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

Abstract

We investigate the boundedness and large time behavior of solutions of the Cauchy-Dirichlet problem for the onedimensional degenerate parabolic equation with gradient nonlinearity:We prove that: either ux blows up in finite time, or u is global and converges in W 1,∞ (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.

Gradient blow-up rates and sharp gradient estimates for diffusive Hamilton-Jacobi equations

“…Results include blowup criteria [1,2,41,27], blowup locations [16,32,43], blowup profiles [4,14,35,36,37,43], continuation after GBU [12,38,37,39,36,21], infinite time GBU [43,42]. See also [15,30,18,25,3,5,20,6,45,7,9,33,10,17] for GBU studies for other equations. As a consequence of interior gradient estimates [43], it is known that GBU for problem (1.1) can only take place on the boundary ∂Ω.…”

Single point gradient blow-up on the boundary for a Hamilton-Jacobi equation with $p$-Laplacian diffusion

“…The large time behavior of global solutions in bounded or unbounded domains has been studied in [16,25,15,5,4,14]. Concerning the asymptotic description of singularities, results on the gradient blow-up rate in one space dimension can be found in [3,26,27]. On the other hand, in any space dimension, it is known [2] that gradient blow-up can take place only on the boundary, i.e.…”

“…and u ≤ Cδ q−p q−p+1 in Q T , where δ(x) = dist(x, ∂Ω), (1.7) and they are sharp in one space dimension [3]. (Our nondegeneracy lemma 7.1 below indicates that they are also sharp in higher dimensions.…”