Asymptotic analysis of the p-Laplacian flow in an exterior domain

Abstract

We consider the Dirichlet problem for the p-Laplacian evolution equation, u t = p u, where p > 2, posed in an exterior domain in R N , with zero Dirichlet boundary condition and with integrable and nonnegative initial data. We are interested in describing the influence of the holes of the domain on the large time behaviour of the solutions. Such behaviour varies depending on the relative values of N and p. We must distinguish between the behaviour near infinity of space (outer analysis), and near the holes (inner analysis). We obtain that the outer analysis is given in all cases by certain self-similar solutions and the inner analysis is given by quasi-stationary states. Logarithmic corrections to exact self-similarity appear in the critical case N = p, which is mathematically more interesting. In this first paper we treat only the cases N > p and N = p, the case N < p will be considered in a companion work.

The first nontrivial eigenvalue for a system ofp-Laplacians with Neumann and Dirichlet boundary conditions

“…On the other hand, by (14) and Lemma 2.2, we have that {v n } n∈N is bounded in W 1,q (Ω). Hence, by the Sobolev embedding theorem, there exist a subsequence, still denoted by {v n } n∈N , and v ∈ W 1,q (Ω) such that (16) v n ⇀ v weakly in W 1,q (Ω),…”

“…Eigenvalues for the p−Laplacian are related to the asymptotic behaviour of solutions to the corresponding evolutions equations, see, for example, [6,16,17].…”

Anomalous large-time behaviour of the $p$-Laplacian flow in an exterior domain in low dimension

“…By standard properties of the theory of the p-Laplacian equation the solution will be bounded for all t > 0, hence we may also assume that u 0 is bounded in the study of large-time behaviour. The study of the cases where N ≥ p was the object of a companion paper [14], where it was proved that for N > p the large-time influence of the holes is moderate since the outflow through ∂ does not exhaust the whole initial mass u 0 (x) dx as t → ∞, and the asymptotic profiles are still of the same type as in the Cauchy problem (Barenblatt profiles). For N = p the mass goes to zero but the asymptotic rates and profiles can be considered as a limit case of the previous situation with the inclusion of logarithmic factors.…”

On the long-time behaviour of solutions of the p-Laplacian parabolic system

“…The paper of S. Kamin and J. L. Vázquez [9] gives a detailed description of the asymptotics of positive solutions of the p-Laplacian equation; Del Pino and Dolbeault [4] establish, for non-negative initial data and under some assumptions on p, not only the convergence to stationary solutions, but also an estimate on the convergence rate. Other results on the asymptotics of solutions in unbounded domains or in the whole space are due to Lee, Petrosyan and Vázquez [12], and Iagar and Vázquez [8].…”