Critical speed for quenching (Q2724365)

A quenching problem is considered for the one-dimensional heat equation in an infinite strip when the nonlinear, concentrated quenching source moves with constant speed through the diffusive medium. This problem is connected with the nonlinear Volterra equation NEWLINE\[NEWLINEu(t)=h(t)+\int_0^t \frac{e^{\frac{-c^2}{4}(t-s)}}{2[\pi(t-s)]^{\frac 12}} f[1-u(s)] ds,\quad t>0,NEWLINE\]NEWLINE where \(u(t)=v(ct,t)\) unknown temperature at \(x=x_0=ct.\) Here \(h(t)=V(ct,t)\) with \(V(x,t)=\frac{1}{2(\pi t)^{\frac 12}}\int_{-\infty}^\infty e^{\frac{-(x-\xi)^2}{4t}}v_0 (\xi) d\xi\) and \(v_0(x)=v(x,0)\) is the initial condition (initial temperature). The nonlinearity \(f\) is assumed to have the properties: \(f(1-v)>0\), \(f_v(1-v)>0\), \(f_{vv}(1-v)>0\) for \(0<v<1\) and \(f(1-v) \to +\infty\) as \(v \to 1^-\). The authors give the bounds for critical speed when quenching will not occur or for the quenching time in opposite case.