Localization of solutions of anisotropic parabolic equations

We study the localization properties of solutions of the Dirichlet problem for the anisotropic parabolic equations u(t) - Sigma(n)(i=1) D-i(a(i)(z, u)vertical bar D(i)u vertical bar(pi-2)D(i)u) = f(z), z = (x, t) is an element of Omega x (0, T), with constant exponents p(i) is an element of (1, infinity), and x is an element of Omega subset of R-n, n >= 2. Such equations arise from the mathematical description of diffusion processes. It is shown that if the equation combines the directions of slow diffusion for which p(i) > 2 and the directions of fast or linear diffusion corresponding to p(i) is an element of (1, 2) or p = 2, then the solutions may simultaneously display the properties intrinsic for the solutions of isotropic equations of fast or slow diffusion. Under the assumptions that f equivalent to 0 for t >= t(f) and u(0) equivalent to 0, f equivalent to 0 for x(1) > s we show, on the one hand, that the solution vanishes in a finite time if n/2 < Sigma(n)(i=1) 1/p(i) <= 1 + n/2 and, on the other hand, that the support of the same solution never reaches the plane x(1) = s + epsilon, provided that 1/n-1 >= 1/n-1 Sigma(n)(i=1) 1/p(i) > 1/p(1). (C) 2008 Elsevier Ltd. All rights reserved.