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Pré-Publication, Document De Travail Année : 2006

Capacitary estimates of solutions of semilinear parabolic equations

Moshe Marcus
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Résumé

We prove that any positive solution of $ \prt_tu-\Delta u+u^q=0$ ($q>1$) in $\BBR^N\ti(0,\infty)$ with initial trace $(F,0)$, where $F$ is a closed subset of $\BBR^N$ can be estimated from above and below and up to two universal multiplicative constants, by a series involving the Bessel capacity $C_{2/q,q'}$. As a consequence we prove that there exists a unique positive solution of the equation with such an initial trace. We also characterize the blow-up set of $u(x,t)$ when $t\downarrow 0$ , by using the "density" of $F$ expressed in terms of the $C_{2/q,q'}$-capacity.
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Dates et versions

hal-00282315 , version 1 (27-05-2008)
hal-00282315 , version 2 (24-09-2010)
hal-00282315 , version 3 (01-07-2011)
hal-00282315 , version 4 (11-03-2012)
hal-00282315 , version 5 (17-06-2012)

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Moshe Marcus, Laurent Veron. Capacitary estimates of solutions of semilinear parabolic equations. 2006. ⟨hal-00282315v5⟩
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