Global existence for wave equations with scale-invariant time-dependent damping and time derivative nonlinearity

TL;DR

This paper establishes the global existence of small data solutions for wave equations with scale-invariant time-dependent damping and nonlinear time derivatives, identifying the critical exponent range in one dimension for the first time.

Contribution

It provides the first known determination of the critical exponent range for such damped wave equations with time-derivative nonlinearities, especially in one dimension.

Findings

Global existence for 1≤n≤3 in low dimensions.

Critical exponent for 1D case identified as p>1+2/μ.

First to determine the critical exponent range for these equations.

Abstract

This paper addresses the Cauchy problem for wave equations with scale-invariant time-dependent damping and nonlinear time-derivative terms, modeled as $$\partial_{t}^2u- \Delta u +\frac{\mu}{1+t}\partial_tu= f(\partial_tu), \quad x\in \mathbb{R}^n, t>0,$$ where $f(\partial_tu)=|\partial_tu|^p $ or $|\partial_tu|^{p-1}\partial_tu$ with $p>1$ and $\mu>0$. We prove global existence of small data solutions in low dimensions $1\leq n\leq 3$ by using energy estimates in appropriate Sobolev spaces. Our primary contribution is an existence result for $p>1+\frac2{\mu}$, in the one-dimensional case, when $\mu \le 2$, which in conjunction with prior blow-up results from \cite{Our2}, establish that the critical exponent for small data solutions in one dimension is $p_G(1,\mu)=1+\frac2{\mu}$, when $\mu \le 2$. To the best of our knowledge, this is the first identification of the critical exponent range for the time-dependent damped wave equations with scale-invariant and time-derivative nonlinearity.