Global well-posedness for the semilinear wave equation with time dependent damping in the overdamping case

Abstract:

We study the global existence of solutions to the Cauchy problem for the wave equation with time-dependent damping and a power nonlinearity in the overdamping case: \begin{align*}\left \{\begin {array}{ll} u_{tt} - \Delta u + b(t) u_t = N(u),&t\in [0,T),\ x\in \mathbb {R}^d,\\ u(0) = u_0,\ u_t(0) = u_1,&x\in \mathbb {R}^d. \end{array}\right . \end{align*} Here, $b(t)$ is a positive $C^1$-function on $[0,\infty )$ satisfying \[ b(t)^{-1} \in L^1(0,\infty ), \]whose case is called overdamping. $N(u)$ denotes the $p$th order power nonlinearities. It is well known that the problem is locally well-posed in the energy space $H^1(\mathbb {R}^d)\times L^2(\mathbb {R}^d)$ in the energy-subcritical or energy-critical case $1\le p\le p_1$, where $p_1:=1+\frac {4}{d-2}$ if $d\ge 3$ or $p_1=\infty$ if $d=1,2$. It is known that when $N(u):=\pm |u|^p$, small data blow-up in $L^1$-framework occurs in the case $b(t)^{-1} \notin L^1(0,\infty )$ and $1<p<p_c(< p_1)$, where $p_c$ is a critical exponent, i.e., threshold exponent dividing the small data global existence and the small data blow-up. The main purpose in the present paper is to prove the global well-posedness to the problem for small data $(u_0,u_1)\in H^1(\mathbb {R}^d)\times L^2(\mathbb {R}^d)$ in the whole energy-subcritical case, i.e., $1\le p<p_1$. This result implies that the small data blow-up does not occur in the overdamping case, different from the other case $b(t)^{-1}\notin L^1(0,\infty )$, i.e., the effective or noneffective damping.