Modified different nonlinearities for weakly coupled systems of semilinear effectively damped waves with different time-dependent coefficients in the dissipation terms

TL;DR

This paper establishes the global existence of small data solutions for weakly coupled semilinear damped wave systems with time-dependent dissipation coefficients, extending understanding of such systems in various regularity classes.

Contribution

It introduces new results on global existence for coupled damped wave systems with time-varying dissipation, considering different nonlinearities and regularity classes.

Findings

Global existence of solutions for all space dimensions.

Solutions exist for small initial data.

Analysis covers multiple regularity classes.

Abstract

We prove the global existence of small data solution in all space dimension for weakly coupled systems of semi-linear effectively damped wave, with different time-dependent coefficients in the dissipation terms. Moreover, nonlinearity terms $ f(t,u) $ and $ g(t,v) $ satisfying some properties of the parabolic equation. We study the problem in several classes of regularity.