Abstract
This article deals with the Cauchy problem for a forward–backward parabolic equation, which is of interest in physical and biological models. Considering such an equation as the singular limit of an appropriate pseudoparabolic third-order regularization, we consider the framework of entropy solutions, namely weak solutions satisfying an additional entropy inequality inherited by the higher order equation. Moreover, we restrict the attention to two-phase solutions, that is solutions taking values in the intervals where the parabolic equation iswell-posed, proving existence and uniqueness of such solutions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Barenblatt G.I., Bertsch M., Dal Passo R., Ughi M.: 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 J. Math. Anal. 24, 1414–1439 (1993)
Bellettini G., Fusco G., Guglielmi N.: A concept of solution and numerical experiments for forward–backward diffusion equations. Discr. Contin. Dyn. Syst. 16, 783–842 (2006)
Binder K., Frisch H.L., Jäckle J.: Kinetics of phase separation in the presence of slowly relaxing structural variables. J. Chem. Phys. 85, 1505–1512 (1986)
Brokate M., Sprekels J.: Hysteresis and Phase Transitions. Applied Mathematical Sciences, vol. 121. Springer, Berlin (1996)
Cannon J.R.: The One-Dimensional Heat Equation. Encyclopedia of Mathematics and Its Applications, vol. 23. Addison-Wesley, Reading, 1984
Evans, L.C.: Weak convergence methods for nonlinear partial differential equations, CBMS Reg. Conf. Ser. Math., vol. 74. American Mathematical Society, Providence, 1990
Evans L.C.: A survey of entropy methods for partial differential equations. Bull. Am. Math. Soc. 41, 409–438 (2004)
Evans L.C., Portilheiro M.: Irreversibility and hysteresis for a forward–backward diffusion equation. Math. Models Methods Appl. Sci. 14, 1599–1620 (2004)
Friedman A.: Free boundary problems for parabolic equations I: Melting of solids. J. Math. Mech. 8, 499–517 (1959)
Friedman A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs (1991)
Ghisi M., Gobbino M.: Gradient estimates for the Perona–Malik equation. Math. Ann. 337, 557–590 (2007)
Gilding B.H.: Hölder continuity of solutions of parabolic equations. J. Lond. Math. Soc. 13, 103–106 (1976)
Gilding, B.H., Tesei, A.: The Riemann problem for a forward–backward parabolic equation, preprint, 2008
Höllig K.: Existence of infinitely many solutions for a forward backward heat equation. Trans. Am. Math. Soc. 278, 299–316 (1983)
Ladyzenskaja, O.A., Solonnikov, V.A., Uralceva, N.N.: Linear and Quasi-linear Equations of Parabolic Type. American Mathematical Society, Providence, 1991
Lafitte, P., Mascia, C.: Numerical exploration of a forward–backward diffusion equation, in preparation
Mascia, C., Terracina, A., Tesei, A.: Evolution of stable phases in forward– backward parabolic equations. Asymptotic Analysis and Singularities (Eds. Kozono, H., Ogawa, T., Tanaka, K., Tsutsumi, Y., Yanagida E.), Advanced Studies in Pure Mathematics 47-2. Mathematical Socity, Japan, 451–478, 2007
Matano H.: Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29, 401–441 (1982)
Novick-Cohen A., Pego R.L.: Stable patterns in a viscous diffusion equation. Trans. Am. Math. Soc. 324, 331–351 (1991)
Padrón V.: Sobolev regularization of a nonlinear ill-posed parabolic problem as a model for aggregating populations. Comm. Partial Differ. Equ. 23, 457–486 (1998)
Padrón V.: Effect of aggregation on population recovery modeled by a forward–backward pseudoparabolic equation. Trans. Am. Math. Soc. 356, 2739–2756 (2003)
Perona P., Malik J.: Scale space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12, 629–639 (1990)
Plotnikov P.I.: Equations with alternating direction of parabolicity and the hysteresis effect. Russian Acad. Sci. Dokl. Math. 47, 604–608 (1993)
Plotnikov P.I.: Passing to the limit with respect to viscosity in an equation with variable parabolicity direction. Differ. Equ. 30, 614–622 (1994)
Plotnikov P.I.: Forward–backward parabolic equations and hysteresis. J. Math. Sci. 93, 747–766 (1999)
Roubíček T., Hoffmann K.-H.: About the concept of measure-valued solutions to distributed parameter systems. Math. Meth. Appl. Sci. 18, 671–685 (1995)
Saks S.: Theory of the Integral. Dover, New York (1964)
Serre D.: Systems of Conservation Laws, vol. 1: Hyperbolicity, Entropies, Shock Waves. Cambridge University Press, Cambridge (1999)
Slemrod M.: Dynamics of measure valued solutions to a backward-forward heat equation. J. Dyn. Differ. Equ. 3, 1–28 (1991)
Smarrazzo, F.: On a class of equations with variable parabolicity direction. Discr. Contin. Dyn. Syst. (to appear)
Tychonov A.N., Samarski A.A.: Partial Differential Equations of Mathematical Physics. Holden-Day, San Francisco (1964)
Visintin A.: Forward–backward parabolic equations and hysteresis. Calc. Var. 15, 115–132 (2002)
Zhang K.: Existence of infinitely many solutions for the one-dimensional Perona–Malik model. Calc. Var. 26, 171–199 (2006)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Mascia, C., Terracina, A. & Tesei, A. Two-phase Entropy Solutions of a Forward–Backward Parabolic Equation. Arch Rational Mech Anal 194, 887–925 (2009). https://doi.org/10.1007/s00205-008-0185-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-008-0185-6