Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Global solutions and blow-up for the wave equation with variable coefficients: II. boundary supercritical source

Tae Gab Ha

Abstract

In this paper, we consider the wave equation with variable coefficients and boundary damping and supercritical source terms. The goal of this work is devoted to prove the local and global existence, and classify decay rate of energy depending on the growth near zero on the damping term. Moreover, we prove the blow-up of the weak solution with positive initial energy as well as nonpositive initial energy.

Global solutions and blow-up for the wave equation with variable coefficients: II. boundary supercritical source

Abstract

In this paper, we consider the wave equation with variable coefficients and boundary damping and supercritical source terms. The goal of this work is devoted to prove the local and global existence, and classify decay rate of energy depending on the growth near zero on the damping term. Moreover, we prove the blow-up of the weak solution with positive initial energy as well as nonpositive initial energy.
Paper Structure (15 sections, 241 equations, 3 figures)

This paper contains 15 sections, 241 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem 2.1 : local existence
  4. Globally Lipschitz source
  5. Locally Lipschitz source
  6. Completion of the proof for the local existence
  7. Energy identity
  8. Proof of Theorem 2.1 : global existence
  9. $\rho \geq \gamma$
  10. The potential well
  11. Proof of Theorem 2.2 : energy decay
  12. Case 1 : $\beta$ is linear.
  13. Case 2 : $\beta$ has polynomial growth near zero.
  14. Case 3 : $\beta$ does not necessarily have polynomial growth near zero.
  15. Proof of Theorem 2.3 : blow-up

Figures (3)

  • Figure 1: The admissible range of the damping parameter $\rho$ and the exponent of the source $\gamma$.
  • Figure 2: The admissible range of parameters $a$ and $b$ with respect to the trace imbedding $H^1_0(\Omega) \hookrightarrow L^{a\gamma + b}(\Gamma_1)$.
  • Figure 3: The figure of $P(\epsilon_8)$

Theorems & Definitions (5)

  • proof
  • proof
  • proof
  • proof
  • proof