We consider parabolic equations of porous medium type of the form $$u_t − div A(x,t,u,Du) = \mu in E_T,$$ in some space time cylinder $E_T$. The most prominent example covered by our assumptions is the classical porous medium equation $$u_t − \Delta u^m = \mu in E_T,$$ with $m\ge1$. We establish a sufficient condition for the continuity of $u$ in terms of a natural Riesz potential of the right-hand side measure $\mu$. As an application we come up with a borderline condition ensuring the continuity of u: more precisely, if $\mu\in L((N+2)/2,1)$, then u is continuous in $E_T$.
Continuity estimates for porous medium type equations with measure data
GIANAZZA, UGO PIETRO
2014-01-01
Abstract
We consider parabolic equations of porous medium type of the form $$u_t − div A(x,t,u,Du) = \mu in E_T,$$ in some space time cylinder $E_T$. The most prominent example covered by our assumptions is the classical porous medium equation $$u_t − \Delta u^m = \mu in E_T,$$ with $m\ge1$. We establish a sufficient condition for the continuity of $u$ in terms of a natural Riesz potential of the right-hand side measure $\mu$. As an application we come up with a borderline condition ensuring the continuity of u: more precisely, if $\mu\in L((N+2)/2,1)$, then u is continuous in $E_T$.File in questo prodotto:
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