Localization of solutions of anisotropic parabolic equations

Abstract

We study the localization properties of solutions of the Dirichlet problem for the anisotropic parabolic equations u t − ∑ i = 1 n D i ( a i ( z , u ) | D i u | p i − 2 D i u ) = f ( z ) , z = ( x , t ) ∈ Ω × ( 0 , T ) , with constant exponents p i ∈ ( 1 , ∞ ) , and x ∈ Ω ⊂ 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 ∈ ( 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 ≡ 0 for t ≥ t f and u 0 ≡ 0 , f ≡ 0 for x 1 > s we show, on the one hand, that the solution vanishes in a finite time if n 2 < ∑ i = 1 n 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 + ϵ , provided that 1 n − 1 ≥ 1 n − 1 ∑ i = 2 n 1 p i > 1 p 1 .