Global behavior for a class of nonlinear evolution equations. (English) Zbl 0572.35062

Sacks, Paul E.

SIAM J. Math. Anal. 16, 233-250 (1985).

The author considers the initial-boundary value problem \[ v_ t=\Delta | v|^{m-1}v+\lambda | v|^{p-1}v\quad in\quad \Omega \times R^+;\quad v=0\quad in\quad \partial \Omega \times R^+;\quad v=v_ 0\quad in\quad \Omega \times \{0\}. \] Here \(\Omega\) is an open bounded domain \(R^ n\), and \(\lambda\geq 0\), \(m>1\), \(p\geq 1\). He provides an exhaustive classification of the global behavior of the solution (including decay estimates in the various cases that may arise, according to the values of m, n and \(\lambda\). Among other things he proves that if \(p<m\) the global existence of the solutions is ensured if \(v_ 0\in L^ 1(\Omega)\); if, moreover, \(v_ 0\geq 0\) and \(v_ 0\neq 0\), then the solution converges, as t goes to infinity, to the (unique) positive steady state. To prove this result, the author extends to the case under consideration a one-sided estimate for the Laplacian of \(v^ m\), which had been first established, in case \(\lambda =0\), by Aronson and Bénilan. Comparison methods and the construction of appropriate super and subsolutions play a major rôle in the proofs.

Reviewer: P.de Mottoni

MSC:

35K60	Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40	Asymptotic behavior of solutions to PDEs
35B05	Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35A05	General existence and uniqueness theorems (PDE) (MSC2000)
76S05	Flows in porous media; filtration; seepage

Keywords:

porous media with absorption; large time behaviour; global behavior; decay estimates; positive steady state; Comparison methods; super and subsolutions