Events

Pavel Drabek

Colloquia

When

Date: Wednesday, November 16, 2022
Time: 4:00 pm - 5:00 pm
Location: Zoom
Pavel Drabek graduated from Faculty of Mathematics and Physics, Charles University, Prague in 1977 and obtained a Candidate of Sciences (equiv. Ph.D.) and Doctor of Sciences (DrSc.) from Czechoslovak Academy of Sciences Prague in 1981 and 1990, respectively. Soon after his graduation, he became a faculty member of the Department of Mathematics of VSSE (Vysoka skola strojni a elektrotechnicka) in Plzen, which is known as University of West Bohemia now. Throughout his career he significantly contributed (and is still actively contributing) to the development of nonlinear functional analysis, nonlinear differential equations, bifurcation theory, critical point theory. His favorite topics (that can be traced back to his years spent in Prague) are nonlinear problems related to the Fucik spectrum and Fredholm Alternative for the p-Laplacian. Pavel Drabek is author or co-author of close to 200 research papers, and several monographs and articles in collective volumes. For his outstanding achievements in science, he obtained several national awards such as The Bernard Bolzano Honorary Medal “For Merits in the Mathematical Sciences” in 2013, awarded by the Academy of Sciences of the Czech Republic.

In this lecture we discuss the existence of monotone travelling wave solutions to the reaction-diffusion equation with rather general density dependent diffusion term. For example, it comprises the porous medium differential operator or the p-Laplacian. The novelty of our approach consists in two facts. First, we focus on the diffusion coefficient which degenerates or has singularities of arbitrary order. In particular, we show that using an appropriate definition of solution the travelling wave profiles exist even in the situations which have not been covered so far. Second, we allow the diffusion coefficient to be discontinuous at a finite number of points with the discontinuities being of the first kind (i.e., finite jumps). As the reaction term concerns, we deal with both bistable and monostable case. In a special case of power-type behavior of diffusion and reaction terms near equilibria we provide detailed asymptotic analysis of monotone travelling wave profiles and classify their shapes. The mutual behavior of both terms, diffusion and reaction, and its influence on the shape of travelling wave is explicitly illustrated and visualized