Qualitative Properties of Positive Solutions of Parabolic Equations: Symmetry, A priori Estimates, and Blow-up Rates
Juraj Földes
PhD thesis advisor: Pavol Quittner

PhD thesis - Full text    (PDF 430 KB)

Summary: In this work, we study qualitative properties of non-negative solutions of quasilinear parabolic equations on bounded and unbounded domains. In the first part, using the method of moving hyperplanes, we establish new nonlinear Liouville theorems for parabolic problems on half spaces. Based on the Liouville theorems and scaling techniques, we derive estimates for the blow-up rates of positive solutions of indefinite parabolic problems in bounded and unbounded domains. As a consequence, we obtain new results on the complete blow-up of these solutions and results for the a priori estimates for positive solutions of indefinite elliptic problems.

In the next part, we employ the method of moving hyperplanes, maximum principle, and Harnack inequality to study symmetry properties of positive solution of asymptotically symmetric quasilinear parabolic problems in the whole space. As the result, we prove that if the problem converges exponentially to a symmetric one, then the solution converges to the space of symmetric functions. We also show that this result does not hold true, if the convergence is not exponential.

Related papers

[1] J. Foldes. Liouville theorems, a priori estimates and blow-up rates for solutions of indefinite superlinear parabolic problems. preprint.

[2] J. Foldes. On symmetry properties of parabolic equations in bounded domains. submitted.

[3] J. Foldes. Symmetry properties of asymptotically symmetric parabolic equations in Rn. submitted.

[4] J. Foldes and P. Polacik. On cooperative parabolic systems: Harnack inequalities and asymptotic symmetry. Discrete Contin. Dyn. Syst., 25(1):133-157, 2009.