Existence and uniqueness of solutions to wave equations with nonlinear degenerate damping and source terms

摘要

In this article we focus on the global well-posedness of the differential equation u_(tt) — Δu+ |u|~kj'(u_t) = |u|~(p-1) u in Ω x (0, T), where j' denotes the derivative of a C~1 convex and real valued function j. The interaction between degenerate damping and a source term constitutes the main challenge of the problem. Problems with non-degenerate damping (k = 0) have been studied in the literature (Georgiev and Todorova, 1994; Levine and Serrin, 1997; Vitillaro, 2003). Thus the degeneracy of monotonicity is the main novelty of this work. Depending on the level of interaction between the source and the damping we characterize the domain of the parameters p, m, k, n (see below) for which one obtains existence, regularity or finite time blow up of solutions. More specifically, when p ≤ m + k global existence of generalized solutions in H~1 x L_2 is proved. For p > m + k, solutions blow up in a finite time. Higher energy solutions are studied as well. For H~2 x H~1 initial data we obtain both local and global solutions with the same regularity. Higher energy solutions are also proved to be unique.