The Cauchy problem for the nonlinear damped wave equation with slowly decaying data

TL;DR

This paper investigates the nonlinear damped wave equation with slowly decaying initial data, establishing well-posedness, asymptotic behavior, and blow-up conditions, including the critical exponent across dimensions.

Contribution

It provides new results on well-posedness, diffusion phenomena, and blow-up for the nonlinear damped wave equation with slowly decaying data, and determines the critical exponent.

Findings

Large data local well-posedness established.

Small data global well-posedness proven.

Asymptotic profile matches a parabolic solution.

Abstract

We study the Cauchy problem for the nonlinear damped wave equation and establish the large data local well-posedness and small data global well-posedness with slowly decaying initial data. We also prove that the asymptotic profile of the global solution is given by a solution of the corresponding parabolic problem, which shows that the solution of the damped wave equation has the diffusion phenomena. Moreover, we show blow-up of solution and give the estimate of the lifespan for a subcritical nonlinearity. In particular, we determine the critical exponent for any space dimension.