On fundamental solutions of the Cauchy problem for a class of degenerate parabolic equations

2000 ◽  
Vol 99 (5) ◽  
pp. 1533-1540
Author(s):  
S. D. Ivasyshen ◽  
O. G. Voznyak
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equations ◽  
Fundamental Solutions ◽  
Degenerate Parabolic Equations ◽  
The Cauchy Problem ◽  
Degenerate Parabolic

scholarly journals Asymptotic Properties of Solutions to the Cauchy Problem for Degenerate Parabolic Equations with Inhomogeneous Density on Manifolds

2021 ◽  
Author(s):  
Daniele Andreucci ◽  
Anatoli F. Tedeev
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equations ◽  
Blow Up ◽  
Asymptotic Properties ◽  
Degenerate Parabolic Equations ◽  
Isoperimetric Function ◽  
Finite Speed Of Propagation ◽  
The Cauchy Problem ◽  
Inhomogeneous Density ◽  
Degenerate Parabolic

AbstractWe consider the Cauchy problem for doubly nonlinear degenerate parabolic equations with inhomogeneous density on noncompact Riemannian manifolds. We give a qualitative classification of the behavior of the solutions of the problem depending on the behavior of the density function at infinity and the geometry of the manifold, which is described in terms of its isoperimetric function. We establish for the solutions properties as: stabilization of the solution to zero for large times, finite speed of propagation, universal bounds of the solution, blow up of the interface. Each one of these behaviors of course takes place in a suitable range of parameters, whose definition involves a universal geometrical characteristic function, depending both on the geometry of the manifold and on the asymptotics of the density at infinity.

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