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: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 spaceIt is known that when N (u) := ±|u| p , small data blow-up in L 1 -framework occurs in the case b(t) −1 / ∈ L 1 (0, ∞) and 1 < p < pc(< p 1 ), where pc 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 ) in the whole energy-subcritical case, i.e. 1 ≤ 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, ∞), i.e. the effective or non-effective damping.Here, d ∈ N, T > 0, u = u(t, x) is a real-valued unknown function of (t, x), b = b(t) is a positive C 1 -function of t ∈ [0, ∞) satisfying b(t) −1 ∈ L 1 (0, ∞) i.e. ∞ 0 b(t) −1 dt < ∞, (1.2)

Sharp lifespan estimates of blowup solutions to semi-linear wave equations with time-dependent effective damping

“…On the other hand, if b(t) = (1 + t) −β with β < −1, then the situation is completely different. In this case, according to the result by Ikeda-Wakasugi [14], the critical exponent disappears, that is, there exists small global solution of (1.1) for every p > 1. This phenomenon is so-called overdamping.…”

The Strichartz estimates for the damped wave equation and the behavior of solutions for the energy critical nonlinear equation

“…We will show the local well-posedness for (NLDW) when 3 ≤ d ≤ 5 by applying the Strichartz estimates. The existence of a local solution has been studied by [17,10,12] (see also [14,15,16]). However, the small data global existence has not been known.…”

“…Remark 1.6. The existence of local solution is well known (see [10,12]). However, the small data global existence has not been known except for low dimension cases.…”

On the nonexistence of global solutions for critical semilinear wave equations with damping in the scattering case

“…Ikeda and Wakasugi [6] have proved that the global existence actually holds for any p > 1 when β < −1.…”