Title | ON A BISTABLE QUASILINEAR PARABOLIC EQUATION: WELL-POSEDNESS AND STATIONARY SOLUTIONS |
Publication Type | Journal Article |
Year of Publication | 2011 |
Authors | BURNS, MARTIN, GRINFELD, MICHAEL |
Secondary Title | Communications in Applied Analysis |
Volume | 15 |
Issue | 2 |
Start Page | 251 |
Pagination | 264 |
Date Published | 04/2011 |
Type of Work | scientific: mathematics |
ISSN | 1083–2564 |
AMS | 35B32, 35D30, 35J62, 35K59 |
Abstract | In this paper we prove existence and uniqueness of variational inequality solutions for a bistable quasilinear parabolic equation arising in the theory of solid-solid phase transitions and discuss its stationary solutions, which can be discontinuous.
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URL | http://www.acadsol.eu/en/articles/15/2/9.pdf |
Short Title | WELL-POSEDNESS AND STATIONARY SOLUTIONS |
Refereed Designation | Refereed |
Full Text | REFERENCES[1] F. Andreu-Vaillo, V. Caselles & J. M. Mazon, Parabolic Quasilinear Equations Minimizing Linear Growth Functionals, vol. 223, Progress in Mathematics, Birkh ̈auser Verlag, Basel, 2004.
[2] M. Bertsch, & R. Dal Passo, Hyperbolic phenomena in a strongly degenerate parabolic equation. Arch. Rat. Mech. and Anal. 117, (1992), 349–387. [3] M. Burns, & M. Grinfeld, Steady state solutions of a bi-stable quasi-linear equation with saturating flux, Euro. J. App. Math. 22, (2011), 317–331. [4] D. Bonheure, P. Habets, F. Obersnel & P. Omari, Classical and non-classical solutions of a prescribed curvature equation. J. Differ. Equ. 243, (2007), 208–237. [5] Burns, M. Reaction-Diffusion Equations with Saturating Flux Arising in the Theory of Solid-Solid Phase Transitions, Ph.D. Thesis. University of Strathclyde, in preparation. [6] R. Chill, On the Lojasiewicz-Simon gradient inequality, J. Funct. Anal. 201, (2003), 572–601. [7] L. Dascal, S. Kamin & N. Sochen, A variational inequality for discontinuous solutions of degenerate parabolic equations. RACSAM 99, (2005), 243–256. [8] F. Demengel & R. Temam, Convex functions of a measure and applications. Indiana Univ. Math. J. 33, (1984), 673–709. [9] P. Fife, Models for phase separation and their mathematics. Electronic J. Differ. Equ. 48, (2000), 1–26. [10] P. Habets & P. Omari, Multiple positive solutions of a one-dimensional prescribed mean curvature problem. Commun. Contem. Math. 9, (2007), 701–730. [11] V. K. Le, Variational method based on finite dimensional approximation in a generalized prescribed mean curvature problem. J. Differ. Equ. 246, (2009), 3559–3578. [12] O.A Ladyzenskaja, V. A. Solonnikov & N. N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, vol 23, Translations of Mathematical Monographs, American Mathematical Society: Providence RI, 1968. [13] F. Obersnel, Classical and non-classical sign changing solutions of a one- dimensional autonomous prescribed curvature equation Adv. Nonlinear Stud. 7, (2007), 1–13. [14] H. Pan, One-dimensional prescribed mean curvature equation with exponential nonlinearity. Nonlin. Anal. 70, (2009) 999–1010. [15] P. Rosenau, Extension of Landau-Ginzburg free-energy functionals to high-gradient domains. Phys. Rev. A 39, (1989), 6614–6617. [16] P. Rosenau, Free-energy functionals at the high-gradient limit. Phys. Rev. A 41, (1990), 2227–2230. [17] J. Smoller & A. Wasserman, Global bifurcation of steady state solutions. J. Differ. Equ 39, (1981), 269–290. |