Extinction of superdiffusions and semilinear partial differential equations
Let \(L\) be a uniformly elliptic second order differential operator in \(\mathbb{R}^d\) with bounded locally Hölder continuous coefficients and let \(\psi\) be a function from \(\mathbb{R}^d\times \mathbb{R}_+\) to \(\mathbb{R}_+\). A measure-valued branching process in \(\mathbb{R}^d\) associated with the nonlinear operator \(Lu-\psi(x,u)\) is called \((L,\psi)\)-superdiffusion. One says that a regular domain \(D\) in \(\mathbb{R}^d\) is special if there exists a strictly positive solution \(u\) of the equation \(Lu=\psi(x,u)\) which is equal to \(0\) on \(\partial D.\) Under the assumption that the function \(L1\) is bounded, the authors describe the class of special domains \(D\) in terms of the extinction time of \((L,\psi)\)-superdiffusion in \(D\). For elliptic operators \(L\) which can be represented in the divergence form, the authors show that a bounded smooth domain \(D\) is special if and only if the principal eigenvalue \(\lambda_1(D)\) of \(-L\) in \(D\) is negative. (In this case, the description of the class of special domains depends only on \(L\) but not on \(\psi\).) The case of the Beltrami-Laplace operator on a complete Riemannian manifold is covered.
scientific article; zbMATH DE number 1274804