Qualitative Aspects of Nonlinear Parabolic Partial Differential Equations and Systems

Abstract: This thesis contains four papers about some aspects of nonlinear parabolic equations and systems. Paper 1. A note on quenching for parabolic equations with dynamic boundary conditions We present a quenching result for semilinear parabolic equations with dynamic boundary conditions in bounded domains with a smooth boundary. Paper 2. On global existence for semilinear parabolic systems We present some results on global existence of classical solutions of certain semilinear parabolic systems with homogeneous Dirichlet boundary conditions in bounded domains with a smooth boundary, relaxing the usual monotonicity assumptions on the nonlinearities. Paper 3. Best constants for Gagliardo-Nirenberg inequalities on the real line We use a variational approach to find the best constants for certain Gagliardo-Nirenberg inequalities on the real line. To show the existence of a minimizer, we use the method of concentration-compactness. Paper 4. On the time evolution of extrema of solutions to nonlinear parabolic equations in unbounded domains We obtain a result about the time evolution of extrema that can be applied to the study of classical solutions to parabolic equations in unbounded domains with a smooth boundary.