[Translate to English:] Seminár z kvalitatívnej teórie diferenciálnych rovníc - Michael Winkler (29.9.2016)
[Translate to English:] vo štvrtok 29.9.2016 o 14:00 hod. v miestnosti M/223
[Translate to English:] Prednášajúci: Michael Winkler (Universität Paderborn)
Názov: Fast decay of solutions to a strongly degenerate parabolic equation of fast diffusion type
Termín: 29.9.2016, 14:00 hod., M/223
Abstrakt:
We study positive solutions of the Cauchy problem in the whole space for the parabolic equation \[ u_t = u^p \Delta u \qquad (\star) \] in the range $p>1$ representing strongly degenerate diffusion. We discuss how spatial decay of the initial data influences the large time behavior of solutions, and thereby investigate the dependence of corresponding growth rates of solutions on growth rates of the initial data in a corresponding fast diffusion equation to which ($\star$) is equivalent.
In particular, we obtain temporal decay at algebraic rates for ($\star$) in cases of suitable algebraic decay of the initial data. For more rapidly decreasing data, we find temporal decay rates which consist of logarithmic, doubly logarithmic or even smaller corrections of an again algebraic limit rate. The methods include comparison with an apparently new family of selfsimilar solutions as well as energy-based arguments involving a refinement of a Gagliardo-Nirenberg-type interpolation inequality in the limit case when one of the summability powers therein approaches zero.
This is joint work with Marek Fila.