Wave equation driven by a
nonlinear dissipative
source and subjected to a
nonlinear damping that is localized on a small region near the boundary is considered. While finite-energy
source" alt="Click to view the MathML source">(H1×L2) solutions to this problem are bounded uniformly for all times
source" alt="Click to view the MathML source">t>0, this property generally fails for higher energy norms (
source" alt="Click to view the MathML source">H2×H1). This is due to a “generation” of higher energy by the
source and the ultimate loss of dissipativity. However, the presence of the damping may turn things around by counteracting the
sources also at the higher energy levels. The benefits of this counteraction depend on the
nonlinear characteristics of dissipation. Any deviation from linearity (be it origin or infinity) causes degradation of the damping, hence of decay rates of finite energy. This, in turn, is shown to have an adverse effect on the stability of higher energies. The main aim of this paper is to provide a quantitative analysis of the interaction between
nonlinearity of the damping and
nonlinearity of the
source. We show that under some correlation between growth rates of the damping and the
source, the norms of topological order
above the finite-energy level remain globally bounded for all times.