Abstract
This work is concerned with a system of nonlinear wave equations with nonlinear damping and source terms acting on both equations. We prove a global nonexistence theorem for certain solutions with positive initial energy.
Highlights
In this paper we study the initial-boundary-value problem utt − div g |∇u|2 ∇u |ut|m−1ut f1 u, v, x, t ∈ Ω × 0, T, vtt − div g |∇v|2 ∇v |vt|r−1vt f2 u, v, x, t ∈ Ω × 0, T, u x, t v x, t 0, x ∈ ∂Ω × 0, T, 1.1 u x, 0 u0 x, v x, 0 v0 x, ut x, 0 vt x, 0u1 x, x ∈ Ω, v1 x, x ∈ Ω, where Ω is a bounded domain in Rn with a smooth boundary ∂Ω, m, r ≥ 1, and fi ·, · : R2 → R i 1, 2 are given functions to be specified later
Wu et al 27 considered problem 1.1 with the nonlinear functions f1 u, v and f2 u, v satisfying appropriate conditions. They proved under some restrictions on the parameters and the initial data several results on global existence of a weak solution
In order to state and prove our result, we introduce the following function space: Z
Summary
This work is concerned with a system of nonlinear wave equations with nonlinear damping and source terms acting on both equations. We prove a global nonexistence theorem for certain solutions with positive initial energy
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