Abstract
We study a quasilinear parabolic equation of forward–backward type in one space dimension, under assumptions on the nonlinearity which hold for a number of important mathematical models (for example, the one-dimensional Perona–Malik equation), using a degenerate pseudoparabolic regularization proposed in Barenblatt et al. (SIAM J Math Anal 24:1414–1439, 1993), which takes time delay effects into account. We prove existence and uniqueness of positive solutions of the regularized problem in a space of Radon measures. We also study qualitative properties of such solutions, in particular concerning their decomposition into an absolutely continuous part and a singular part with respect to the Lebesgue measure. In this respect, the existence of a family of viscous entropy inequalities plays an important role.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Amann H.: Time-delayed Perona–Malik type problems. Acta. Math. Univ. Comenianae 76, 15–38 (2007)
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)
Brokate, M., Sprekels, J.: Hysteresis and Phase Transitions. Appl. Math. Sci., vol. 121. Springer, 1996
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. Mod. Methods Appl. Sci. 14, 1599–1620 (2004)
Giaquinta, M., Modica, G., Soucek, J.: Cartesian Currents in the Calculus of Variations. Springer, 1998
Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order. Springer, 1983
Mascia C., Terracina A., Tesei A.: Two-phase entropy solutions of a forward–backward parabolic equation. Arch. Rational Mech. Anal. 194, 887–925 (2009)
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. Commun. Partial Differ. Equ. 23, 457–486 (1998)
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.: Passing to the limit with respect to viscosity in an equation with variable parabolicity direction. Differ. Equ. 30, 614–622 (1994)
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.: Forward–backward parabolic equations and hysteresis. J. Math. Sci. 93, 747–766 (1999)
Smarrazzo F.: On a class of equations with variable parabolicity direction. Discrete Contin. Dyn. Syst. 22, 729–758 (2008)
Smarrazzo F., Tesei A.: Long-time behaviour of solutions to a class of forward–backward parabolic equations. SIAM J. Math. Anal. 42, 1046–1093 (2010)
Smarrazzo, F., Tesei, A.: Degenerate Regularization of Forward–Backward Parabolic Equations: The Vanishing Viscosity Limit
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Bressan
Rights and permissions
About this article
Cite this article
Smarrazzo, F., Tesei, A. Degenerate Regularization of Forward–Backward Parabolic Equations: The Regularized Problem. Arch Rational Mech Anal 204, 85–139 (2012). https://doi.org/10.1007/s00205-011-0470-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-011-0470-7