Global Solvability and Asymptotic Stability for the Wave Equation with Nonlinear Boundary Damping and Source Term

Abstract

We study the global existence and uniform decay rates of solutions of the problem

$$ \begin{array}{*{20}c} {(P)} & {\left\{ \begin{gathered} u_{tt} - \Delta u = \left| u \right|^\rho uin\Omega \times ]0, + \infty [ \hfill \\ u = 0on\Gamma _0 \times ]0, + \infty [ \hfill \\ \partial _\nu u + g\left( {u_t } \right) = 0on\Gamma _1 \times ]0, + \infty [ \hfill \\ u\left( {x,0} \right) = u^0 \left( x \right),u_t \left( {x,0} \right) = u^1 \left( x \right); \hfill \\ \end{gathered} \right.} \\ \end{array} $$

where Ω is a bounded domain of R ⁿ, n≥1, with a smooth boundary $\Gamma = \Gamma _0 \cup \Gamma _1$ and 0<ρ< ²_n−2 , n≥3; ρ>0, n=1, 2.

Assuming that no growth assumption is imposed on the function g near the origin and, moreover, that the initial data are taken inside the Potential Well, we prove existence and uniqueness of regular and weak solutions to problem (P). For this end we use nonlinear semigroup theory arguments inspired in the work of the authors Chueshov, Eller and Lasiecka [5]. Furthermore, uniform decay rates of the energy related to problem (P) are also obtained by considering a similar approach firstly introduced by Lasiecka and Tataru [15]. The present work generalizes the work of the authors Cavalcanti, Domingos Cavalcanti and Martinez [4] and complements the work of Vitillaro [30]. It is important to mention that in [30] no decay result is proved and the dissipative term on the boundary is of a preassigned polynomial growth at the origin.

Partially supported by a grant of CNPq, Brazil.