Capacitary estimates of solutions of semilinear parabolic equations |
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Authors: | Moshe Marcus Laurent Veron |
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Institution: | 1. Department of Mathematics, Technion, Haifa, Israel 2. Department of Mathematics, University of Tours, Tours, France
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Abstract: | We prove that any positive solution of ${\partial_tu-\Delta u+u^q=0 (q > 1)}$ in ${\mathbb{R}^N \times (0, \infty)}$ with initial trace (F, 0), where F is a closed subset of ${\mathbb{R}^{N}}$ can be represented, up to two universal multiplicative constants, by a series involving the Bessel capacity ${C_{2/q, q^{\prime}}}$ . 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 ${C_{2/q, q^{\prime}}}$ -Bessel capacity. |
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