Global existence of small data weak solutions to the semilinear wave equations with time-dependent scale-invariant damping

TL;DR

This paper proves the global existence of small data weak solutions for a class of semilinear wave equations with time-dependent damping, extending results to generalized Tricomi equations and identifying critical exponents for solution existence.

Contribution

It establishes new global existence results for small data solutions to semilinear wave equations with scale-invariant damping and generalizes these results to Tricomi-type equations with explicit conditions.

Findings

Global solutions exist for $eta o -2$ in higher dimensions.

Existence depends on the conformal exponent $p_{conf}$ and parameters $n,m,eta$.

Solutions are shown to exist under specific restrictions on $p$ and parameters.

Abstract

In this paper, we are concerned with the global existence of small data weak solutions to the $n-$dimensional semilinear wave equation $\partial_t^2u-\Delta u+\frac{\mu}{t}\partial_tu=|u|^p$ with time-dependent scale-invariant damping, where $n\geq 2$, $t\geq 1$, $\mu\in(0,1)\cup(1,2]$ and $p>1$. This equation can be changed into the semilinear generalized Tricomi equation $\partial_t^2u-t^m\Delta u=t^{\alpha(m)}|u|^p$, where $m=m(\mu)>0$ and $\alpha(m)\in\Bbb R$ are two suitable constants. At first, for the more general semilinear Tricomi equation $\partial_t^2v-t^m\Delta v=t^{\alpha}|v|^p$ with any fixed constant $m>0$ and arbitrary parameter $\alpha\in\Bbb R$, we shall show that in the case of $\alpha\leq -2$, $n\geq 3$ and $p>1$, the small data weak solution $v$ exists globally; in the case of $\alpha>-2$, through determining the conformal exponent $p_{conf}(n,m,\alpha)>1$, the global small data weak solution $v$ exists when some extra restrictions of $p\geq p_{conf}(n,m,\alpha)$ are given. Returning to the original equation $\partial_t^2u-\Delta u+\frac{\mu}{t}\partial_tu=|u|^p$, the corresponding global existence results on the small data solution $u$ can be obtained.