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A probabilistic approach to the trace at the boundary for solutions of a semilinear parabolic partial differential equation. (English) Zbl 0872.35052
The author investigates the trace at the boundary of nonnegative solutions of the semilinear parabolic partial differential equation \[ {\partial u\over\partial t}= {1\over 2} \Delta u- u^2.\tag{\(*\)} \] He author proves the following main theorem: Let \(\Omega\) be a domain in \(\mathbb{R}^d\), for \(y\in\mathbb{R}^d\) and \(r>0\) denote by \(B(y,r)\) the open ball of radius \(r\) centered at \(y\), and let \(u\in C^{1/2}((0,\infty)\times\Omega)\) be a nonnegative solution of \((*)\) in \((0,\infty)\times\Omega\). Set \[ \Lambda=\Biggl\{y\in \Omega:\lim_{t\to 0} \int_{B(y,r)} u(t,z)dz=\infty\text{ for all }r>0\Biggr\}. \] Then there exists a Radon measure \(\nu\) on \(\Omega\backslash\Lambda\) such that, for every \(\varphi\in C_c(\Omega\backslash\Lambda)\), \[ \langle\nu,\varphi\rangle= \lim_{t\to 0}\int_{\Omega\backslash\Lambda} u(t,z)\varphi(z)dz. \]

35K55 Nonlinear parabolic equations
60J45 Probabilistic potential theory
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