Seminar on Qualitative Theory of Differential Equations - Petra Macková (15.10.2020)
Thursday 15.10.2020 at 14:00, Lecture room C
By: Pavol Quittner
Solutions with moving singularities for nonlinear diffusion equations
There are various results on solutions with moving singularities for the heat equation, semilinear heat equations, or the Navier-Stokes system. This talk will be concerned with the existence of positive solutions of equations of porous medium type with a singularity that moves in time along a prescribed curve and keeps the spatial profile of singular stationary solutions, which have a substantially different form in two space dimensions and dimensions higher than two. It turns out that there appears a critical exponent for the existence of such solutions. I will give a sketch of the proof of the existence of such solutions, which is based on finding suitable sub- and supersolutions.