Abstract
This paper examines the asymptotic behavior of measure valued solutions to the initial value problem for the nonlinear heat conduction equation
,xεΩ, t>0 in a bounded domainΩ⊂R N with boundary conditions of the form
In particular, use of the Young measure representation of composite weak limits allows proof of a general trend to equilibrium. No linearity or monotonicity is assumed forq; the only major restriction onq is that it satisfies the Fourier inequalityq(λ)·λ⩾0 for allλε R N. Applications are given to problems whereq is not monotone.
Similar content being viewed by others
References
Aubin, J. P. (1984).Applied Nonlinear Analysis, John Wiley, New York.
Ball, J. M. (1988). A version of the fundamental theorem for Young measures. In Rascle, M., and Serre, D. (eds.),Proceedings, CNRS-NSF Workshop on Continuum Theory of Phase Transitions, Nice, France, January 1988, Springer Lecture Notes in Mathematics, Springer-Verlag, New York (in press).
Dellacherie, C., and Meyer, P.-A. (1975).Probabilities et Potentel, Hermann, Paris.
DiPerna, R. J. (1983a). Convergence of approximate solutions to conservation laws.Arch. Ration. Mech. Anal. 82, 27–70.
DiPerna, R. J. (1983b). Convergence of the viscosity method for isentropic gas dynamics.Comm. Math. Phys. 91, 1–30.
DiPerna, R. J. (1983c). Generalized solutions to conservation laws. In Ball, J. M. (ed.),Systems of Nonlinear Partial Differential Equations, NATO ASI Series, D. Reidel, Dordrecht.
DiPerna, R. J. (1985). Measure-valued solutions to conservation laws.Arch. Ration. Anal. Mech. 88, 223–270.
DiPerna, R. J., and Majda, A. J. (1987a). Concentrations and regularizations in weak solutions of the incompressible fluid equations.Comm. Math. Phys. 108, 667–689.
DiPerna, R. J., and Majda, A. J. (1987b). Concentrations in regularizations for 2-D incompressible flow.Comm. Pure Appl. Math. 40, 301–345.
Dunford, N., and Schwartz, J. (1958).Linear Operators, Part 1. General Theory, Interscience, New York.
Fujita, H., and Kalo, T. (1964). On the Navier-Stokes initial value problem, I.Arch. Ration. Mech. Anal. 16, 269–315.
Hollig, K. (1983). Existence of infinitely many solutions for a forward backward heat equation.Trans. Am. Math. Soc. 278, 299–316.
Hollig, K., and Nohel J. (1983). A diffusion equation with a non monotone constitutive function. In Ball J. M. (ed.),Systems of Nonlinear Partial Differential Equations, NATO ASI Series Vol. CIII, Reidel, pp. 409–422.
Kato, T. (1966).Perturbation Theory for Linear Operators, Springer-Verlag, New York.
MacShane, E. J. (1947).Integration, Princeton University Press, Princeton.
Maxwell, J. C. (1876). On stresses in rarified gases arising from inequalities of temperature.Phil. Trans. Roy. Soc. London 170, 231–256; 680–712.
Natanson, I. P. (1955).Theory of Functions of a Real Variable, Vol. 1, F. Unger Publishing Co., New York.
Nicolaenko, B., Scheuer, B., and Temam, R. (1988).Some global dynamical properties of a class of pattern formation equations, University of Minnesota IMA Preprint Series #381.
Pazy, A. (1983).Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York.
Schonbek, M. E. (1982). Convergence of solutions to nonlinear dispersive equations,Comm. Part. Dif. Eg. 7, 959–1000.
Slemrod, M. (1989a). Weak asymptotic decay via a “relaxed invariance principle” for a wave equation with nonlinear, nonmonotone clamping, to appear inProc. Royal Soc. Edinburgh.
Slemrod, M. (1989b). Trend to equilibrium in the Becker-Döring cluster equations, to appear inNonlinearity.
Slemrod, M. (1989c). The relaxed invariance principle and weakly dissipative infinite dimensional dynamical systems, to appear inProc. Conf. on Mixed Problems, (K. Kirschgassner, (ed.), Springer Lecture Notes).
Tartar, L. (1979). Compensated compactness and applications to partial differential equations. InNonlinear Analysis and Mechanics, Herior-Watt Symposium IV, Pitman Research Notes in Mathematics, pp. 136–192.
Temam, R. (1988).Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York.
Truesdell, C. (1984).Rational Thermodynamics, 2nd ed., Springer-Verlag, New York.
Truesdell, C., and Noll, W. (1965). The non-linear field theories of mechanics. In Flugge, S. (ed.),Encyclopedia of Physics, Vol. III/3, Springer-Verlag, Berlin-Heidelberg-New York.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Slemrod, M. Dynamics of measured valued solutions to a backward-forward heat equation. J Dyn Diff Equat 3, 1–28 (1991). https://doi.org/10.1007/BF01049487
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01049487