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

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Universitat Autònoma de Barcelona
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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.
Proceedings of the second conference on differential equations and their applications, II (Valldoreix, 1979)
P. A. Adams: “Sobolev Spaces”, Academic Press, New York (1975) H. Brezis: “Operateurs maximaux monotones et semigroupes de contractions dans les espaces de Hilbert” Notas de Matematica, North Holland (1973) H. Brezis: “Monotonicity methods in Hilbert Spaces and some applications to nonlinear partial differential equations”, en Contributions to Nonlinear Functional Analysis, E. Zarantonello, ed. (1971) H. Brezis – A. Friedman: “Estimates on the support of solutions of parabolic variational inequalities” III. Journal of Math. 20 (1976), pag. 82-99 G. Díaz – I. Díaz: “Finite extinction time for a class of nonlinear parabolic equations” (Aparecerá) I. Díaz – M. Herrero: “Proprietes de support compact pour certaines equations elliptiques et paraboliques non lineaires”, Compt. Rend. Acad. Sci. (Paris), Serie (1978), pag. 812-815 L. C. Evans – B. F. Knerr: “Instantaneous shrinking of the support of nonnegative solutions to certain nonlinear parabolic equations and variational inequalities”. (Aparecerá) M. Herrero: “Sobre el comportamiento de las soluciones de ciertos problemas parabólicos no lineales”, (aparecerá) M. Herrero – J. L. Vázquez: “On a class of nonlinear parabolic equations” (aparecerá) A. S. Kalashnikov: “The propagation of disturbances in problems of nonlinear heat conduction with absorption”. Zh. Vychsl. Mat. Mat Fiz 14 (1974) pag. 891-905 O. Ladyzenska’ia: “Sur de nouvelles equations dans la dynamique des fluids visqueux et leur resolution globale” Troudi. Mat. Inst. Stekloff (1976), pag. 85-104 J. L. Lions: “Quelques methods de resolution des problems aux limites non lineaires”, Ed. Dunod (1968) M. C. Pelissier: “Sur quelques problemas non lineaires en glaciologie”, These 3eme cycle. Publications mathematiques d’Orsay (1977)