*(English)*Zbl 1072.35098

The paper deals with the Cauchy–Dirichlet problem for one-dimensional semilinear parabolic equations with a gradient nonlinearity of the form

where $p>2,$ $M\ge 0$ and ${u}_{0}\in X:=\{v\in {C}^{1}[0,1]:\phantom{\rule{4pt}{0ex}}v\left(0\right)=0,\phantom{\rule{4pt}{0ex}}v\left(1\right)=M\}\xb7$

The authors provide a complete classification of large time behaviour of the classical solutions $u\xb7$ Precisely, either the space derivative ${u}_{x}$ blows up in a finite time with $u$ itself remaining bounded, or $u$ is global and converges in ${C}^{1}$-norm to the unique steady state. The main difficulty concerns the proof of ${C}^{1}$-boundedness of all global solutions, and in avoinding it the authors compute a nontrivial Lyapunov functional by carrying out a method introduced by T. Zelenyak. After deriving precise estimates on the solutions and on the Lyapunov functional, they argue by contradiction by showing that any ${C}^{1}$ unbounded global solution should converge to a singular stationary solution, which does not exist.