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: { u t t − Δ u + b ( t ) u t = N ( u ) , a m p ; t ∈ [ 0 , T ) , x ∈ R d , u ( 0 ) = u 0 , u t ( 0 ) = u 1 , a m p ; x ∈ R d . \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 ) b(t) is a positive C 1 C^1 -function on [ 0 , ∞ ) [0,\infty ) satisfying \[ b ( t ) − 1 ∈ L 1 ( 0 , ∞ ) , b(t)^{-1} \in L^1(0,\infty ), \] whose case is called overdamping. N ( u ) N(u) denotes the p p th order power nonlinearities. It is well known that the problem is locally well-posed in the energy space H 1 ( R d ) × L 2 ( R d ) H^1(\mathbb {R}^d)\times L^2(\mathbb {R}^d) in the energy-subcritical or energy-critical case 1 ≤ p ≤ p 1 1\le p\le p_1 , where p 1 := 1 + 4 d − 2 p_1:=1+\frac {4}{d-2} if d ≥ 3 d\ge 3 or p 1 = ∞ p_1=\infty if d = 1 , 2 d=1,2 . It is known that when N ( u ) := ± | u | p N(u):=\pm |u|^p , small data blow-up in L 1 L^1 -framework occurs in the case b ( t ) − 1 ∉ L 1 ( 0 , ∞ ) b(t)^{-1} \notin L^1(0,\infty ) and 1 > p > p c ( > p 1 ) 1>p>p_c(> p_1) , where p c 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 ) ∈ H 1 ( R d ) × L 2 ( R d ) (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 ≤ p > p 1 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 ∉ L 1 ( 0 , ∞ ) b(t)^{-1}\notin L^1(0,\infty ) , i.e., the effective or noneffective damping.