The blow-up estimate for heat equations with nonlinear boundary conditions
The paper is devoted to the study of the blow-up estimates of solutions to the following problem: \[ \begin{alignedat}{3} u_t &= \Delta u, &\qquad v_t &= \Delta v &&\qquad (x \in \Omega,\;t>0),\\ \frac{\partial u}{\partial \eta} &= u^{\alpha} v^p, &\qquad \frac{\partial v}{\partial \eta} &= u^q v^{\beta} &&\qquad (x \in \partial \Omega,\;t>0),\\ u(x,0)&=u_o(x), &\qquad v(x,0) &= v_o(x) &&\qquad (x \in \Omega). \end{alignedat} \] Here \(\Omega\) is a bounded domain in \(\mathbb{R}^n\) with smooth boundary and \(\eta\) is the exterior normal vector on the boundary. It is supposed that \(\alpha, \beta \geq 0\) and \(p,q >0\) and that the initial data \(u_o, \, v_o\) are continuous nonnegative functions satisfying the compatibility conditions. The paper is closely related to [\textit{M. Pedersen} and \textit{Z. G. Lin}, J. Math. Anal. Appl. 255, 551--563 (2001; Zbl 0980.35058)]. Using different methods the author generalizes the results presented there.
scientific article; zbMATH DE number 2118021