Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition (Q998150)

The author studies the following initial boundary value problem \[ \begin{alignedat}{2} u_t&=u^{1+\gamma}u_{xx}, &\quad& 0<x<1,\;t>0,\\ u_x(0,t)&=-u^q(0,t), &\quad& u_x(1,t)=0,\;t>0,\\ u(x,0) &=u_0(x), &\quad& 0\leqslant x\leqslant 1, \end{alignedat} \] where \(\gamma>0\) and \(q>0\) are given constants, \(u_0\) is a positive bounded smooth function defined on \([0,1]\) such that \(u'_0(0)=-u^q_0(0)\) and \(u'_0(1)=0\). The author first shows that any positive solution blows up in finite time. He also proves the single point blow-up for monotone solutions when \(q>1\) and estimates the blow-up rate for the case \(q\in (0,1)\). Then, in the single blow-up point case, the existence of a unique self-similar profile is proven. Moreover, by constructing a Lyapunov function, the author proves the convergence of the solution to the unique self-similar solution as \(t\) approaches the blow-up time.