On the Cauchy problem and initial traces for a degenerate parabolic equation
| dc.contributor.author | Di Benedetto, E. | |
| dc.contributor.author | Herrero, Miguel A. | |
| dc.date.accessioned | 2023-06-20T17:05:58Z | |
| dc.date.available | 2023-06-20T17:05:58Z | |
| dc.date.issued | 1989-07 | |
| dc.description.abstract | The authors study the Cauchy problem for the degenerate parabolic equation ut = div(|Du| p−2 Du)(p<2), and find sufficient conditions on the initial trace u0 (and in particular on its behaviour as |x|→∞) for existence of a solution in some strip RN × (0,T). Using a Harnack type inequality they show that these conditions are optimal in the case of nonnegative solutions. Uniqueness of solutions is shown if u0 belongs to L1loc(RN), but is left open in the case that u0 is merely a locally bounded measure. The results are closely related to papers by Aronson-Caffarelli, Benilan-Crandall-Pierre, and Dahlberg-Kenig about the porous medium equation ut = Δum. The proofs are different and allow one to generalize some of the above results to equations with variable coefficients. | |
| dc.description.department | Depto. de Análisis Matemático y Matemática Aplicada | |
| dc.description.faculty | Fac. de Ciencias Matemáticas | |
| dc.description.faculty | Instituto de Matemática Interdisciplinar (IMI) | |
| dc.description.refereed | TRUE | |
| dc.description.sponsorship | NSF | |
| dc.description.sponsorship | CICYT | |
| dc.description.status | pub | |
| dc.eprint.id | https://eprints.ucm.es/id/eprint/17482 | |
| dc.identifier.doi | 10.1090/S0002-9947-1989-0962278-5 | |
| dc.identifier.issn | 0002-9947 | |
| dc.identifier.officialurl | http://www.ams.org/journals/tran/1989-314-01/S0002-9947-1989-0962278-5/S0002-9947-1989-0962278-5.pdf | |
| dc.identifier.relatedurl | http://www.ams.org | |
| dc.identifier.uri | https://hdl.handle.net/20.500.14352/57767 | |
| dc.issue.number | 1 | |
| dc.journal.title | Transactions of the American Mathematical Society | |
| dc.language.iso | eng | |
| dc.page.final | 224 | |
| dc.page.initial | 187 | |
| dc.publisher | American Mathematical Society | |
| dc.relation.projectID | DMS-8502297 | |
| dc.relation.projectID | PB86-0112-C02-0 | |
| dc.rights.accessRights | restricted access | |
| dc.subject.cdu | 517.9 | |
| dc.subject.cdu | 517.955 | |
| dc.subject.keyword | Cauchy problem | |
| dc.subject.keyword | porous medium equation | |
| dc.subject.keyword | existence | |
| dc.subject.keyword | Harnack-type inequality | |
| dc.subject.keyword | Uniqueness | |
| dc.subject.ucm | Ecuaciones diferenciales | |
| dc.subject.unesco | 1202.07 Ecuaciones en Diferencias | |
| dc.title | On the Cauchy problem and initial traces for a degenerate parabolic equation | |
| dc.type | journal article | |
| dc.volume.number | 314 | |
| dcterms.references | N. D. Alikakos and R. Rostamian, Gradient estimates for degenerate diffusion equations II, Proc. Roy. Soc. Edinburgh Sect. A 91 (1981/82), 335-346. D. G. Aronson, Widder's inversion theorem and the initial distribution problem, SIAM J. Math. Anal. 12, 4 (1981), 639-651. D. G. Aronson and L. A. Caffarelli, The initial trace of a solution of the porous medium equation, Trans. Amer. Math. Soc. 380 1 (1983), 351-366. Ph. Bènilan and M. G. Crandall, Regularizing effects of homogeneous evolution equations, MRC Tech. Rep. 2076, Madison, Wis., 1980. Ph. Bènilan, M. G. Crandall, and M. Pierre, Solutions of the porous medium equation in RN under optimal conditions on initial values, Indiana Univ. Math. J. 33 (1984), 51-87. B. E. J. Dahlberg and C. E. Kenig, Nonnegative solutions of the porous medium equation, Comm. Partial Differential Equations 9 (1984), 409-437. E. DiBenedetto, On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 13, 3 (1986), 487-535. E. DiBenedetto, Intrinsic Harnack type inequalities for solutions of certain degenerate parabolic equations, Arch. Rational Mech. Anal. 100 (1988), 129-147. E. DiBenedetto and A. Friedman, Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math. 357 (1985), 1-22. M. A. Herrero and J. L. Vázquez, Asymptotic behaviour of the solutions of a strongly nonlinear parabolic problem, Ann. Fac. Sci. Toulouse Math. 3 (1981), 113-127. A. S. Kalashnikov, Cauchy's problem in classes of increasing functions for certain quasi-linear degenerate parabolic equations of the second order, Differencial'nye Uravnenija 9 (1973), 682-691. A. S. Kalashnikov, On uniqueness conditions for the generalized solutions of the Cauchy problem for a class of quasi-linear degenerate parabolic equations, Differencial'nye Uravnenija 9 (1973), 2207-2212. O. A. Ladyzenskaja, V. A. Solonnikov, and N. N. Ural'tseva, Linear and quasi-linear equations of parabolic type, Transl. Math. Monographs, vol. 23, Amer. Math. Soc, Providence, R. I., 1968. A. N. Tychonov, Théorèmes d'unicité pour l'équation de la chaleur, Mat. Sb. 42 (1935), 199-216. D. V. Widder, Positive temperatures in an infinite rod, Trans. Amer. Math. Soc. 55 (1944), 85-95. G. I. Barenblatt, On some unsteady motions of a liquid or a gas in a porous medium, Prikl. Mat. Meh. 16 (1952), pp. 67-78. M. Porzio and P. Vincenzotti, A priori bounds for weak solutions of certain degenerate parabolic equations (to appear). Chen Ya-zhe and E. DiBenedetto, Boundary estimates for solutions of nonlinear degenerate parabolic systems, J. Reine Angew. Math. 395 (1989), 102-131. | |
| dspace.entity.type | Publication |
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