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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 $$\varphi\in C_c(\Omega\backslash\Lambda)$$, $\langle\nu,\varphi\rangle= \lim_{t\to 0}\int_{\Omega\backslash\Lambda} u(t,z)\varphi(z)dz.$

##### MSC:
 35K55 Nonlinear parabolic equations 60J45 Probabilistic potential theory
##### Keywords:
nonnegative solutions
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