Home  About us  Contact
 

Laplacian Equations (laplacian + equation)

Distribution by Scientific Domains
Mathematics and Statistics100%

Selected Abstracts

Power concavity on nonlinear parabolic flows

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 11 2005
Ki-Ahm Lee
Our object in this paper is to show that the concavity of the power of a solution is preserved in the parabolic p -Laplace equation, called power concavity, and that the power is determined by the homogeneity of the parabolic operator. In the parabolic p -Laplace equation for the density u, the concavity of u(p,2)/p is considered, which indicates why the log-concavity has been considered in heat flow, p = 2. In addition, the long time existence of the classical solution of the parabolic p -Laplacian equation can be obtained if the initial smooth data has -concavity and a nondegenerate gradient along the initial boundary. © 2004 Wiley Periodicals, Inc. [source]

Evolution completeness of separable solutions of non-linear diffusion equations in bounded domains

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 15 2004
V. A. Galaktionov
Abstract As a basic example, we consider the porous medium equation (m > 1) (1) where , , ,N is a bounded domain with the smooth boundary ,,, and initial data . It is well-known from the 1970s that the PME admits separable solutions , where each ,k , 0 satisfies a non-linear elliptic equation . Existence of at least a countable subset , = {,k} of such non-linear eigenfunctions follows from the Lusternik,Schnirel'man variational theory from the 1930s. The first similarity pattern t,1/(m,1),0(x), where ,0 > 0 in ,, is known to be asymptotically stable as t , , and attracts all nontrivial solutions with u0 , 0 (Aronson and Peletier, 1981). We show that if , is discrete, then it is evolutionary complete, i.e. describes the asymptotics of arbitrary global solutions of the PME. For m = 1 (the heat equation), the evolution completeness follows from the completeness-closure of the orthonormal subset , = {,k} of eigenfunctions of the Laplacian , in L2. The analysis applies to the perturbed PME and to the p -Laplacian equations of second and higher order. Copyright © 2004 John Wiley & Sons, Ltd. [source]

Limit behaviour of solutions to equivalued surface boundary value problem for p -Laplacian equations: II

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 10 2001
Li Fengquan
Abstract In this paper (which is a continuation of Part-I), we discuss the limit behaviour of solutions to boundary value problem with equivalued surface for p -Laplacian equations in the case of 1[source]