Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Semilinear parabolic equations with distributions as initial data

Pages: 305 - 316, Volume 3, Issue 3, July 1997      doi:10.3934/dcds.1997.3.305

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Francis Ribaud - C.M.L.A., U.R.A. 1611, E.N.S. de Cachan, 61 Av. du Président Wilson, 94235 Cachan Cedex, France (email)

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.

Received: October 1996;      Available Online: April 1997.