Boundedness of global solutions of a p-Laplacian evolution equation with a nonlinear gradient term

Article type: Research Article

Authors: Attouchi, Amal

Affiliations: Université Paris 13, Sorbonne Paris Cité, Laboratoire Analyse, Géométrie et Applications, CNRS, UMR 7539, 93430 Villetaneuse, France. E-mail: [email protected]

Abstract: We investigate the boundedness and large time behavior of solutions of the Cauchy–Dirichlet problem for the one-dimensional degenerate parabolic equation with gradient nonlinearity: ut=(|ux|p−2ux)x+|ux|q in (0,∞)×(0,1),q>p>2. We prove that: either ux blows up in finite time, or u is global and converges in W1,∞(0,1) to the unique steady state. This in particular eliminates the possibility of global solutions with unbounded gradient. For that purpose a Lyapunov functional is constructed by the approach of Zelenyak.

Keywords: asymptotic behavior, p-Laplacian, gradient term, boundedness, Lyapunov functional

DOI: 10.3233/ASY-141263

Journal: Asymptotic Analysis, vol. 91, no. 3-4, pp. 233-251, 2015