Semilinear parabolic equations with distributions as initial data

Francis Ribaud

doi:10.3934/dcds.1997.3.305

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1997, Volume 3, Issue 3: 305-316. Doi: 10.3934/dcds.1997.3.305

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Semilinear parabolic equations with distributions as initial data

Francis Ribaud^1,

1.
C.M.L.A., U.R.A. 1611, E.N.S. de Cachan, 61 Av. du Président Wilson, 94235 Cachan Cedex

Received: September 30, 1996

Published: June 1997

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Abstract

We study the local Cauchy problem for the semilinear parabolic equations

$\partial _t U-\Delta U=P(D)F(U), \quad (t,x) \in [0,T[ \times \mathbb{R}^n $

with initial data in Sobolev spaces of fractional order $H^s_p(\mathbb{R}^n)$. The techniques that we use allow us to consider measures but also distributions as initial data ($s<0$). We also prove some smoothing effects and $L^q([0,T[,L^p)$ estimates for the solutions of such equations.

Keywords:

Local Cauchy problem,
semilinear parabolic equations.

Mathematics Subject Classification: 35K45, 35K55.

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