Critical curve for a weakly coupled system of semi-linear $\sigma$-evolution equations with different damping types

TL;DR

This paper investigates the critical exponents for a coupled system of semi-linear sigma-evolution equations with different damping types, establishing conditions for global existence and finite-time blow-up of solutions.

Contribution

It introduces a new analysis of the critical curve for coupled sigma-evolution equations with mixed damping, extending previous results to fractional sigma and diverse damping mechanisms.

Findings

Identified the critical curve between nonlinear exponents p and q.

Proved global well-posedness for small data solutions.

Established finite-time blow-up and lifespan estimates for solutions.

Abstract

In this paper, we would like to consider the Cauchy problem for a weakly coupled system of semi linear $sigma$ evolution equations with different damping mechanisms for any $\sigma>1$, parabolic like damping and $\sigma$ evolution like damping. Motivated strongly by the well known Nakao's problem, the main goal of this work is to determine the critical curve between the power exponents $p$ and $q$ of nonlinear terms by not only establishing the global well posedness property of small data solutions but also indicating blow up in finite time solutions. We want to point out that the application of a modified test function associated with a judicious choice of test functions really plays an essential role to show a blow up result for solutions and upper bound estimates for lifespan of solutions, where $\sigma$ is assumed to be any fractional number. To end this paper, lower bound estimates for lifespan of solutions are also shown to verify their sharp results in some spatial dimensions