Entropy methods for reaction-diffusion equations: Slowly growing a-priori bounds

Revista Matematica Iberoamericana

Volumn 24, Issue 2, 2008, Pages 407-431

  Desvillettes, Laurent ^a   Fellner, Klemens ^b,c  

^a Paris Descartes University   (France)

^b UNIVERSITY OF CAMBRIDGE   (United Kingdom)

^c UNIVERSITY OF VIENNA   (Austria)

Author keywords

Entropy method; Exponential decay; Reaction diffusion; Slowly growing a priori estimates

Indexed keywords

EID: 54049156080     PISSN: 02132230     EISSN: 22350616     Source Type: Journal    
DOI: 10.4171/RMI/541     Document Type: Article

References (37)

1. AMANN, H.: Global existence for semilinear parabolic systems. J. Reine Angew. Math. 360 (1985), 47-83.
2. CÁCERES, M., CARRILLO, J., TOSCANI, G.: Long-time behavior for a nonlinear fourth order parabolic equation. Trans. Amer. Math. Soc. 357 (2005), no. 3, 1161-1175.
3. CÁCERES, M., CARRILLO, J., GOUDON, T.: Equilibration rate for the linear inhomogeneous relaxation-time Boltzmann equation for charged particles. Comm. Partial Differential Equations 28 (2003), no. 5-6, 969-989.
4. CONWAY, E., HOFF, D., SMOLLER, J.: Large time behaviour of solutions of systems of nonlinear reaction-diffusion equations. SIAM J: Appl. Math. 35 (1978), no. 1, 1-16.
5. CSISZAR, I.: Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis von Markoffschen Ketten. Magyar Tud. Akad. Mat. Kutató Int. Közl 8 (1963), 85-108.
6. CARRILLO, J., VAZQUEZ, J. L.: Fine asymptotics for fast diffusion equations. Comm. Partial Differential Equations 28 (2003), no. 5-6, 1023-1056.
7. DEL PINO, M., DOLBEAULT, J.: Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions. J. Math. Pures Appl. 81 (2002), no. 9, 847-875.
8. DESVILLETTES, L., FELLNER, K.: Exponential Decay toward Equilibrium via Entropy Methods for Reaction-Diffusion Equations. J. Math. Anal. Appl. 319 (2006), no. 1, 157-176.
9. DESVILLETTES, L., FELLNER, K., PIERRE, M., VOVELLE, J.: About Global Existence for Quadratic Systems of Reaction-Diffusion. Adv. Nonlinear Stud. 7 (2007), no. 3, 491-511.
10. DESVILLETTES, L., MOUHOT, C.: Large time Behavior of the a priori bounds for the solutions to the spatially homogeneous Boltzmann equations with soft potentials. Asymptot. Anal. 54 (2007), no. 3-4, 235-245.
11. DESVILLETTES, L., VILLANI, C.: On the spatially homogeneous Landau equation for hard potentials. II. H-theorem and applications. Comm. Partial Differential Equations 25 (2000), no. 1-2, 261-298.
12. DESVILLETTES, L., VILLANI, C.: On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation. Comm. Pure Appl. Math. 54 (2001), no. 1, 1-42.
13. DESVILLETTES, L., VILLANI, C.: On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation. Invent. Math. 159 (2005), no. 2, 245-316.
14. FELLNER, K., NEUMANN, L., SCHMEISER, C.: Convergence to global equilibrium for spatially inhomogeneous kinetic models of non-micro-reversible processes. Monatsh. Math. 141 (2004), no. 4, 289-299.
15. FITZGIBBON, W. E., MORGAN, J., SANDERS, R.: Global existence and boundedness for a class of inhomogeneous semilinear parabolic systems. Nonlinear Anal. 19 (1992), no. 9, 885-899.
16. FITZGIBBON, W., HOLLIS, S., MORGAN, J.: Stability and Lyapunov functions for reaction-diffusion systems. SIAM J. Math. Anal. 28 (1997), no. 3, 595-610.
17. GLITZKY, A., GROCER, K., HÜNLICH, R.: Free energy and dissipation rate for reaction-diffusion processes of electrically charged species. Appl. Anal. 60 (1996), no. 3-4, 201-217.
18. GLITZKY, A., HÜNLICH, R.: Energetic estimates and asymptotics for electro-reaction-diffusion systems. Z. Angew. Math. Mech. 77 (1997), no. 11, 823-832.
19. GROGER, K.: Free energy estimates and asymptotic behaviour of reaction-diffusion processes. Preprint 20, Institut für Angewandte Analysis und Stochastik, Berlin, 1992.
20. HARAUX, A., YOUKANA, A.: On a result of K. Masuda concerning reaction-diffusion equations. Tôhoku Math. J. (2) 40 (1988), no. 1, 159-163.
21. HOLLIS, S., MARTIN, R., PIERRE, M.: Global existence and boundedness in reaction-diffusion systems. SIAM J. Math. Appl. 18 (1987), 744-761.
22. HOSHINO, H., YAMADA, Y.: Asymptotic behavior of global solutions for some reaction-diffusion systems. Nonlinear Anal. 23 (1994), no. 5, 639-650.
23. MASUDA, K.: On the global existence and asymptotic behavior of solution of reaction-diffusion equations. Hokkaido Math. J. 12 (1983), 360-370.
24. MARTIN, R. H., PIERRE, M.: Nonlinear Reaction-Diffusion Systems. In Nonlinear Equations in Applied Sciences, 363-398. Math. Sci. Engrg. 185. Academic Press, Boston, MA, 1992.
25. MORGAN, J.: Global existence for semilinear parabolic systems. SIAM J. Math. Anal. 20 (1989), no. 5, 1128-1144.
26. MULONE, G.: Nonlinear stability in fluid-dynamics in the presence of stabilizing effects and the choice of a measure of perturbations. In WASCOM 2003, 12th Conference on Waves and Stability in Continuous Media, 352-365. World Sci. Publ., River Edge, NJ, 2004.
27. KANEL, Y. I., KIRANE, M.: Global solutions of Reaction-Diffusion Systems with a Balance Law and Nonlinearities of Exponential Growth. J. Differential Equations 165 (2000), no. 1, 24-41.
28. 1 for reaction-diffusion systems. J. Evol. Eq. 3 (2003), no. 1, 153-168.
29. PIERRE, M., SCHMITT, D.: Blowup in reaction-diffusion systems with dissipation of mass. SIAM Rev. 42 (2000), no. 1, 93-106 (electronic).
30. RIONERO, S.: On the stability of binary reaction-diffusion systems. Nuovo Cimento Soc. Ital. Fis. B 119 (2004), no. 7-9, 773-784.
31. ROTHE, F.: Global Solutions of Reaction-Diffusion Systems. Lecture Notes in Mathematics 1072. Springer-Verlag, Berlin, 1984.
32. TAYLOR, M. E.: Partial Differential Equation III - Nonlinear Equations. Applied Mathematical Sciences 117. Springer-Verlag, New York, 1997.
33. TOSCANI, G., VILLANI, C.: Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous Boltzmann equation. Comm. Math. Phys. 203 (1999), no. 3, 667-706.
34. TOSCANI, G., VILLANI, C.: On the trend to equilibrium for some dissipative systems with slowly increasing a priori bounds. J. Statist. Phys. 98 (2000), no. 5-6, 1279-1309.
35. VILLANI, C.: Cercignani's conjecture is sometimes true and always almost true. Comm. Math. Phys. 234 (2003), no. 3, 455-490.
36. WEBB, G. F.: Theory of nonlinear age-dependent population dynamics. Monographs and Textbooks in Pure and Applied Mathematics 89. Marcel Dekker, Inc., New York, 1985.
37. ZELENYAK, T.: Stabilization of solutions of boundary value problems for second-order parabolic equations in one space variable. Differentsial'nye Uravneniya 4 (1968), 34-45.