with Reactions

Bendong Lou, Maolin Zhou

Abstract

Consider the Cauchy problem of one dimensional porous medium equation (PME) with reactions. We first prove a general convergence result, that is, any bounded global solution starting at a nonnegative compactly supported initial data converges as $t\to \infty$ to a nonnegative zero of the reaction term or a ground state stationary solution. Based on it, we give out a complete classification on the asymptotic behaviors of the solutions for PME with monostable, bistable and combustion types of nonlinearities.

Convergence of Solutions of the Porous Medium Equation with Reactions

Abstract

Consider the Cauchy problem of one dimensional porous medium equation (PME) with reactions. We first prove a general convergence result, that is, any bounded global solution starting at a nonnegative compactly supported initial data converges as to a nonnegative zero of the reaction term or a ground state stationary solution. Based on it, we give out a complete classification on the asymptotic behaviors of the solutions for PME with monostable, bistable and combustion types of nonlinearities.
Paper Structure (23 sections, 36 theorems, 370 equations, 5 figures)

This paper contains 23 sections, 36 theorems, 370 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Well-posedness and a priori estimates
  4. Waiting times of the free boundaries and compactly supported stationary solutions
  5. Lipschitz continuity, Darcy law and strict monotonicity
  6. Zero Number Argument and General Convergence Result
  7. Zero number and intersection number properties
  8. Monotonicity outside of the initial support
  9. General convergence
  10. Monostable RPME
  11. Stationary solutions
  12. Hair-trigger effect
  13. Combustion RPME
  14. Stationary solutions
  15. Asymptotic behavior of the solutions
  16. ...and 8 more sections

Key Result

Theorem 1.1

Assume (F) and (I). Let $u(x,t)$ be a bounded, nonnegative, time-global solution of (CP). Then $u(\cdot,t)$ converges as $t\to \infty$, in the sense of def-conv, to a stationary solution of (CP), which is one of the following types:

Figures (5)

  • Figure 1: An example of the initial data.
  • Figure 2: Type I ground state solution.
  • Figure 3: Type II ground state solution.
  • Figure 4: Vanishing of zero at $t=T$.
  • Figure 5: Generation of zero at $t=t_1$.

Theorems & Definitions (72)

  • Theorem 1.1: General convergence theorem
  • Theorem 1.2: Intersection number properties
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 62 more