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

TL;DR

This paper investigates the wave equation with variable coefficients, boundary damping, and supercritical source terms, establishing conditions for existence, decay, and blow-up of solutions based on initial energy and damping growth.

Contribution

It provides new results on local and global existence, energy decay rates, and blow-up phenomena for wave equations with complex boundary and source conditions.

Findings

Proved local and global existence of solutions.

Classified decay rates depending on damping growth.

Established blow-up conditions for solutions with positive and nonpositive initial energy.

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.