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

Abstract

We consider the Cauchy problem of the semilinear wave equation with a damping term $$\begin{aligned} \left\{ \begin{array}{ll} u_{tt} - \Delta u + c(t,x) u_t = |u|^p,&{}(t,x)\in (0,\infty )\times {\mathbb {R}}^N,\\ u(0,x) = \varepsilon u_0(x), \quad u_t(0,x) = \varepsilon u_1(x),&{} x\in {\mathbb {R}}^N, \end{array}\right. \end{aligned}$$where \(p>1\) and the coefficient of the damping term has the form $$\begin{aligned} c(t,x) = a_0 (1+|x|^2)^{-\alpha /2} (1+t)^{-\beta } \end{aligned}$$with some \(a_0 > 0\), \(\alpha < 0\), \(\beta \in (-1, 1]\). In particular, we mainly consider the cases $$\begin{aligned} \alpha< 0, \beta =0 \quad \text{ or } \quad \alpha < 0, \beta = 1, \end{aligned}$$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 $$\begin{aligned} p = 1+ \frac{2}{N-\alpha }. \end{aligned}$$This shows that the critical exponent is the same as that of the corresponding parabolic equation $$\begin{aligned} c(t,x) v_t - \Delta v = |v|^p. \end{aligned}$$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 [15]. We also give an upper estimate of the lifespan.