Critical exponent for the semilinear wave equations with a damping increasing in the far field

TL;DR

This paper determines the critical exponent for semilinear wave equations with spatially increasing damping, showing it matches the exponent of the related parabolic equation, and proves global existence and blow-up results.

Contribution

It establishes the critical exponent for wave equations with increasing damping in the far field, extending understanding of damping effects on solution behavior.

Findings

Critical exponent is p = 1 + 2/(N - α).

Global existence proved using weighted energy estimates and Caffarelli-Kohn-Nirenberg inequality.

Blow-up results obtained via test-function method and lifespan estimates.

Abstract

We consider the Cauchy problem of the semilinear wave equation with a damping term \begin{align*} u_{tt} - \Delta u + c(t,x) u_t = |u|^p, \quad (t,x)\in (0,\infty)\times \mathbb{R}^N,\quad u(0,x) = \varepsilon u_0(x), \ u_t(0,x) = \varepsilon u_1(x), \quad x\in \mathbb{R}^N, \end{align*} where $p>1$ and the coefficient of the damping term has the form \begin{align*} c(t,x) = a_0 (1+|x|^2)^{-\alpha/2} (1+t)^{-\beta} \end{align*} with some $a_0 > 0$, $\alpha < 0$, $\beta \in (-1, 1]$. In particular, we mainly consider the cases $ \alpha < 0, \beta =0$ or $\alpha < 0, \beta = 1$, which imply $\alpha + \beta < 1$, namely, the damping is spatially increasing and effective. Our aim is to prove that the critical exponent is given by $ p = 1+ \frac{2}{N-\alpha}$. This shows that the critical exponent is the same as that of the corresponding parabolic equation $c(t,x) v_t - \Delta v = |v|^p$. The global existence part is proved by a weighted energy estimates with an exponential-type weight function and a special case of the Caffarelli-Kohn-Nirenberg inequality. The blow-up part is proved by a test-function method introduced by Ikeda and Sobajima (arXiv:1710.06780v1). We also give an upper estimate of the lifespan.