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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

u t =u xx +|u x | p ,t>0,0<x<1,u(t,0)=0,u(t,1)=M,t>0,u(0,x)=u 0 (x),0<x<1,

where p>2, M0 and u 0 X:={vC 1 [0,1]: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.

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