The Derrida-Lebowitz-Speer-Spohn equation: Existence, nonuniqueness, and decay rates of the solutions

SIAM Journal on Mathematical Analysis

Volumn 39, Issue 6, 2008, Pages 1996-2015

  Jüngel, Ansgar ^a   Matthes, Daniel ^b  

^a VIENNA UNIVERSITY OF TECHNOLOGY   (Austria)

^b UNIVERSITY OF PAVIA   (Italy)

Author keywords

Decay rates; Entropy entropy dissipation method; Existence of weak solutions; Logarithmic fourth order equation; Long time behavior of solutions; Nonuniqueness of solutions

Indexed keywords

BOUNDARY VALUE PROBLEMS; DECAY (ORGANIC); ENTROPY; NONLINEAR EQUATIONS;

DECAY RATES; EXISTENCE OF WEAK SOLUTIONS; LOGARITHMIC FOURTH-ORDER EQUATION; LONG-TIME BEHAVIOR OF SOLUTIONS; NONUNIQUENESS OF SOLUTIONS;

SOLUTIONS;

EID: 47649125184     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/060676878     Document Type: Article

References (23)

1. A. ARNOLD, P. MARKOWICH, G. TOSCANI, AND A. UNTERREITER, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck-type equations, Comm. Partial Differential Equations, 26 (2001), pp. 43-100.
2. E. BERETTA, M. BERTSCH, AND R. DAL PASSO, Nonnegative solutions of a fourth-order non-linear degenerate parabolic equation, Arch. Ration. Mech. Anal., 129 (1995), pp. 175-200.
3. W. BECKNER, A generalized Poincaré inequality for Gaussian measures, Proc. Amer. Math. Soc, 105 (1989), pp. 397-400.
4. P. BLEHER, J. LEBOWITZ, AND E. SPEER, Existence and positivity of solutions of a fourth-order nonlinear PDE describing interface fluctuations, Comm. Pure Appl. Math., 47 (1994), pp. 923-942.
5. S. BOBKOV AND P. TETALI, Modified log-Sobolev inequalities in discrete settings, J. Theoret. Probab., to appear.
6. M. CÁCERES, J. CARRILLO, AND G. TOSCANI, Long-time behavior for a nonlinear fourth order parabolic equation, Trans. Amer. Math. Soc., 357 (2004), pp. 1161-1175.
7. J. A. CARRILLO, J. DOLBEAULT, I. GENTIL, AND A. JÜNGEL, Entropy-energy inequalities and improved convergence rates for nonlinear parabolic equations, Discrete Contin. Dyn. Systet. Ser. B, 6 (2006), pp. 1027-1050.
8. J. A. CARRILLO, A. JÜNGEL, AND S. TANG, Positive entropie schemes for a nonlinear fourthorder equation, Discrete Contin. Dynam. Syst. Ser. B, 3 (2003), pp. 1-20.
9. I. CSISZÁR, Information-type measures of difference of probability distributions and indirect observations, Studia Sci. Math. Hungar., 2 (1967), pp. 299-318.
10. B. DERRIDA, J. LEBOWITZ, E. SPEER, AND H. SPOHN, Dynamics of an anchored Toom interface, J. Phys. A, 24 (1991), pp. 4805-4834.
11. B. DERRIDA, J. LEBOWITZ, E. SPEER, AND H. SPOHN, Fluctuations of a stationary nonequilibrium interface, Phys. Rev. Lett., 67 (1991), pp. 165-168.
12. J. DOLBEAULT, I. GENTIL, AND A. JÜNGEL, A logarithmic fourth-order parabolic equation and related logarithmic Sobolev inequalities, Commun. Math. Sci., 4 (2006), pp. 275-290.
13. U. GIANAZZA, G. SAVARÉ, AND G. TOSCANI, The Wasserstein Gradient Flow of the Fisher Information and the Quantum Drift-Diffusion Equation, Arch. Ration. Mech. Anal., 2008, to appear.
14. M. P. GUALDANI, A. JÜNGEL, AND G. TOSCANI, A nonlinear fourth-order parabolic equation with non-homogeneous boundary conditions, SIAM J. Math. Anal., 37 (2006), pp. 1761-1779.
15. D. HENRY, How to remember the Sobolev inequalities, in Differential Equations, São Paolo 1981, Lecture Notes in Math. 957, Springer, Berlin, 1982, pp. 97-109.
16. A. JÜNGEL AND D. MATTHES, An algorithmic construction of entropies in higher-order non-linear PDEs, Nonlinearity, 19 (2006), pp. 633-659.
17. A. JÜNGEL AND R. PINNAU, Global nonnegative solutions of a nonlinear fourth-oder parabolic equation for quantum systems, SIAM J. Math. Anal., 32 (2000), pp. 760-777.
18. A. JÜNGEL AND R. PINNAU, A positivity preserving numerical scheme for a nonlinear fourthorder parabolic equation, SIAM J. Numer. Anal., 39 (2001), pp. 385-406.
19. A. JÜNGEL AND G. TOSCANI, Exponential decay in time of solutions to a nonlinear fourth-order parabolic equation, Z. Angew. Math. Phys., 54 (2003), pp. 377-386.
20. S. KULLBACK, A lower bound for discrimination information in terms of variation, IEEE Trans. Inform. Theory, 4 (1967), pp. 126-127.
21. M. LEDOUX, On Talagrand's deviation inequalities for product measures, ESAIM Probab. Statist., 1 (1995/97), pp. 63-87.
22. p(0, T; B), Ann. Mat. Pura Appl. (4), 146 (1987), pp. 65-96.
23. M. TAYLOR, Partial Differential Equations. III. Nonlinear Equations, Springer, New York, 1997.