Doubly nonlinear parabolic equations

This thesis is devoted to the study of the existence and regularity of weak solutions to some doubly nonlinear parabolic equations. Such equations are nonlinear both with respect to the solution and its gradient, and are thus more difficult to handle than the parabolic p-Laplace equation and the porous medium equation, which are special cases of the equations considered here. Much of the work in this thesis concerns the so-called diffusive shallow medium equation, which presents an additional technical difficulty due to the given profile function appearing in the equation. For the diffusive shallow medium equation we use a De Giorgi type iteration to prove that weak solutions are locally bounded. We show that bounded weak solutions are locally Hölder continuous using the method of intrinsic scaling introduced originally by DiBenedetto. We prove the existence of a bounded solution to a Cauchy-Dirichlet problem associated with the diffusive shallow medium equation by introducing regularized problems for which the existence of solutions is easy to prove by means of Galerkin's method. In order to show that these solutions converge to a solution of the original problem, we have proved some new compactness results which might be useful also for other nonlinear equations. There are various definitions for weak solutions appearing in the literature, and in this thesis we have devoted special attention to showing that the adopted definitions allow rigorous proofs of the existence and regularity results without any additional assumptions. For the diffusive shallow medium equation, a definition which is natural in this sense has not previously been used. For doubly singular parabolic equations we have proved a range of regularity results such as local boundedness, an integral Harnack inequality, local Hölder continuity, expansion of positivity and a Harnack inequality in properly scaled space-time cylinders.