Quenching problem in nonhomogeneous media (Q1979020)

The author studies critical values of \(\lambda \) for which the problem \[ u_t=\Delta u+\lambda k(x)(1-u)^{-\beta},\quad u(0,x)=u_0(x), \quad u|_{\partial \Omega}=0 \] has a global solution, generalizing similar results obtained for \(k\equiv 1\). Existence of a finite critical value \(\lambda ^\ast \) as well as some estimates of \(\lambda ^\ast \) depending on \(k\) are proved and a comparison of the corresponding minimal solutions is given.