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hal-00282315, version 5

## Capacitary estimates of solutions of semilinear parabolic equations

Moshe Marcus () 1, Laurent Veron () 2

(2006)

Abstract: 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.

• 1:  Department of Mathematics (TECHNION)
• Technion - Israel Institute of Technology
• 2:  Laboratoire de Mathématiques et Physique Théorique (LMPT)
• CNRS : UMR6083 – Université François Rabelais - Tours
• Domain : Mathematics/Analysis of PDEs
• Keywords : Heat equation – singularities – Borel measures – Besov spaces – real interpolation – Bessel capacities – quasi-additivity – capacitary measures – Wiener type test – initial trace.type test – initial trace.
• Comment : à paraître Calculus of Variations and Partial Differential Equations 2012 – DOI: 10.1007/s00526-012-0545-9
• Available versions :  v1 (2008-05-27) v2 (2010-09-24) v3 (2011-07-01) v4 (2012-03-11) v5 (2012-06-17)

• hal-00282315, version 5
• oai:hal.archives-ouvertes.fr:hal-00282315
• From:
• Submitted on: Sunday, 17 June 2012 10:04:35
• Updated on: Tuesday, 28 August 2012 16:30:36