Degenerate parabolic equations with initial data measures
HTML articles powered by AMS MathViewer
- by Daniele Andreucci PDF
- Trans. Amer. Math. Soc. 349 (1997), 3911-3923 Request permission
Abstract:
We address the problem of existence of solutions to degenerate (and nondegenerate) parabolic equations under optimal assumptions on the initial data, which are assumed to be measures. The requirements imposed on the initial data are connected both with the degeneracy of the principal part of the equation, and with the form of the nonlinear forcing term. The latter depends on the space gradient of a power of the solution. Applications to related problems are also outlined.References
- Herbert Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function spaces, differential operators and nonlinear analysis (Friedrichroda, 1992) Teubner-Texte Math., vol. 133, Teubner, Stuttgart, 1993, pp. 9–126. MR 1242579, DOI 10.1007/978-3-663-11336-2_{1}
- Daniele Andreucci, New results on the Cauchy problem for parabolic systems and equations with strongly nonlinear sources, Manuscripta Math. 77 (1992), no. 2-3, 127–159. MR 1188577, DOI 10.1007/BF02567050
- D. Andreucci and E. DiBenedetto, A new approach to initial traces in nonlinear filtration, Ann. Inst. H. Poincaré C Anal. Non Linéaire 7 (1990), no. 4, 305–334 (English, with French summary). MR 1067778, DOI 10.1016/S0294-1449(16)30294-3
- D. Andreucci and E. DiBenedetto, On the Cauchy problem and initial traces for a class of evolution equations with strongly nonlinear sources, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 18 (1991), no. 3, 363–441. MR 1145316
- Alfred Rosenblatt, Sur les points singuliers des équations différentielles, C. R. Acad. Sci. Paris 209 (1939), 10–11 (French). MR 85
- Pierre Baras and Michel Pierre, Critère d’existence de solutions positives pour des équations semi-linéaires non monotones, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), no. 3, 185–212 (French, with English summary). MR 797270, DOI 10.1016/S0294-1449(16)30402-4
- Philippe Bénilan, Michael G. Crandall, and Michel Pierre, Solutions of the porous medium equation in $\textbf {R}^{N}$ under optimal conditions on initial values, Indiana Univ. Math. J. 33 (1984), no. 1, 51–87. MR 726106, DOI 10.1512/iumj.1984.33.33003
- Lucio Boccardo and Thierry Gallouët, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal. 87 (1989), no. 1, 149–169. MR 1025884, DOI 10.1016/0022-1236(89)90005-0
- Emmanuele DiBenedetto and Avner Friedman, Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math. 357 (1985), 1–22. MR 783531, DOI 10.1515/crll.1985.357.1
- Yoshikazu Giga and Tetsuro Miyakawa, Navier-Stokes flow in $\mathbf R^3$ with measures as initial vorticity and Morrey spaces, Comm. Partial Differential Equations 14 (1989), no. 5, 577–618. MR 993821, DOI 10.1080/03605308908820621
- A. L. Gladkov, On the Cauchy problem in classes of growing functions for the filtration equation with convection, Mat. Sb. 186 (1995), no. 6, 35–56 (Russian, with Russian summary); English transl., Sb. Math. 186 (1995), no. 6, 803–825. MR 1349013, DOI 10.1070/SM1995v186n06ABEH000044
- Hiroki Hoshino and Yoshio Yamada, Solvability and smoothing effect for semilinear parabolic equations, Funkcial. Ekvac. 34 (1991), no. 3, 475–494. MR 1150875
- K. Kobayasi, Semilinear parabolic equations with nonmonotone nonlinearity, Memoirs Sagami Inst. of Technology, 23 (1989), 83–99.
- Hideo Kozono and Masao Yamazaki, Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data, Comm. Partial Differential Equations 19 (1994), no. 5-6, 959–1014. MR 1274547, DOI 10.1080/03605309408821042
- O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva, Lineĭ nye i kvazilineĭ nye uravneniya parabolicheskogo tipa, Izdat. “Nauka”, Moscow, 1967 (Russian). MR 0241821
- Y. Niwa, Semilinear heat equations with measures as initial data, Thesis, University of Tokyo, 1986.
- Michael E. Taylor, Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations, Comm. Partial Differential Equations 17 (1992), no. 9-10, 1407–1456. MR 1187618, DOI 10.1080/03605309208820892
Additional Information
- Daniele Andreucci
- Affiliation: Università “La Sapienza”, Dipartimento di Metodi e Modelli Matematici, via A. Scarpa 16, 00161 Roma, Italy
- Email: andreucc@dmmm.uniroma1.it
- Received by editor(s): August 15, 1994
- Received by editor(s) in revised form: March 20, 1995
- Additional Notes: The author is a member of GNFM of Italian CNR. Work supported by MURST project “Problemi non lineari...”
- © Copyright 1997 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 349 (1997), 3911-3923
- MSC (1991): Primary 35K65, 35R05; Secondary 35K55, 35K15
- DOI: https://doi.org/10.1090/S0002-9947-97-01530-4
- MathSciNet review: 1333384