Seminár z kvalitatívnej teórie diferenciálnych rovníc

Seminar on Qualitative Theory of Differential Equations

Thursday 29.9.2016 at 14:00 Lecture room M-223

Michael Winkler (Universität Paderborn):

Fast decay of solutions to a strongly degenerate parabolic equation of fast diffusion type

Abstract:

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.