Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Existence and uniqueness for a degenerate parabolic equation with $L^{1}$-data


Authors: F. Andreu, J. M. Mazón, S. Segura de León and J. Toledo
Journal: Trans. Amer. Math. Soc. 351 (1999), 285-306
MSC (1991): Primary 35K65, 47H20
DOI: https://doi.org/10.1090/S0002-9947-99-01981-9
MathSciNet review: 1433108
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we study existence and uniqueness of solutions for the boundary-value problem, with initial datum in $L^{1}(\Omega )$,

\begin{equation*}u_{t} = \mathrm{div} \ {\hbox{$\mathbf a$}}(x,Du) \quad \text{in } (0, \infty ) \times \Omega, \end{equation*}

\begin{equation*}-{\frac{{\partial u} }{{\partial \eta _{a}}}} \in \beta (u) \quad \text{on } (0, \infty ) \times \partial \Omega,\end{equation*}

\begin{equation*}u(x, 0) = u_{0}(x) \quad \text{in }\Omega ,\end{equation*}

where a is a Carathéodory function satisfying the classical Leray-Lions hypothesis, $\partial / {\partial \eta _{a}}$ is the Neumann boundary operator associated to ${\hbox{$\mathbf a$}}$, $Du$ the gradient of $u$ and $\beta $ is a maximal monotone graph in ${\mathbb{R}}\times {\mathbb{R}}$ with $0 \in \beta (0)$.


References [Enhancements On Off] (What's this?)

  • [AMST] F. Andreu, J. M. Mazón, S. Segura de León and J. Toledo, Quasi-linear elliptic and parabolic equations in $L^{1}$ with non-linear boundary conditions, Adv. Math. Sci. Appl. 7 (1997), 183-213. MR 98f:35079
  • [Ba] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976. MR 52:11166
  • [Be] Ph. Bénilan, Equations d'évolution dans un espace de Banach quelconque et applications, Thèse Orsay, 1972.
  • [B-V] Ph. Bénilan, L. Boccardo, Th. Gallouët, R. Gariepy, M. Pierre and J. L. Vazquez, An $L^{1}$-Theory of Existence and Uniqueness of Solutions of Nonlinear Elliptic Equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22 (1995), 241-273. MR 96k:35052
  • [BBC] Ph. Bénilan, H. Brezis and M. G. Crandall, A semilinear equation in $L^{1}$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), 523-555. MR 52:11299
  • [BCr-1] Ph. Bénilan and M. G. Crandall, Regularizing effects of homogeneous evolution equations, in Contribution to Analysis and Geometry (D. N. Clark et al., eds.), Johns Hopkins University Press, 1981, pp. 23-39. MR 83g:47063
  • [BCr-2] Ph. Bénilan and M. G. Crandall, Completely accretive operators, in Semigroup Theory and Evolution Equations (Ph. Clement et al., eds.), Marcel Dekker, 1991, pp. 41-76. MR 93e:47071
  • [BCP] Ph. Bénilan, M. G. Crandall and A. Pazy, Evolution Equations Governed by Accretive Operators, Forthcoming.
  • [BCS] Ph. Bénilan, M. G. Crandall and P. Sacks, Some $L^{1}$ Existence and Dependence Results for Semilinear Elliptic Equations under Nonlinear Boundary Conditions, Appl. Math. Optim. 17 (1988), 203-224. MR 89d:35055
  • [BG-1] L. Boccardo and Th. Gallouët, Non-linear Elliptic and Parabolic Equations Involving Measure Data, J. Funct. Anal. 87 (1989), 149-169. MR 92d:35286
  • [BG-2] L. Boccardo and Th. Gallouët, Nonlinear elliptic equations with right-hand side measures, Comm. in Partial Diff. Equations 17 (1992), 641-655. MR 94c:35083
  • [Br-1] H. Brézis, Problèmes Unilatéraux, J. Math. Pures et Appl. 51 (1972), 1-168. MR 55:1166
  • [Br-2] H. Brézis, Opérateur maximaux monotone et semi-groupes de contractions dans les espaces de Hilbert, North-Holland, 1973. MR 50:1060
  • [Cr] M. G. Crandall, Nonlinear Semigroups and Evolution Governed by Accretive Operators, Proc. Symposia in Pure Math., vol. 45, Amer. Math. Soc., 1986, pp. 305-336. MR 87h:47140
  • [Di-1] E. Di Benedetto, Degenerate Parabolic Equations, Springer-Verlag, 1993. MR 94h:35130
  • [Di-2] E. Di Benedetto, Degenerate and singular parabolic equations, in Recent Advances in Partial Differential Equations (M. A. Herrero and E. Zuazua eds., eds.), Wiley-Masson, 1994, pp. 55-84. MR 95a:35078
  • [DiH-1] E. Di Benedetto and M. A. Herrero, On the Cauchy problem and initial traces for a degenerate parabolic equations, Trans. Amer. Math. Soc. 314 (1989), 187-224. MR 90d:35139
  • [DiH-2] E. Di Benedetto and M. A. Herrero, Non-negative Solutions of the Evolution p-Laplacian Equation. Initial Traces and Cauchy problem when $1 < p < 2$, Arch. Rat. Mech. Anal. 111 (1990), 225-290. MR 92g:35088
  • [DH] J. I. Diaz and M. A. Herrero, Estimates on the support of the solutions of some nonlinear elliptic and parabolic problems, Proc. Royal Soc. Edinburgh 89A (1981), 249-258. MR 83i:35019
  • [DL] G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, 1976. MR 58:25191
  • [Fr] A. Friedman, Generalized Heat Transfer between Solids and Gases under Nonlinear Boundary Conditions, J. Math. Mech. 51 (1959), 161-183. MR 21:1138
  • [KS] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications, Academic Press, 1980. MR 81g:49013
  • [K] M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press, 1964. MR 28:2414
  • [L] J. L. Lions, Quelques méthodes de résolution de problémes aux limites non linéaires, Dunod/Gauthier-Vilars, 1968. MR 41:4326
  • [M] C. B. Morrey, Jr., Multiple Integrals in the Calculus of Variations, Springer-Verlag, 1966. MR 34:2380
  • [N] J. Ne\v{c}as, Les Méthodes Directes en Théorie des Equations Elliptiques, Masson et Cie, Paris, 1967. MR 37:3168
  • [Ra-1] J. M. Rakotoson, Some Quasilinear Parabolic Equations, Nonlinear Analysis T. M. A. 17 (1991), 1163-1175. MR 93a:35074
  • [Ra-2] J. M. Rakotoson, A Compactness Lemma for Quasilinear Problems: Application to Parabolic Equations, J. Funct. Anal. 106 (1992), 358-374. MR 94a:35067
  • [X] X. Xu, A p-Laplacian problem in $L^{1}$ with nonlinear boundary conditions, Commun. in Partial Differential Equations 19 (1994), 143-176. MR 95b:35075
  • [Zi] W. P. Ziemer, Weakly Differentiable Functions, Springer-Verlag, 1989. MR 91e:46046

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 35K65, 47H20

Retrieve articles in all journals with MSC (1991): 35K65, 47H20


Additional Information

F. Andreu
Affiliation: Departamento de Análisis Matemático, Universitat de València, 46100 Burjassot, Valencia, Spain
Email: Fuensanta.Andreu@uv.es

J. M. Mazón
Affiliation: Departamento de Análisis Matemático, Universitat de València, 46100 Burjassot, Valencia, Spain
Email: Mazon@uv.es

S. Segura de León
Affiliation: Departamento de Análisis Matemático, Universitat de València, 46100 Burjassot, Valencia, Spain
Email: Sergio.Segura@uv.es

J. Toledo
Affiliation: Departamento de Análisis Matemático, Universitat de València, 46100 Burjassot, Valencia, Spain
Email: Jose.Toledo@uv.es

DOI: https://doi.org/10.1090/S0002-9947-99-01981-9
Keywords: Non-linear parabolic equations, non-linear boundary conditions, $p$-Laplacian, accretive operators, mild solutions
Received by editor(s): September 11, 1995
Received by editor(s) in revised form: December 2, 1996
Additional Notes: This research has been partially supported by DGICYT, Project PB94-0960
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society