Quenching, nonquenching, and beyond quenching for solution of some parabolic equations (Q1812907)

The author gives a detailed study of the \(1D\)-parabolic equation \(u_ t- u_{xx}=\varepsilon(1-u)^{-\beta}\) on \((0,1)\times[0,T)\) with \(0\leq u_ 0<1\) and \(u=0\) on \(\partial[0,1]\times[0,T)\), depending on the positive parameters \(\varepsilon\) and \(\beta\). He presents first a complete description of the stationary solutions, showing that there are two of them, \(0\leq f_ -\leq f_ +<1\), if \(\varepsilon\) belongs to some open intervall \((\varepsilon_ 0(\beta)\), \(\varepsilon(\beta))\) with \(\varepsilon_ 0(\beta)=\max\{0,8(1-\beta)/(1+\beta)^ 2\}\), one for \(\varepsilon=\varepsilon(\beta)\) and none for \(\varepsilon>\varepsilon(\beta)\). Also for \(\varepsilon<\varepsilon_ 0(\beta)\) in case of \(\beta<1\), there is exactly one stationary solution of class \(C_ 2\). Concerning the time dependent problem he shows for \(\beta \geq 1\), that \(f_ -\) is stable with \(u\to f_ -\), provided \(u_ 0\leq f_ +\), while there are initial values in an arbitrary neighbourhood of \(f_ +\), such that \(u\) quenches (meaning \(u\to 1\) in finite time). Further quenching always occurs if there is no stationary solution (the case \(\varepsilon>\varepsilon(\beta))\). The complete study for the case \(\beta<1\) and the similar results for the related problem \(u_ t- u_{xx}=0\), \(0\leq u_ 0<1\), \(u(0,t)=0\), but \(u_ x(1,t)=\varepsilon(1- u(1,t))^{-\beta}\) are also given.