S. Toutefois, X. On-ne-suppose-pas, and . Complet, on a encore : 1. si ?(t) est complet, alors (a) les assertions 1., 2. et 3

A. Ambrosetti and A. Malchiodi, Nonlinear analysis and semilinear elliptic problems, volume 104 de Cambridge Studies in Advanced Mathematics, 2007.

D. D. Ang and T. Tran, Synopsis, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, vol.840, issue.1-2, pp.119-133, 1990.
DOI : 10.1090/S0025-5718-1981-0595041-4

S. N. Antontsev, G. Gagneux, R. Luce, and G. Vallet, New unilateral problems in stratigraphy, ESAIM: Mathematical Modelling and Numerical Analysis, vol.40, issue.4, pp.765-784, 2006.
DOI : 10.1051/m2an:2006029

S. N. Antontsev, G. Gagneux, R. Luce, and G. Vallet, A NON-STANDARD FREE BOUNDARY PROBLEM ARISING FROM STRATIGRAPHY, Analysis and Applications, vol.04, issue.03, pp.209-236, 2006.
DOI : 10.1142/S0219530506000759

S. N. Antontsev, G. Gagneux, R. Luce, and G. Vallet, On a pseudoparabolic problem with constraint, Differential Integral Equations, vol.19, issue.12, pp.1391-1412, 2006.

S. N. Antontsev, G. Gagneux, A. Mokrani, and G. Vallet, Stratigraphic modelling by the way of a pseudoparabolic problem with constraint, Advances in Mathematical Science and Applications, vol.19, pp.195-209, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00866255

S. N. Antontsev, G. Gagneux, and G. Vallet, Some Stratigraphic Control Problems, Journal of Applied Mechanics and Technical Physics, vol.44, issue.6, pp.85-94, 2003.
DOI : 10.1023/A:1026287705015

S. N. Antontsev, G. Gagneux, and G. Vallet, A compactness result for a pseudoparabolic conservation law with constraint, Ninth International Conference Zaragoza- Pau on Applied Mathematics and Statistics de Monogr. Semin. Mat. García Galdeano, pp.403-410, 2006.

D. Ba?-inov and P. Simeonov, Integral inequalities and applications, volume 57 de Mathematics and its Applications (East European Series), 1992.

G. I. Barenblatt, M. Bertsch, R. Dal-passo, and M. Ughi, A Degenerate Pseudoparabolic Regularization of a Nonlinear Forward-Backward Heat Equation Arising in the Theory of Heat and Mass Exchange in Stably Stratified Turbulent Shear Flow, SIAM Journal on Mathematical Analysis, vol.24, issue.6, pp.1414-1439, 1993.
DOI : 10.1137/0524082

G. I. Barenblatt, J. Garcia-azorero, A. De-pablo, and J. Vazquez, Mathematical model of the non equilibrium water oil displacement in porous strata, Applicable Analysis, vol.2, issue.1-2, pp.19-45, 1997.
DOI : 10.1063/1.1745403

G. I. Barenblatt, Y. P. Zheltov, and I. N. Kochina, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata], Journal of Applied Mathematics and Mechanics, vol.24, issue.5, pp.1286-1303, 1960.
DOI : 10.1016/0021-8928(60)90107-6

A. Beliaev, Positive solutions of the porous medium equation with hysteresis, Journal of Mathematical Analysis and Applications, vol.281, issue.1, pp.125-137, 2003.
DOI : 10.1016/S0022-247X(02)00568-1

. Ph, D. Bénilan, H. Blanchard, and . Ghidouche, On a nonlinear system for shape memory alloys, Contin. Mech. Thermodyn, vol.2, issue.1, pp.65-76, 1990.

P. Bénilan, M. G. Crandall, and A. Pazy, Bonnes solutions " d'un problème d'évolution semi-linéaire. ( " Mild solutions " of a semilinear evolution problem), C. R. Acad. Sci, issue.13, pp.306527-530, 1988.

T. Benjamin and J. Bona, Model Equations for Long Waves in Nonlinear Dispersive Systems, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.272, issue.1220, pp.47-78, 1972.
DOI : 10.1098/rsta.1972.0032

H. Berliocchi and J. Lasry, Int??grandes normales et mesures param??tr??es en calcul des variations, Bulletin de la Société mathématique de France, vol.79, pp.129-184, 1973.
DOI : 10.24033/bsmf.1755

D. Blanchard, A. Damlamian, and H. Ghidouche, A nonlinear system for phase change with dissipation, Differential Integral Equations, pp.344-362, 1989.

D. Blanchard and H. Ghidouche, A nonlinear system for irreversible phase changes, European Journal of Applied Mathematics, vol.301, issue.01, pp.91-100, 1990.
DOI : 10.2307/2373376

M. Böhm and R. E. Showalter, A Nonlinear Pseudoparabolic Diffusion Equation, SIAM Journal on Mathematical Analysis, vol.16, issue.5, pp.980-999, 1985.
DOI : 10.1137/0516074

A. Bouziani and N. Merazga, Solution to a semilinear pseudoparabolic problem with integral conditions, Electron. J. Differential Equations, vol.18, issue.115, p.pp, 2006.

F. Boyer and P. Fabrie, Eléments d'analyse pour l'étude de quelques modèles d'écoulements de fluides visqueux incompressibles, de Mathématiques & Applications (Berlin) [Mathematics & Applications, 2006.
DOI : 10.1007/3-540-29819-3

H. Brezis, Analyse fonctionnelle Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master's Degree], Théorie et applications. [Theory and applications], 1983.

H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Mathematics Studies

C. Castaing, Valadier : Convex analysis and measurable multifunctions, Lecture Notes in Mathematics, vol.580, 1977.
DOI : 10.1007/bfb0087685

R. Cauty, Solution du probl??me de point fixe de Schauder, Fundamenta Mathematicae, vol.170, issue.3, pp.231-246, 2001.
DOI : 10.4064/fm170-3-2

. Th, A. Cazenave, and . Haraux, Introduction aux problèmes d'évolution semi-linéaires, de Mathématiques & Applications (Paris) [Mathematics and Applications]. Ellipses, 1990.

C. Chen and G. Cheng, Anomalous dimension in the solution of the Barenblatt???s equation, Journal of Mathematical Physics, vol.39, issue.3, pp.1589-1600, 1998.
DOI : 10.1063/1.532399

P. Colli, F. Luterotti, G. Schimperna, and U. Stefanelli, Global existence for a class of generalized systems for irreversible phase changes, NoDEA : Nonlinear Differential Equations and Applications, vol.9, issue.3, pp.255-276, 2002.
DOI : 10.1007/s00030-002-8127-8

P. Colli, F. Luterotti, G. Schimperna, and U. Stefanelli, Global existence for a class of generalized systems for irreversible phase changes, NoDEA : Nonlinear Differential Equations and Applications, vol.9, issue.3, pp.255-276, 2002.
DOI : 10.1007/s00030-002-8127-8

C. Cuesta and J. Hulshof, A model problem for groundwater flow with dynamic capillary pressure: stability of travelling waves, Nonlinear Analysis: Theory, Methods & Applications, vol.52, issue.4, pp.1199-1218, 2003.
DOI : 10.1016/S0362-546X(02)00160-8

J. E. Dennis, J. Robert, and B. , Schnabel : Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Classics in Applied Mathematics, vol.16, 1996.

J. Droniou, Intégration et espaces de sobolev à valeurs vectorielles, 2001.

J. Droniou, Quelques résultats sur les espaces de sobolev, 2001.

W. Düll, Some Qualitative Properties of Solutions to a Pseudoparabolic Equation Modeling Solvent Uptake in Polymeric Solids, Communications in Partial Differential Equations, vol.11, issue.8, pp.7-91117, 2006.
DOI : 10.1007/BF01474161

G. Duvaut and J. , Lions : Les inéquations en mécanique et en physique, Travaux et Recherches Mathématiques, issue.21, 1972.

I. Ekeland and R. Témam, Convex analysis and variational problems, Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), vol.28, 1999.
DOI : 10.1137/1.9781611971088

C. Lawrence and . Evans, A survey of entropy methods for partial differential equations, Bull. Amer. Math. Soc. (N.S.), vol.41, issue.4, pp.409-438, 2004.

B. Ewald, M. Petcu, and R. Temam, STOCHASTIC SOLUTIONS OF THE TWO-DIMENSIONAL PRIMITIVE EQUATIONS OF THE OCEAN AND ATMOSPHERE WITH AN ADDITIVE NOISE, Analysis and Applications, vol.05, issue.02, pp.183-198, 2007.
DOI : 10.1142/S0219530507000948

R. E. Ewing, The Approximation of Certain Parabolic Equations Backward in Time by Sobolev Equations, SIAM Journal on Mathematical Analysis, vol.6, issue.2, pp.283-294, 1975.
DOI : 10.1137/0506029

R. E. Ewing, Time-Stepping Galerkin Methods for Nonlinear Sobolev Partial Differential Equations, SIAM Journal on Numerical Analysis, vol.15, issue.6, pp.1125-1150, 1978.
DOI : 10.1137/0715075

R. Eymard and . Th, Analytical and Numerical Study of a Model of Erosion and Sedimentation, SIAM Journal on Numerical Analysis, vol.43, issue.6, pp.2344-2370, 2006.
DOI : 10.1137/040605874

R. Eymard and . Th, A Partial Differential Inequality in Geological Models, Chinese Annals of Mathematics, Series B, vol.41, issue.6, pp.709-736, 2007.
DOI : 10.1007/s11401-006-0215-3

R. Eymard, . Th, V. Gallouët, R. Gervais, and . Masson, Convergence of a Numerical Scheme for Stratigraphic Modeling, SIAM Journal on Numerical Analysis, vol.43, issue.2, 2004.
DOI : 10.1137/S0036142903426208

R. Eymard, . Th, D. Gallouët, R. Granjeon, Q. H. Masson et al., Multi-lithology stratigraphic model under maximum erosion rate constraint, International Journal for Numerical Methods in Engineering, vol.60, issue.2, pp.527-548, 2004.
DOI : 10.1002/nme.974

I. Fonseca and G. Leoni, Modern methods in the calculus of variations : L p spaces, 2007.

A. Friedman and Z. Schuss, Degenerate evolution equations in Hilbert space, Transactions of the American Mathematical Society, vol.161, pp.401-427, 1971.
DOI : 10.1090/S0002-9947-1971-0283623-9

G. Gagneux and M. Madaune-tort, Analyse mathématique de modèles non linéaires de l'ingénierie pétrolière, de Mathématiques & Applications (Berlin) [Mathematics & Applications, 1996.

G. Gagneux and G. Vallet, Sur des probl??mes d'asservissements stratigraphiques, ESAIM: Control, Optimisation and Calculus of Variations, vol.8, pp.715-739, 2002.
DOI : 10.1051/cocv:2002055

G. Gagneux and G. Vallet, A result of existence for an original convection-diffusion equation, RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat, vol.99, issue.1, pp.125-131, 2005.

J. Garcia-azorero and A. D. Pablo, Finite propagation for a pseudoparabolic equation: Two-phase non-equilibrium flows in porous media, Nonlinear Analysis: Theory, Methods & Applications, vol.33, issue.6, pp.551-573, 1998.
DOI : 10.1016/S0362-546X(97)00678-0

V. Gervais and R. Masson, Mathematical and numerical analysis of a stratigraphic model, ESAIM: Mathematical Modelling and Numerical Analysis, vol.38, issue.4, pp.585-611, 2004.
DOI : 10.1051/m2an:2004035

D. Granjeon, Modélisation stratigraphique déterministe ; conception et applications d'un modèle diffusif 3d multilithologique, 1997.

W. Grecksch and C. Tudor, Stochastic evolution equations : A Hilbert space approach, de Mathematical Research. Akademie-Verlag, 1995.

J. Hulshof and J. R. King, Analysis of a Darcy flow model with a dynamic pressure saturation relation, SIAM J. Appl. Math, vol.59, issue.1, pp.318-346, 1999.

J. Hulshof and J. Vázquez, Self-similar solutions of the second kind for the modified porous medium equation, European Journal of Applied Mathematics, vol.1224, issue.03, pp.391-403, 1994.
DOI : 10.1103/PhysRevA.44.6544

N. Igbida, Solutions auto-similaires pour une équation de, Barenblatt. Rev. Mat. Apl, vol.17, issue.1, pp.21-36, 1996.

E. I. Kaikina, Nonlinear pseudoparabolic type equations on a half-line with large initial data, Nonlinear Analysis: Theory, Methods & Applications, vol.67, issue.10, pp.2839-2858, 2007.
DOI : 10.1016/j.na.2006.09.044

N. David, M. A. Keck, and . Mckibben, On a McKean-Vlasov stochastic integro-differential evolution equation of Sobolev-type. Stochastic Anal, Appl, vol.21, issue.5, pp.1115-1139, 2003.

J. U. Kim, On a stochastic scalar conservation law, Indiana University Mathematics Journal, vol.52, issue.1, pp.227-256, 2003.
DOI : 10.1512/iumj.2003.52.2310

A. N. Kolmogorov and S. V. , Fomin : Éléments de la théorie des fonctions et de l'analyse fonctionnelle, 1998.

M. O. Korpusov, Global solvability of an initial-boundary value problem for a semilinear system of equations, Zh. Vychisl. Mat. Mat. Fiz, vol.42, issue.7, pp.1039-1050, 2002.

L. Thibault, Personal communication, 2009.

K. B. Liaskos, I. G. Stratis, and A. N. Yannacopoulos, Pseudoparabolic equations with additive noise and applications, Mathematical Methods in the Applied Sciences, vol.2, issue.5, pp.963-985, 2009.
DOI : 10.1002/mma.1077

J. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Travaux et Recherches Mathématiques, 1968.

V. Padrón, Effect of aggregation on population revovery modeled by a forward-backward pseudoparabolic equation, Transactions of the American Mathematical Society, vol.356, issue.07, pp.2739-2756, 2004.
DOI : 10.1090/S0002-9947-03-03340-3

M. Ptashnyk, Nonlinear pseudoparabolic equations and variational inequalities, 2004.

M. Ptashnyk, Nonlinear pseudoparabolic equations as singular limit of reaction???diffusion equations, Applicable Analysis, vol.85, issue.10, pp.1285-1299, 2006.
DOI : 10.1080/00036810600871909

M. Ptashnyk, Degenerate quasilinear pseudoparabolic equations with memory terms and variational inequalities, Nonlinear Analysis: Theory, Methods & Applications, vol.66, issue.12, pp.2653-2675, 2007.
DOI : 10.1016/j.na.2006.03.046

A. Quarteroni and A. Valli, Numerical approximation of partial differential equations, Series in Computational Mathematics, 1994.

G. Schimperna, A. Segatti, and U. Stefanelli, Well-posedness and long-time behavior for a class of doubly nonlinear equations, Discrete Contin. Dyn. Syst, vol.18, issue.1, pp.15-38, 2007.

B. Schweizer, Averaging of flows with capillary hysteresis in stochastic porous media, European Journal of Applied Mathematics, vol.18, issue.03, pp.389-415, 2007.
DOI : 10.1017/S0956792507007000

A. Segatti, Global attractor for a class of doubly nonlinear abstract evolution equations, Discrete and Continuous Dynamical Systems, vol.14, issue.4, pp.801-820, 2006.
DOI : 10.3934/dcds.2006.14.801

R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, volume 49 de Mathematical Surveys and Monographs, 1997.

R. E. Showalter and T. W. Ting, Pseudoparabolic Partial Differential Equations, SIAM Journal on Mathematical Analysis, vol.1, issue.1, pp.1-26, 1970.
DOI : 10.1137/0501001

M. Sibony and J. Mardon, Analyse numérique I : Systèmes linéaires et non linéaires. 2 e tirage corr, Actualités Scientifiques et Industrielles, 1405. Paris : Hermann. IX, 515 p. (pagination non continue), 1984.

J. Simon, Compact sets in the spaceL p (O,T; B), Annali di Matematica Pura ed Applicata, vol.287, issue.1, pp.65-96, 1987.
DOI : 10.1007/BF01762360

G. A. Sviridyuk and A. F. Karamova, On the Phase Space Fold of a Nonclassical Equation, Differential Equations, vol.62, issue.3, pp.1400-1405, 1438.
DOI : 10.1007/s10625-005-0300-5

G. Vallet, Sur une loi de conservation issue de la g??ologie, Comptes Rendus Mathematique, vol.337, issue.8, pp.559-564, 2003.
DOI : 10.1016/j.crma.2003.08.012

G. Vallet, On a new conservation law resulting from sedimentary basin dynamics, VIII Journées Zaragoza-Pau de Mathématiques Appliquées et de Statistiques, volume 31 de Monogr. Semin. Mat. García Galdeano, pp.247-255, 2004.

G. Vallet, Dirichlet problem for a degenerated hyperbolic-parabolic equation, Adv. Math. Sci. Appl, vol.15, issue.2, pp.423-450, 2005.

G. Vallet, On a Degenerated Parabolic-Hyperbolic Problem Arising From Stratigraphy, Numerical mathematics and advanced applications, pp.412-420, 2006.
DOI : 10.1007/978-3-540-34288-5_36

G. Vallet, Stochastic perturbation of nonlinear degenerate parabolic problems. Differential and integral equation

G. Vallet and P. Wittbold, ON A STOCHASTIC FIRST-ORDER HYPERBOLIC EQUATION IN A BOUNDED DOMAIN, Infinite Dimensional Analysis, Quantum Probability and Related Topics, vol.12, issue.04, pp.1-39, 2009.
DOI : 10.1142/S0219025709003872
URL : https://hal.archives-ouvertes.fr/hal-00866266

C. J. Van-duijn and L. A. , A New Class of Entropy Solutions of the Buckley???Leverett Equation, SIAM Journal on Mathematical Analysis, vol.39, issue.2, pp.507-536, 2007.
DOI : 10.1137/05064518X

Z. Wang and J. Yin, Uniqueness of bounded solutions to a viscous diffusion equation, Electronic Journal of Qualitative Theory of Differential Equations, vol.8, issue.17, p.pp, 2003.
DOI : 10.14232/ejqtde.2003.1.17

K. Yosida, Functional analysis, Die Grundlehren der mathematischen Wissenschaften, 1974.

E. Zeidler, Nonlinear functional analysis and its applications. I Fixed-point theorems, 1986.

É. De, P. Aux, . Non, . De, . Pseudo-parabolique et al., Keywords Pseudo parabolic problems, implicit time-semidiscretization

I. Bâtiment, Adour Avenue de l'Université -BP 1155 64013 PAU CEDEX ISBN : 10PAUU305 Études de problèmes aux limites non linéaires de type pseudo-parabolique