PhD thesis advisor:

PhD thesis - Full text (PDF 430 KB)

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.

[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.