Quenching for a diffusive equation with a concentrated singularity. (English) Zbl 0891.35061

The authors deal with the diffusion equation \(v_t=v_{xx}+\varepsilon\delta(x-a)f(v)\) for \(0<x<1\), \(t>0\), where \(\delta (x)\) is the Dirac distribution and \(f(v)\) a nonlinear positive increasing convex function on \((0,c)\) with \(f(c-)=\infty \). The so called quenching is studied, i.e. the phenomenon when for some time the solution remains bounded but its derivatives blow up. By means of Green’s function the initial boundary value problem is transformed into a nonlinear Volterra integral equation. In case of Dirichlet boundary condition for small \(\varepsilon \leq \varepsilon^*\) there exists a global solution, while for sufficiently \(\varepsilon >\varepsilon^*\) quenching always occurs in finite time. In case of Neuman boundary condition the solution quenches in finite time for every \(\varepsilon >0\). The asymptotic analysis of the solution at quenching time closes the paper.


35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35R05 PDEs with low regular coefficients and/or low regular data
45D05 Volterra integral equations