On positive solutions of the heat equation satisfying a nonlinear boundary condition
The following inequality is considered: \[ \begin{alignedat}{2} {\partial u\over \partial t} &\geq \Delta u&\quad& \text{in } \Omega\times (0,\infty),\\ {\partial u\over \partial {\mathbf n}} &\geq {u^{\sigma}\over (1+t)^{\alpha}}&\quad & \text{on } \partial \Omega\times (0,\infty).\end{alignedat} \] Here, \(\Omega\) is an unbounded domain in \(\mathbb{R}^n\), \(\partial/\partial {\mathbf n}\) is the directional derivative along the outward normal to \(\partial \Omega\), and \(\sigma>1\) and \(\alpha\in \mathbb{R}\) are constants. For each dimension \(n\), the paper provides sharp conditions on the exponents \(\sigma\) and \(\alpha\) for which the above inequality has a classical positive solution. Note that the nonexistence of such a solution implies the nonexistence of a global solution to the corresponding heat equation because the blow-up of the solution takes place.
scientific article; zbMATH DE number 5820359