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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 $$ \left\{\eqalign{& u_t=u_{xx}+|u_x|^p,\quad t>0,\ 0<x<1,\cr & u(t,0)=0,\quad u(t,1)=M,\qquad t>0,\cr & u(0,x)=u_0(x),\qquad 0<x<1,\cr }\right. $$ where $p>2,$ $M\geq0$ and $u_0\in X:=\{v\in C^1[0,1]\colon\ v(0)=0,\ v(1)=M\}.$ 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.

35K60Nonlinear initial value problems for linear parabolic equations
35K65Parabolic equations of degenerate type
35B45A priori estimates for solutions of PDE
Full Text: EuDML
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