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Semilinear parabolic equations with distributions as initial data

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  • 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.

    Mathematics Subject Classification: 35K45, 35K55.


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