Energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms
In this paper we develop an intrinsic approach to derivation of energy decay rates for the semilinear wave equation with localized interior nonlinear monotone damping g(ut) and a source term f(u). The proposed approach allows to consider, in an unified way, much more general classes of hyperbolic problems than addressed before in the literature. These generalizations refer to both geometric and topological aspects of the problem. The method leads to optimal decay rates for solutions of semilinear hyperbolic equations driven by a source of critical exponent and subjected to a nonlinear damping localized in a small region adjacent to a portion of the boundary. The distinct features of the model include: (i) Neumann boundary conditions are assumed and, (ii) no growth conditions are imposed on the damping g(s). It is well known that Neumann boundary does not satisfy Lopatinski condition and, therefore, the analysis of propagation of energy in the absence of the damping on the Neumann part of the boundary requires special geometric considerations. In addition, the sole conditions assumed on g(s) are monotonicity, continuity and g(0)=0. In particular, no differentiability and no growth conditions are imposed on the damping both at the origin and at the infinity. The asymptotic decay rates for the energy function are obtained from an intrinsic algorithm driven by solutions of simple ODE. Several examples illustrate the theory by exhibiting various decay rates (exponential, algebraic, rational, logarithmic, etc.) for the energy functional. An important corollary of our energy decay theorem is a stability result which shows that, under certain conditions, when dissipation is sublinear at infinity, the solution of the system remains uniformly bounded for all time in the norms above the finite energy level, even in the presence of a nonlinear source term.
Nonlinear Analysis, Theory, Methods and Applications
Lasiecka, I., & Toundykov, D. (2006). Energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms. Nonlinear Analysis, Theory, Methods and Applications, 64 (8), 1757-1797. https://doi.org/10.1016/j.na.2005.07.024