Seminár z kvalitatívnej teórie diferenciálnych rovníc

Seminar on Qualitative Theory of Differential Equations

Thursday 30.4.2015 at 14:00 Lecture room M-223

Pavol Quittner (KAMŠ FMFI UK):

Liouville theorems for a class of superlinear parabolic problems without gradient structure

Abstract:

We are interested in Liouville-type theorems for entire solutions of scaling invariant superlinear parabolic problems.

Such theorems guarantee universal estimates for solutions of more general problems.

We consider a class of two-component parabolic systems without gradient structure and show that the components

of any positive bounded entire solution have to be proportional, hence the problem can be reduced to a scalar problem.

The universal estimates guaranteed by our Liouville theorems imply global existence and boundedness of threshold

solutions lying on the boundary between global existence and blow-up and also yield optimal blow-up rate estimates

of solutions which blow-up in finite time. Finally, we use the universal estimates to prove the existence of positive

periodic solutions of strongly cooperative parabolic Lotka-Volterra systems with equal diffusion.