A multi-dimensional quenching problem due to a concentrated nonlinear source in \(\mathbb R^N\)
The authors consider the Cauchy problem for the equation \(u_t-\Delta u=\alpha\delta_b(|x|)f(u)\) in \(\mathbb R^N\times(0,T]\), where \(\alpha>0\) and \(\delta_b\) is the one-dimensional Delta distribution at \(b>0\). Moreover \(f\), \(f'\) and \(f''\) are positive in the interval \([0,c)\) and \(\lim_{u\to c^-}f(u)=\infty\). Under suitable assumptions they show that a vanishing initial datum evolves into a nonnegative solution which quenches, that is reaches height \(c\), in finite time and exactly on the sphere with radius \(b\). Some related results are also given.
scientific article; zbMATH DE number 5316883