Sharp lifespan estimates for the weakly coupled system of semilinear damped wave equations in the critical case

TL;DR

This paper derives sharp lifespan estimates for solutions to the critical weakly coupled semilinear damped wave equations, using test function methods and Sobolev spaces, with applications to reaction-diffusion systems.

Contribution

It provides the first sharp upper and lower lifespan estimates in the critical case for these coupled equations, advancing understanding of their long-term behavior.

Findings

Established upper bound estimates for lifespan.

Derived lower bound estimates using polynomial-logarithmic Sobolev spaces.

Applied results to reaction-diffusion systems in the critical case.

Abstract

The open question, which seems to be also the final part, in terms of studying the Cauchy problem for the weakly coupled system of damped wave equations or reaction-diffusion equations, is so far known as the sharp lifespan estimates in the critical case. In this paper, we mainly investigate lifespan estimates for solutions to the weakly coupled system of semilinear damped wave equations in the critical case. By using a suitable test function method associated with nonlinear differential inequalities, we catch upper bound estimates for the lifespan. Moreover, we establish polynomial-logarithmic type time-weighted Sobolev spaces to obtain lower bound estimates for the lifespan in low spatial dimensions. Then, together with the derived lifespan estimates, new and sharp results on estimates for the lifespan in the critical case are claimed. Finally, we give an application of our results to the semilinear reaction-diffusion system in the critical case.