Existence for a nonlinear wave equation with damping and source terms

By J. Serrin, Grozdena Todorova, Enzo Vitillaro. Differential and Integral Equations, 2003

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utt −∆u+Q(t, x, ut) = f(x, u), u(0) = u0, ut(0) = u1, (1.1) in (0, T ) × Rn, where 0 < T ≤ ∞, n ≥ 1. The data u0 and u1 are in the energy space and are compactly supported in Rn. The terms Q and f are nonlinear damping and source terms respectively. Problems like (1.1) have been studied in many papers. For the case Q ≡ 0 see [3], [7], [11], [12], [20], [30]; for the case f ≡ 0 see [8], [10]; when both terms Q and f interact, see [6], [9], [13], [14], [15], [18], [19], [25], [26], [27], [31], [32], [33]. For the sake of simplicity we illustrate our results by considering the model power–like equation with damping term

Subfields: Computational Theory and Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/52619add92009ffa286fa6cdbcbc6ede5e3ef05e

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Citations in the corpus

On nonlinear wave equations with degenerate damping and source terms (2005, background): “We refer the reader to [6, 16, 18, 22, 24, 27, 30] and the references therein.”
Uniqueness of weak solutions for the semilinear wave equations with supercritical boundary/interior sources and damping (2008, background): “In addition, the same uniqueness question has been unresolved [14, 28] in the case of interior but supercritical sources, which fail to satisfy the locally Lipschitz condition.”
Blow-up of weak solutions for the semilinear wave equations with nonlinear boundary and interior sources and damping (2008, background): “This general topic has attracted a lot of attention in recent years ([14, 30, 36, 33, 27, 40, 29] and references therein).”
On a wave equation with supercritical interior and boundary sources and damping terms (2011, background): “However, even without nonlinear boundary dynamics, an interior source of a super-critical order (3 < p ≤ 5) substantially perturbs the evolution generator so that the presence of damping becomes essential even for the local existence theory [6], [18], [32], [39]; the latter references along with [3]”