Generalized solutions of a nonlinear parabolic equation with generalized functions as initial data |
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Authors: | Jorge Aragona Antnio Ronaldo Gomes Garcia Stanley Orlando Juriaans |
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Institution: | aUniversidade de São Paulo, Instituto de Matemática e Estatística, Brazil;bUniversidade do Estado do Rio Grande do Norte, Departamento de Matemática e Estatística, Brazil;cUniversidade Federal Rural do Semi-Árido, Departamento de Ciências Exatas e Naturais, Brazil |
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Abstract: | In H. Brézis, A. Friedman, Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pure Appl. (9) (1983) 73–97.] Brézis and Friedman prove that certain nonlinear parabolic equations, with the δ-measure as initial data, have no solution. However in J.F. Colombeau, M. Langlais, Generalized solutions of nonlinear parabolic equations with distributions as initial conditions, J. Math. Anal. Appl (1990) 186–196.] Colombeau and Langlais prove that these equations have a unique solution even if the δ-measure is substituted by any Colombeau generalized function of compact support. Here we generalize Colombeau and Langlais’ result proving that we may take any generalized function as the initial data. Our approach relies on recent algebraic and topological developments of the theory of Colombeau generalized functions and results from J. Aragona, Colombeau generalized functions on quasi-regular sets, Publ. Math. Debrecen (2006) 371–399.]. |
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Keywords: | Colombeau algebra Generalized function Initial data Parabolic equation |
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