Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping–source interaction

Abstract

We establish, subject to some natural additional assumptions imposed on the relation between the source and the damping, both well-posedness and effective optimal decay rates for the solutions of a semilinear model of the wave equation. The theory presented allows to consider both superlinear and sublinear behaviours of the dissipation in the presence of unstructured sources.

Global existence and blow‐up of solutions for nonlinear viscoelastic wave equation with degenerate damping and source

“…The basic idea comes from [11], and has been used in [5], [6], [10], [20]. In the sequel, we devote to the proof of Theorem 2.8.…”

“…For the case without source term in (1.4), it is known that the damping term assures the global existence of solutions for (1.4), for example, see [1], [2], [4], [13], [16]. For the analysis of the interaction between various damping and source terms in (1.4), we refer the reader to [5], [8], [10], [11], [19], [26], [28] and the references therein. It is worth noting that k = 0 in (1.1) the corresponding problem can be solved by monotonicity argument and approaches in [11].…”

Weak solutions to the initial boundary value problem for a semilinear wave equation with damping and source terms

“…The decisive factor in establishing long time behavior of solutions under the interaction of these two effects appears to be the power of the nonlinearity: a higher power of damping would yield long time existence, while a dominant source term will cause the solutions to go to infinity in finite time. This relationship has been studied on both bounded and unbounded domains [2,3,4,6,9,11,13,16,19].…”

“…When the mapping f is locally Lipschitz (in other words, f has subcritical exponent p ≤ 3 for n = 3), local existence of solutions can be proved through monotone semigroup theory (see the Appendix of [4]); for global existence the source term has to additionally satisfy a coercivitytype condition involving the first eigenvalue of the Laplacian. Under these Lipschitz assumptions for the interior source terms the authors of [3,9,4] study existence of solutions and stability issues, where nonlinear source and damping terms act on the boundary of the domain. This setup is different from ours since problems with nonlinear damping on the boundary in general do not satisfy the Lopatinski condition.…”

Global existence and nonexistence of solutions for a viscoelastic wave equation with nonlinear boundary source term

“…He obtained the local existence, blow up and global existence results of solutions. More results on the initial boundary value problem for the wave equations in the context of nonlinear boundary damping and source terms, we refer readers to see (Aassila et al [1], Cavalcanti et al [6], Feng and Li [9], [10]) and the papers cited therein.…”