Blow-up sets for linear diffusion equations in one dimension (Q1881500)

The authors consider the heat equation in the half-line with Dirichlet boundary data which blow up in finite time. Though the blow-up set may be any interval \([0,a],\) \(a\in [0,\infty],\) depending on the Dirichlet data, it is proved that the effective blow-up set, that is, the set of points \(x\geq 0\) where the solution behaves like \(u(0,t),\) consists always only of the origin. As an application, the authors consider a system of two heat equations with a nontrivial nonlinear flux coupling at the boundary. It is shown that by prescribing the nonlinearities the two components may have different blow-up sets. However, the effective blow-up sets do not depend on the coupling and coincide with the origin for both components.