Critical exponent for the semilinear wave equation with scale invariant damping

TL;DR

This paper investigates the critical exponent for the semilinear damped wave equation with scale-invariant damping, establishing conditions for global existence and blow-up depending on the nonlinearity power and coefficient size.

Contribution

It provides new results on the critical exponent and solution behavior for scale-invariant damped wave equations with time-dependent coefficients.

Findings

Global solutions exist for nonlinearity power above Fujita exponent with large coefficients.

Blow-up can occur even with small coefficients under certain conditions.

Asymptotic behavior is highly sensitive to coefficient size and nonlinearity power.

Abstract

In this paper we consider the critical exponent problem for the semilinear damped wave equation with time-dependent coefficients. We treat the scale invariant cases. In this case the asymptotic behavior of the solution is very delicate and the size of coefficient plays an essential role. We shall prove that if the power of the nonlinearity is greater than the Fujita exponent, then there exists a unique global solution with small data, provided that the size of the coefficient is sufficiently large. We shall also prove some blow-up results even in the case that the coefficient is sufficiently small.