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