Nonlinear Singular Parabolic Equations
Piotr Cezary Biler, Tadeusz Nadzieja, Andrzej Raczyński · 2020
Recently much attention has been paid to mathematical questions related to solvability of the initial-boundary value problems for related parabolic-elliptic systems, regularity and asymptotic behavior of solutions. Solutions that cease to exist after a concentration of mass at the origin are shown to exist in (Herrero, Velázquez, 1996). Their construction is based on matched asymptotic expansions (the formal part), and on a delicate topological argument showing the existence of solutions with prescribed asymptotics in different regions (the rigorous part of the proof). The solutions blowing up without the concentration of mass are constructed as well as those exploding after a concentration at the origin of an arbitrarily prescribed mass. A new ingredient compared to the preceding analysis of radial solution is a physically motivated functional which will play the role of a Lyapunov function. The importance of self-similar solutions is connected with their role in describing the long time behavior of arbitrary global solutions.