Remarks on Hölder continuity for parabolic equations and convergence to global attractors (Q1585024)

The author considers solutions to quasilinear parabolic equations of the form \[ {\partial u\over\partial t}= \text{div}(a(x, t,u,Du))+ b(x,t,u,Du),\quad (x,t)\in\Omega\times [0,T],\tag{\(*\)} \] where \(\Omega\) is a bounded open subset of \(\mathbb{R}^N\). Both nondegenerate and degenerate diffusion terms are allowed. The `logarithmic functions' technique is used to show interior Hölder continuity of bounded weak solutions to \((*)\). The results concerning boundary regularity in case of Dirichlet or Neumann boundary conditions and convergence of the solutions to a global attractor in the uniform supremum norm are also included.