Decay property of solutions to the wave equation with space-dependent damping, absorbing nonlinearity, and polynomially decaying data

TL;DR

This paper investigates how space-dependent damping, absorbing nonlinearity, and initial data decay influence the long-term decay behavior of solutions to the semilinear wave equation in unbounded domains.

Contribution

It provides a detailed analysis of the decay rates of solutions based on damping strength, nonlinearity, and initial data decay, extending understanding of wave equations with complex damping and nonlinear effects.

Findings

Decay rates depend on damping coefficient and nonlinearity power

Initial data decay influences solution energy decay

Results apply to whole space and exterior domain problems

Abstract

We study the large time behavior of solutions to the semilinear wave equation with space-dependent damping and absorbing nonlinearity in the whole space or exterior domains. Our result shows how the amplitude of the damping coefficient, the power of the nonlinearity, and the decay rate of the initial data at the spatial infinity determine the decay rates of the energy and the $L^2$-norm of the solution. In Appendix, we also give a survey of basic results on the local and global existence of solutions and the properties of weight functions used in the energy method.