Existence and Uniqueness of Solutions of a Nonlinear Heat Equation

M.A. Rincon, J. Límaco, I. Shih Liu

Abstract


A nonlinear partial differential equation of the following form is considered: u′ − div a(u)∇u + b(u) |∇u|2 = 0, which arises from the heat conduction problems with strong temperature-dependent material parameters, such as mass density, specific heat and heat conductivity. Existence, uniqueness and asymptotic behavior of initial boundary value problems under appropriate assumptions on the material parameters are established.

References


[1] N. Alaa and M. Iguername, Weak periodic solution of some quasilinear parabolic equations with data measures, Journal of Inequalities in Pure and Applied Mathematics, 3 (2002), 1-14.

H. Amann, Periodic solutions of semilinear parabolic equations, Nonlinear Analysis, (1978), 1-29.

L. Boccardo, F. Murat, and J.P. Puel, Existence results for some quasilinear parabolic equations, Nonlinear Analysis, 13 (1989), 373-392.

L. Boccardo and T. Gallouet, Nonlinear elliptic and parabolic equations involving measure data, Journal of Functional Analysis, 87 (1989), 149-168.

M. Chipot and F.B. Weissler, Some blow up results for a nonlinear parabolic equation with a gradient term, SIAM J. Math. Anal., 20 (1989), 886-907.

E. Feireisl, H. Petzeltová, and F. Simondon, Admissible solutions for a class of nonlinear parabolic problems with non-negative data, Proceedings of the Royal Society of Edinburgh, Section A - Mathematics, 131, No. 5 (2001), 857-883.

J.L. Lions, “Quelques méthodes de résolution des probl`emes aux limites non linéares”, First edition, Dunod-Gauthier Villars, Paris, 1969.

I-S. Liu, “Continuum Mechanics”, Spring-Verlag, Berlin-Heidelberg, 2002.

M. Nakao, On boundedness periodicity and almost periodicity of solutions of some nonlinear parabolic equations, J. Differential Equations, 19 (1975), 371-385.

G. Prodi, Un Teorema di unicita per le equazioni di Navier-Stokes, Annali di Mat., 48 (1959), 173-182.




DOI: https://doi.org/10.5540/tema.2005.06.02.0273

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Trends in Computational and Applied Mathematics

A publication of the Brazilian Society of Applied and Computational Mathematics (SBMAC)

 

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