Asymptotic Stability of Wave Equation With Non‐Complementary Local Memory Effects and Frictional Damping on Riemannian Manifold

ABSTRACT

In this paper, we investigate the well‐posedness and stability of a class of wave equations characterized nonlinearities of arbitrary growth, along with the combined effects of non‐complementary local memory and frictional damping. The problem is formulated on an open, bounded, and connected subset of a two‐dimensional compact Riemannian manifold. On one portion of its smooth boundary, homogeneous Dirichlet conditions are imposed, while acoustic boundary conditions with nonlinear dissipation are prescribed on the remainder. Combining a contradiction argument, microlocal analysis techniques, and the method introduced by Lasiecka and Tataru, we carry out the following analysis. First, we derive an observability inequality for a truncated problem. A density argument then allows us to extend this result to the original problem. Within this framework, we further construct a damping region, distributed throughout the interior of the domain, whose measure can be made arbitrarily small. Finally, we obtain the asymptotic stability result without any prior quantification of decay rates for these two non‐complementary dissipative effects. This work relaxes the distributional assumptions on the two damping mechanisms, reduces the damping region, and improves previous studies on the interaction between the two dissipative effects.