Long-term dynamics of semilinear wave equation with nonlinear localized interior damping and a source term of critical exponent
This article addresses long-term behavior of solutions to a semilin-ear damped wave equation with a critical source term. A distinctive feature of the model is the geometrically constrained dissipation: it only affects a small subset of the domain adjacent to a connected portion of the boundary. The main result of the paper provides an affirmative answer to the open question whether global attractors for a wave equation with critical source and geometrically constrained damping are smooth and finite-dimensional. A positive answer to the same question in the case of subcritical sources was given in . However, critical exponent of the source term combined with weak geometrically restricted dissipation constitutes the major new difficulty of the problem. To overcome this issue we develop a new version of Carleman's estimates and apply them in the context of recent results  on fractal dimension of global attractors.
Discrete and Continuous Dynamical Systems
Chueshov, I., Lasiecka, I., & Toundykov, D. (2008). Long-term dynamics of semilinear wave equation with nonlinear localized interior damping and a source term of critical exponent. Discrete and Continuous Dynamical Systems, 20 (3), 459-509. https://doi.org/10.3934/dcds.2008.20.459