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Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena. (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

$\left\{\begin{array}{cc}& {u}_{t}={u}_{xx}+{|{u}_{x}|}^{p},\phantom{\rule{1.em}{0ex}}t>0,\phantom{\rule{4pt}{0ex}}00,\hfill \\ & u\left(0,x\right)={u}_{0}\left(x\right),\phantom{\rule{2.em}{0ex}}0

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

The authors provide a complete classification of large time behaviour of the classical solutions $u·$ 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.

##### MSC:
 35K60 Nonlinear initial value problems for linear parabolic equations 35K65 Parabolic equations of degenerate type 35B45 A priori estimates for solutions of PDE