Long-term dynamics of a semilinear wave equation with localized nonlinear dissipation, critical source term, and mixed boundary conditions

This dissertation addresses long-term behavior of solutions to a semilinear damped wave equation. A distinctive feature of our model is a nonlinear critical source term acting in conjunction with a geometrically constrained dissipation, which only affects a subset of a boundary collar. The main result provides an affirmative answer to the open question whether global attractors for a wave equation with a critical source and localized damping are finite-dimensional and smooth. A positive answer to the same question in the case of subcritical sources was given in [9]. However, criticality of the source term combined with weak geometrically restricted damping 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 [11] on fractal dimension of global attractors.