Evolution of the solutions of some diffusion problems with absorption (Spanish: Evolución de las soluciones de ciertos problemas de difusión con absorción)

Abstract

This note is an account of results obtained by the author [Rev. Real Acad. Cienc. Exact. Fís. Natur. Madrid 75 (1981), no. 5, 1165–1183; MR0649591 (83m:35076)], and the author and J. L. Vázquez ["On a class of nonlinear parabolic equations'', to appear] about the property of compact support of solutions of the Cauchy problem ut=∑(∂/∂xi)(|∂u/∂xi|p−2∂u/∂xi)+α(u) in RN×(0,T), 1<p<+∞, u(0)=u0(x) in RN. The assumptions on the initial datum are u0∈L2(RN)∩L∞(RN), u0≥0, u0(x)→0 uniformly as |x|→∞, and on the absorption term α(u) they are ∫10ds/[sα(s)]1/p<∞ when p>2, and ∫10ds/α(s)<∞ when 1<p≤2. It is shown, by means of comparison with suitable supersolutions, that for t>0 the support of x↦u(t,x) is compact (even if the initial datum is not compactly supported) and that the solution disappears in finite time, i.e., u(x,t)≡0 if t>t0, where t0 is a positive number depending upon u0.