Minimal blow-up asymptotics of quasilinear heat equations (Q2784200)

The authors study the problem \(u_t=(k(u)u_x)_x\) for \(x>0\) and \(0<t<1\) with boundary data \(u(0,t)=\psi(t)\) with \(\psi\) an increasing, nonnegative function with \(\psi(1)=\infty\) and \(k\) a smooth nonnegative function which is positive and increasing on \((0,\infty)\). They give a precise rate at which the solution \(u\) becomes infinite near the point \(x=0\), \(t=1\) under suitable conditions on \(\psi\) and \(k\). In fact, for their class of functions \(\pi\) and \(k\), this behavior is related in a simple fashion to the asymptotic behavior for solutions of the heat equation.