On the regularity properties for solutions of the Cauchy problem for the porous media equation

Hayasida, Kazuya

doi:10.1090/S0002-9939-1989-0948150-0

On the regularity properties for solutions of the Cauchy problem for the porous media equation
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Proc. Amer. Math. Soc. 107 (1989), 107-112 Request permission

Abstract:

We consider the Cauchy problem for the equation ${\partial _t}u = \Delta {u^m}$ in ${R^N} \times (0,T)$. We assume that $1 < m < 3N/(3N - 2)$ and the initial data ${u_0}$ is in $C_0^1({R^N})$ and ${u_0} \geq 0$ in ${R^N}$. Then we prove that the second derivatives of ${u^m}$ with respect to the space-variable are in ${L^2}({R^N} \times (0,T))$.

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