ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

Vol. LXXII, 1(2003)
p. 129 139

Boundary Behavior in Strongly Degenerate Parabolic Equations
M. Winkler


Abstract.  The paper deals with the initial value problem with zero Dirichlet boundary data for $$ u_t = u^p \Delta u \quad \mbox{in } \Omega \times (0,\infty) $$ with $p \ge 1$. The behavior of positive solutions near the boundary is discussed and significant differences from the case of the heat equation ($p=0$) and the porous medium equation ($p \in (0,1)$) are found. In particular, for $p \ge 1$ there is a large class of initial data for which the corresponding solution will never enter the cone $\{ v: \Omega \to \R \ | \ \exists \, c>0: \ v(x) \ge c \dist(x,\rO) \}$.\\ Finally, for $p>2$ a solution $u$ with $u(t) \in C_0^\infty(\Omega) \ \forall \, t \ge 0$ is constructed.

AMS subject classification:  35K55, 35K65, 35B65
Keywords:  Degenerate diffusion, regularity

Download:     Adobe PDF     Compressed Postscript      

Acta Mathematica Universitatis Comenianae
Institute of Applied Mathematics
Faculty of Mathematics, Physics and Informatics
Comenius University
842 48 Bratislava, Slovak Republic  

Telephone: + 421-2-60295111 Fax: + 421-2-65425882  
e-Mail: amuc@fmph.uniba.sk   Internet: www.iam.fmph.uniba.sk/amuc

© Copyright 2003, ACTA MATHEMATICA UNIVERSITATIS COMENIANAE