Optimal sub- or supersolutions in reaction-diffusion problems (Q5957787)

The author considers the semilinear parabolic problem NEWLINE\[NEWLINE \begin{aligned} & {{\partial u}\over{\partial t}} = \Delta u + f(u) \quad\text{in}\quad \Omega\times(0,T), \\& {{\partial u}\over{\partial n}} + g(u) = 0 \quad\text{on}\quad \partial\Omega\times(0,T), \\ & u(x,0) = u_0(x),\end{aligned}\tag \(*\) NEWLINE\]NEWLINE where \(\Omega\) is a sufficiently smooth bounded domain in \({\mathbb R}^N\), \(n\) is the exterior normal vector field to \(\partial\Omega\), and \(f,g\) are sufficiently smooth functions. He constructs sub- and supersolutions that are optimal in the sense that they are solutions of \((*)\) if \(\Omega\) is the ball in \({\mathbb R}^N\) of suitable radius depending on various quantities related to an associated linear elliptic problem on \(\Omega\) such as the torsion problem or the fixed membrane problem. NEWLINENEWLINENEWLINEThe results are applied to the estimation of blowup or vanishing times for solutions of certain parabolic problems, as well as to some elliptic problems.