Geodesic
Non-local quasi-linear parabolic equations
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 60 (2005) no. 6, pp. 1021-1033 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

This is a survey of the most common approaches to quasi-linear parabolic evolution equations, a discussion of their advantages and drawbacks, and a presentation of an entirely new approach based on maximal $L_p$ regularity. The general results here apply, above all, to parabolic initial-boundary value problems that are non-local in time. This is illustrated by indicating their relevance for quasi-linear parabolic equations with memory and, in particular, for time-regularized versions of the Perona–Malik equation of image processing. 
@article{RM_2005_60_6_a2,
     author = {H. Amann},
     title = {Non-local quasi-linear parabolic equations},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {1021--1033},
     year = {2005},
     volume = {60},
     number = {6},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/RM_2005_60_6_a2/}
}
TY  - JOUR
AU  - H. Amann
TI  - Non-local quasi-linear parabolic equations
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2005
SP  - 1021
EP  - 1033
VL  - 60
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/RM_2005_60_6_a2/
LA  - en
ID  - RM_2005_60_6_a2
ER  - 
%0 Journal Article
%A H. Amann
%T Non-local quasi-linear parabolic equations
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2005
%P 1021-1033
%V 60
%N 6
%U http://geodesic.mathdoc.fr/item/RM_2005_60_6_a2/
%G en
%F RM_2005_60_6_a2
H. Amann. Non-local quasi-linear parabolic equations. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 60 (2005) no. 6, pp. 1021-1033. http://geodesic.mathdoc.fr/item/RM_2005_60_6_a2/

[1] H. Amann, “Hopf bifurcation in quasilinear reaction-diffusion systems”, Delay Differential Equations and Dynamical Systems, Proceedings (Claremont, 1990), Lecture Notes in Math., 1475, eds. S. Busenberg and M. Martelli, 1991, 53–63 | MR | Zbl

[2] H. Amann, “Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems”, Function Spaces, Differential Operators and Nonlinear Analysis (Friedrichroda, 1992), Teubner-Texte Math., 133, Teubner, Stuttgart, 1993, 9–126 | MR | Zbl

[3] H. Amann, Linear and Quasilinear Parabolic Problems. V. I: Abstract Linear Theory, Monogr. Math., 89, Birkhäuser, Boston, 1995 | MR | Zbl

[4] H. Amann, “Coagulation-fragmentation processes”, Arch. Ration. Mech. Anal., 151:4 (2000), 339–366 | DOI | MR | Zbl

[5] H. Amann, “Maximal regularity for nonautonomous evolution equations”, Adv. Nonlinear Stud., 4:4 (2004), 417–430 | MR | Zbl

[6] H. Amann, “Quasilinear parabolic problems via maximal regularity”, Adv. Differential Equations, 10:10 (2005), 1081–1110 | MR | Zbl

[7] H. Amann, Time-delayed Perona-Malik type problems, Preprint | MR

[8] H. Amann, P. Quittner, “Semilinear parabolic equations involving measures and low regularity data”, Trans. Amer. Math. Soc., 356:3 (2004), 1045–1119 | DOI | MR | Zbl

[9] A. Belahmidi, Équations aux dérivées partielles appliquées à la restoration et à l'agrandissement des images, PhD thesis, CEREMADE, Université Paris-Dauphin, Paris, 2003; http://tel.ccsd.cnrs.fr

[10] Ph. Bénilan, Équations d'évolution dans un espace de Banach quelconque et applications, PhD thesis, Univ. Paris, Orsay, 1972

[11] H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland, Amsterdam, 1973 | MR

[12] F. Catté, P.-L. Lions, J.-M. Morel, T. Coll, “Image selective smoothing and edge detection by nonlinear diffusion”, SIAM J. Numer. Anal., 29:1 (1992), 182–193 | DOI | MR | Zbl

[13] M. Chipot, Elements of Nonlinear Analysis, Birkhäuser, Basel, 2000 | MR | Zbl

[14] Ph. Clément, Sh. Li, “Abstract parabolic quasilinear equations and application to a groundwater flow problem”, Adv. Math. Sci. Appl., 3, Special Issue (1993/94), 17–32 | MR

[15] Ph. Clément, G. Simonett, “Maximal regularity in continuous interpolation spaces and quasilinear parabolic equations”, J. Evol. Equ., 1:1 (2001), 39–67 | DOI | MR | Zbl

[16] M. G. Crandall, “Nonlinear semigroups and evolution governed by accretive operators”, Nonlinear Functional Analysis and Its Applications, Part 1 (Berkeley, Calif., 1983), Proc. Sympos. Pure Math., 45, Amer. Math. Soc., Providence, RI, 1986, 305–337 | MR

[17] R. Denk, M. Hieber, J. Prüss, $\mathscr R$-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166, no. 788, 2003 | MR

[18] J. Escher, “Quasilinear parabolic systems with dynamical boundary conditions”, Comm. Partial Differential Equations, 18:7–8 (1993), 1309–1364 | DOI | MR | Zbl

[19] J. Escher, G. Simonett, “Moving surfaces and abstract parabolic evolution equations”, Topics in Nonlinear Analysis, Progr. Nonlinear Differential Equations Appl., 35, Birkhäuser, Basel, 1999, 183–212 | MR | Zbl

[20] P. Guidotti, “Singular quasilinear abstract Cauchy problems”, Nonlinear Anal., 32:5 (1998), 667–695 | DOI | MR | Zbl

[21] M. E. Gurtin, A. C. Pipkin, “A general theory of heat conduction with finite wave speeds”, Arch. Ration. Mech. Anal., 31 (1968), 113–126 | DOI | MR | Zbl

[22] D. Khenri, Geometricheskaya teoriya polulineinykh parabolicheskikh uravnenii, Mir, M., 1985 | MR

[23] G. Hetzer, “Global existence, uniqueness, and continuous dependence for a reaction-diffusion equation with memory”, Electron. J. Differential Equations, 1996, no. 05 (electronic) | MR

[24] E. Hopf, “Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen”, Math. Nachr., 4 (1951), 213–231 | MR | Zbl

[25] T. Kato, “Abstract evolution equations of parabolic type in Banach and Hilbert spaces”, Nagoya Math. J., 19 (1961), 93–125 | MR | Zbl

[26] O. A. Ladyzhenskaya, Matematicheskie voprosy dinamiki vyzkoi neszhimaemoi zhidkosti, Nauka, M., 1970 | MR

[27] O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Uraltseva, Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967 | MR | Zbl

[28] J.-B. Leblond, “Mathematical results for a model of diffusion and precipitation of chemical elements in solid matrices”, Nonlinear Anal. Real World Appl., 6:2 (2005), 297–322 | DOI | MR | Zbl

[29] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, River Edge, NJ, 1996 | MR | Zbl

[30] J.-L. Lions, Équations différentielles opérationelles et problèmes aux limites, Springer-Verlag, Berlin, 1961 | MR | Zbl

[31] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progr. Nonlinear Differential Equations Appl., 16, Birkhäuser, Basel, 1995 | MR | Zbl

[32] J. Meixner, “On the linear theory of heat conduction”, Arch. Ration. Mech. Anal., 39 (1971), 108–130 | DOI | MR

[33] M. Nitzberg, T. Shiota, “Nonlinear image filtering with edge and corner enhancement”, IEEE Trans. Pattern Anal. Machine Intelligence, 14:8 (1992), 826–833 | DOI

[34] J. W. Nunziato, “On heat conduction in materials with memory”, Quart. Appl. Math., 29 (1971), 187–204 | MR | Zbl

[35] P. Perona, J. Malik, “Scale-space and edge detection using anisotropic diffusion”, IEEE Trans. Pattern Anal. Machine Intelligence, 12:7 (1990), 629–639 | DOI

[36] J. Prüss, “Maximal regularity for evolution equations in $L_p$-spaces”, Conf. Semin. Mat. Univ. Bari, 2003, no. 285 (2002), 1–39 | MR

[37] G. Simonett, “Center manifolds for quasilinear reaction-diffusion systems”, Differential Integral Equations, 8:4 (1995), 753–796 | MR | Zbl

[38] G. Simonett, “Invariant manifolds and bifurcation for quasilinear reaction-diffusion systems”, Nonlinear Anal., 23:4 (1994), 515–544 | DOI | MR | Zbl

[39] P. E. Sobolevskii, “Uravneniya parabolicheskogo tipa v banakhovom prostranstve”, Tr. MMO, 10 (1961), 297–350 | MR | Zbl

[40] M. I. Vishik, O. A. Ladyzhenskaya, “Kraevye zadachi dlya uravnenii v chastnykh proizvodnykh i nekotorykh klassov operatornykh uravnenii”, UMN, 11:6 (1956), 41–97 | MR | Zbl

[41] L. Weis, “Operator-valued Fourier multiplier theorems and maximal $L_p$-regularity”, Math. Ann., 319:4 (2001), 735–758 | DOI | MR | Zbl

[42] S. Zheng, Nonlinear Evolution Equations, Chapman Hall/CRC Monogr. Surv. Pure Appl. Math., 133, Chapman Hall/CRC, Boca Raton, FL, 2004 | MR | Zbl