Global existence for small amplitude semilinear wave equations with time-dependent scale-invariant damping

TL;DR

This paper proves a sharp global existence result for small amplitude solutions to semilinear wave equations with time-dependent scale-invariant damping, using a transformation to a generalized Tricomi equation and establishing weighted Strichartz estimates.

Contribution

It introduces a novel approach by transforming the wave equation into a generalized Tricomi form and derives new weighted Strichartz estimates for this class.

Findings

Global existence for $p_{crit}(n,ta)<pq p_{conf}(n,ta)$ in dimensions $n\u00a0 extgreatera03$.

Established weighted Strichartz estimates for generalized Tricomi equations.

Connected the analysis of damped wave equations to Tricomi-type equations for broader applicability.

Abstract

In this paper we prove a sharp global existence result for semilinear wave equations with time-dependent scale-invariant damping terms if the initial data is small. More specifically, we consider Cauchy problem of $\partial_t^2u-\Delta u+\frac{\mu}{t}\partial_tu=|u|^p$, where $n\ge 3$, $t\ge 1$ and $\mu\in(0,1)\cup(1,2)$. For critical exponent $p_{crit}(n,\mu)$ which is the positive root of $(n+\mu-1)p^2-(n+\mu+1)p-2=0$ and conformal exponent $p_{conf}(n,\mu)=\frac{n+\mu+3}{n+\mu-1}$, we establish global existence for $n\geq3$ and $p_{crit}(n,\mu)<p\leq p_{conf}(n,\mu)$. The proof is based on changing the wave equation 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, then we investigate more general semilinear Tricomi equation $\partial_t^2v-t^m\Delta v=t^{\alpha}|v|^p$ and establish related weighted Strichartz estimates. Returning to the original wave equation, the corresponding global existence results on the small data solution $u$ can be obtained.