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Communications on Pure and Applied Analysis (CPAA)
 

Regularity for solutions of the two-phase Stefan problem

Pages: 591 - 600, Volume 7, Issue 3, May 2008

doi:10.3934/cpaa.2008.7.591       Abstract        Full Text (152.1K)       Related Articles

Marianne Korten - Department of Mathematics, Kansas State University, Manhattan, KS 66506, United States (email)
Charles N. Moore - Department of Mathematics, Kansas State University, Manhattan, KS 66506, United States (email)

Abstract: We consider the two-phase Stefan problem $u_t=\Delta\alpha(u)$ where $\alpha(u) =u+1$ for $u<-1$, $\alpha(u) =0$ for $-1 \leq u \leq 1$, and $\alpha(u)=u-1$ for $u \geq 1$. We show that if $u$ is an $L_{l o c}^2$ distributional solution then $\alpha(u)$ has $L_{l o c}^2$ derivatives in time and space. We also show $|\alpha(u)|$ is subcaloric and conclude that $\alpha(u)$ is continuous.

Keywords:  Stefan problem, degenerate parabolic equation.
Mathematics Subject Classification:  Primary: 35K65; Secondary: 80A22.

Received: March 2007;      Revised: October 2007;      Published: February 2008.