The interface blow-up phenomenon and local estimates for doubly degenerate parabolic equations

Abstract

We study two classes of degenerate parabolic equations of the forms

where λ >0, m+λ−2 >0 and a(s) and ρ(s) are positive continuous functions on (0,∞).We examine under which conditions on behaviour of a(s) and ρ(s), corresponding non-negative solutions of the Cauchy problems possess the finite speed of propagations or the interface blow-up phenomena. Moreover, we study local behaviour of solutions of the corresponding Cauchy problems under the optimal assumptions of initial data.

Keywords:

• Degenerate parabolic equations
• Finite speed of propagation
• Interface blow-up phenomenon
• Local behaviour

Keywords:

• AMS 2000 Mathematics Subject Classifications35B33
• 35B45
• 35K55
• 35K65

Acknowledgment

The research of the author is partially supported by INTAS grant 03-51-5007.