Localization of solutions of anisotropic parabolic equations Original Research Article

We study the localization properties of solutions of the Dirichlet problem for the anisotropic parabolic equations View the MathML sourceut−∑i=1nDi(ai(z,u)|Diu|pi−2Diu)=f(z),z=(x,t)∈Ω×(0,T), Turn MathJax on with constant exponents pi∈(1,∞)pi∈(1,∞), and x∈Ω⊂Rnx∈Ω⊂Rn, n≥2n≥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 pi>2pi>2 and the directions of fast or linear diffusion corresponding to pi∈(1,2)pi∈(1,2) or p=2p=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≡0f≡0 for t≥tft≥tf and u0≡0u0≡0, f≡0f≡0 for x1>sx1>s we show, on the one hand, that the solution vanishes in a finite time if View the MathML sourcen2<∑i=1n1pi≤1+n2 and, on the other hand, that the support of the same solution never reaches the plane x1=s+ϵx1=s+ϵ, provided that View the MathML source1n−1≥1n−1∑i=2n1pi>1p1.