Seminar 30.4.2015: Pavol Quittner

Seminar on Qualitative Theory of Differential Equations
organized by P.Quittner, M.Fila and R.Kollar

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Seminar 30.4.2015: Pavol Quittner

Postby quittner » Wed Apr 22, 2015 1:04 pm

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

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.
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