## Existence and boundary stabilization of a nonlinear hyperbolic equation with time-dependent coefficients

### Summary

Summary: In this article, we study the hyperbolic problem $$K(x,t)u_{tt} - \sum_{j=1}^n\left(a(x,t)u_{x_j}\right) + F(x,t,u,\nabla u) = 0$$ coupled with boundary conditions $$u=0,\quad\hbox{on }\Gamma_1\,, \quad {\partial u \over\partial\nu} + \beta(x)u_t =0\quad\hbox{ on }\Gamma_0\,.$$ Here the variable $x$ belongs to a bounded region of ${\Bbb R}^n$, whose boundary is partitioned into two disjoint sets $\Gamma_0,\Gamma_1$. We prove existence, uniqueness, and uniform stability of strong and weak solutions when the coefficients and the boundary conditions provide a damping effect.

35B40, 35L80

### Keywords/Phrases

boundary stabilization, asymptotic behaviour