On nonlinear wave equations with degenerate damping and source terms

Abstract

In this article we focus on the global well-posedness of the differential equation u t t − Δ u + | u | k ∂ j ( u t ) = | u | p − 1 u  in  Ω × ( 0 , T ) u_{tt}- \Delta u + |u|^k\partial j(u_t) = |u|^{ p-1}u \, \text { in } \Omega \times (0,T) , where ∂ j \partial j is a sub-differential of a continuous convex function j j . Under some conditions on j j and the parameters in the equations, we obtain several results on the existence of global solutions, uniqueness, nonexistence and propagation of regularity. Under nominal assumptions on the parameters we establish the existence of global generalized solutions. With further restrictions on the parameters we prove the existence and uniqueness of a global weak solution. In addition, we obtain a result on the nonexistence of global weak solutions to the equation whenever the exponent p p is greater than the critical value k + m k+m , and the initial energy is negative. We also address the issue of propagation of regularity. Specifically, under some restriction on the parameters, we prove that solutions that correspond to any regular initial data such that u 0 ∈ H 2 ( Ω ) ∩ H 0 1 ( Ω ) u_0\in H^2(\Omega )\cap H^1_0(\Omega ) , u 1 ∈ H 0 1 ( Ω ) u_1 \in H^1_0(\Omega ) are indeed strong solutions.