On gradient estimates and other qualitative properties of solutions of nonlinear non autonomous parabolic systems

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Díaz Díaz, Jesús Ildefonso and Antontsev, S.N. (2009) On gradient estimates and other qualitative properties of solutions of nonlinear non autonomous parabolic systems. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A: Matemáticas , 103 (1). pp. 201-214. ISSN 1578-7303

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Abstract

We prove several uniform L(1)-estimates on solutions of a general class of one-dimensional parabolic systems, mainly coupled in the diffusion term, which, in fact, can be of degenerate type. They are uniform in the sense that they don't depend on the coefficients, nor on the size of the spatial domain. The estimates concern the own Solution or/and its spatial gradient. This paper extends some previous results by the authors to the case of nonautonomous coefficients and possibly non homogeneous boundary conditions. Moreover, an application to the asymptotic decay of the L(1)-norm of solutions, as t -> +infinity, is also given.

Item Type: Article uniform gradient estimates; quasilinear parabolic onedimensional systems; uniform l(1)-estimates independent on the spatial domain; equations; gas Sciences > Mathematics > Differential equations 15174 Alt, H. W. and Luckhaus, S., (1983). Quasilinear Elliptic-Parabolic Differential Equations, Math. Z., 183, 311–341. Amann, H., (1990). Dynamic theory of quasilinear parabolic equations: II. Reaction-diffusion systems, Diff. Int. Equ.,3, 13–75. Andreu, F., Caselles, V. and Mazón, J. M., (2004). Parabolic Quasilinear Equations Minimizing Linear Growth Functions, Birkhäuser, Basel. Antontsev, S. N. and Díaz, J.I., (2007). Interfaces generated by discharge of a hot gas in a cold atmosphere, in Abstracts of Fourth International Conference of Applied Mathematics and Computing, August 12–18, 2007, Plovdiv, Bulgaria, 21–22. Antontsev, S. N. and Díaz, J. I., (2007). Mathematical analysis of the discharge of a hot gas in a colder atmosphere, in Book of abstracts. XXII joint session of Moscow Mathematical Society and I. G. Petrovskii seminar, Moscow, May 21–26. 2007, 19–20. Antontsev, S. N. and Díaz, J. I., (2007). Mathematical analysis of the discharge of a laminar hot gas in a cold atmosphere, in Abstracts of All-Russian Conference "Problems of continuum mechanics and physics of detonation", Novosibirsk, Russia, September 17–22. 2007, 189–190. Antontsev, S. N. and Díaz, J. I., (2007). Mathematical analysis of the discharge of a laminar hot gas in a colder atmosphere, RACSAM. Rev. R. Acad. Cien. Serie A. Mat, 101(1), 119–124. Antontsev, S. N. and Díaz, J. I., (2008). On thermal and stagnation interfaces generated by the discharge of a laminar hot gas in a stagnant colder atmosphere, in preparation for IFB. Antontsev, S. N. and Díaz, J. I., (2008). Mathematical analysis of the discharge of a laminar hot gas in a colder atmosphere, J. Appl. Mech. Tech. Phys., (Russian version, 49, 4), 1–14. Antontsev, S. N. and Díaz, J. I., Uniform $L^{1}$L1-gradient estimates of solutions solutions to quasilinear parabolic systems in higher dimensions. Article in preparation. Antontsev, S. N. and Díaz, J. I., New $L^{1}$L1-gradient type estimates of solutions to one dimensional quasilinear parabolic systems. To appear in Contemporary Mathematics. cf. Antontsev, S. N., Díaz, J. I., and S. Shmarev, (2002). Energy Methods for Free Boundary Problems: Applications to Non-linear PDEs and Fluid Mechanics, Bikhäuser, Boston. Benilan, Ph., (1972). Equation d'évolution dans un espace de Banach quelconque et applications, Théese, Université de Paris-Sud. Benilan, Ph., Crandall, M. G. and Pazy, A., Evolution Equations Governed by Accretive Operators. Book in preparation. Brezis, H. (1983). Analyse fonctionnelle Théorie et applications, Masson, Paris. Brezis, H. and Strauss. W. A., (1973). Semilinear second-order elliptic equations in $\roman{L}^{1}$L1, J. Math.Soc. Japan, 25, 565–590. Chipot, M., (2002). I goes to plus infinity. Birkhäuser, Bassel. Díaz, J. I., (1985). Nonlinear Partial Differential Equations and Free Boundaries, Pitman, London. Díaz, J. I. and Padial, J. F., (1996). Uniqueness and existence of solutions in the $BV(Q)$BV(Q) space to a doubly nonlinear parabolic problem, Publications Matematiques, 40, 527–560. Egorov, Yu. V., Kondrat'ev, V. A. and Oleinik, O. A., (1998). Asymptotic behavior of solutions to nonlinear elliptic and parabolic systems in cylindrical domains, Mat. Sb., 3, 45–68. Evans, L. C. and Gariepy, R. F., (1992). Measure Theory and Fine Properties of Functions, CRC Press. Gilding, B. H., (1989). Improved Theory for a Nonlinear Degenerate Parabolic Equation, Annali Scu. Norm. Sup. Pisa, Serie IV, 14, 165–224. Hopf, E., (1950). The partial differential equation $u_{t}+uu_{x}=\mu u_{xx}$ut+uux=μuxx, Comm. Pure Appl. Math, 3, 201–230. Kalashnikov, A. S., (1987). Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations, Uspekhi Mat. Nauk, 42, 135–176. Kruzhkov, S. N. (1970). First-order quasilinear equations in several independent variables. Mat. Sbornik., 81, 228–255. Ladženskaja, O. A., Solonnikov, V. A. and Ural'tseva, N. N., (1967). Linear and quasilinear equations of parabolic type, American Mathematical Society, Providence, R, I. Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23. Lions, J.-L., (1969). Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod. Málek, J., Nečas, J., Rokyta, M. and Růžička, M.,(1996). Weak and measure-valued solutions to evolutionary PDEs, Chapman Hall, London. Pai, S., (1949). Two-dimensional jet mixing of a compressible fluid, J. Aeronaut. Sci., 16, Pai, S., (1952). Axially symmetrical jet mixing of a compressible fluid, Quart. Appl. Math., 10. 141–148. Pai, S., (1954). Fluid dynamics of jets, D. Van Nostrand Company, Inc., Toronto-New York-London, Publishers, New York-London. Quittner, P. and Souplet, Ph., (2007). Superlinear Parabolic Problems, Birkhäuser, Basel. Sánchez-Sanz, M., Sánchez, A. and Li~nán, A., (2006). Front solutions in high-temperature laminar gas jets, J. Fluid. Mech., 547, 257–266. Volpert, A. I. and Khudayaev, S. I., (1985). Analysis in classes of discontinuous functions and equations of mathematical physics, Nijhoff, Dordrecht. Wu, Z., Zhao, J. Yin, J. and Li, H., (2001). Nonlinear Diffusion Equations, World Scientific, New Jersey. 10 May 2012 08:29 16 May 2013 18:03

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