Desvillettes, Laurent; Fellner, Klemens Entropy methods for reaction-diffusion equations: Slowly growing a-priori bounds. (English) Zbl 1171.35330 Rev. Mat. Iberoam. 24, No. 2, 407-431 (2008). Authors’ abstract: In the continuation of [J. Math. Anal. Appl. 319, No. 1, 157–176 (2006; Zbl 1096.35018)], we study reversible reaction-diffusion equations via entropy methods (based on the free energy functional) for a 1D system of four species. We improve the existing theory by getting 1) almost exponential convergence in \(L^1\) to the steady state via a precise entropy-entropy dissipation estimate, 2) an explicit global \(L^\infty\) bound via interpolation of a polynomially growing \(H^1\) bound with the almost exponential \(L^1\) convergence, and 3), finally, explicit exponential convergence to the steady state in all Sobolev norms. Reviewer: Pigong Han (Beijing, China) Cited in 41 Documents MSC: 35B40 Asymptotic behavior of solutions to PDEs 35K57 Reaction-diffusion equations Keywords:reaction-diffusion; entropy method; exponential decay; slowly growing a priori estimates; reversible reaction-diffusion equations; 1D system of four species Citations:Zbl 1096.35018 × Cite Format Result Cite Review PDF Full Text: DOI Euclid EuDML References: [1] Amann, H.: Global existence for semilinear parabolic systems. J. Reine Angew. Math. 360 (1985), 47-83. · Zbl 0564.35060 · doi:10.1515/crll.1985.360.47 [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. · Zbl 1077.35028 · doi:10.1090/S0002-9947-04-03528-7 [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. · Zbl 1045.35094 · doi:10.1081/PDE-120021182 [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. · Zbl 0383.35035 · doi:10.1137/0135001 [5] Csiszár, I.: Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis von Markoffschen Ketten. Magyar Tud. Akad. Mat. Kutató Int. Közl 8 (1963), 85-108. · Zbl 0124.08703 [6] Carrillo, J., Vazquez, J. L.: Fine asymptotics for fast diffusion equations. Comm. Partial Differential Equations 28 (2003), no. 5-6, 1023-1056. · Zbl 1036.35100 · doi:10.1081/PDE-120021185 [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. · Zbl 1112.35310 · doi:10.1016/S0021-7824(02)01266-7 [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. · Zbl 1096.35018 · doi:10.1016/j.jmaa.2005.07.003 [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. · Zbl 1330.35211 [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. · Zbl 1141.35337 [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. · Zbl 0951.35130 · doi:10.1080/03605300008821513 [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. · Zbl 1029.82032 · doi:10.1002/1097-0312(200101)54:1<1::AID-CPA1>3.0.CO;2-Q [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. · Zbl 1162.82316 · doi:10.1007/s00222-004-0389-9 [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. · Zbl 1112.82038 · doi:10.1007/s00605-003-0058-2 [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. · Zbl 0801.35041 · doi:10.1016/0362-546X(92)90057-L [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. · Zbl 0876.35014 · doi:10.1137/S0036141094272241 [17] Glitzky, A., Gröger, 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. · Zbl 0871.35017 · doi:10.1080/00036819608840428 [18] Glitzky, A., Hünlich, R.: Energetic estimates and asymptotics for electro-reaction-diffusion systems. Z. Angew. Math. Mech. 77 (1997), no. 11, 823-832. · Zbl 0887.35024 · doi:10.1002/zamm.19970771105 [19] Gröger, 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. · Zbl 0689.35041 · doi:10.2748/tmj/1178228084 [21] Hollis, S., Martin, R., Pierre, M.: Global existence and boundedness in reaction-diffusion systems. SIAM J. Math. Appl. 18 (1987), no. 3, 744-761. · Zbl 0655.35045 · doi:10.1137/0518057 [22] Hoshino, H., Yamada, Y.: Asymptotic behavior of global solutions for some reaction-diffusion systems. Nonlinear Anal. 23 (1994), no. 5, 639-650. · Zbl 0811.35062 · doi:10.1016/0362-546X(94)90243-7 [23] Masuda, K.: On the global existence and asymptotic behavior of solution of reaction-diffusion equations. Hokkaido Math. J. 12 (1983), 360-370. · Zbl 0529.35037 [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. · Zbl 0781.35030 [25] Morgan, J.: Global existence for semilinear parabolic systems. SIAM J. Math. Anal. 20 (1989), no. 5, 1128-1144. · Zbl 0692.35055 · doi:10.1137/0520075 [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., Ricer Edge, NJ, 2004. · Zbl 1069.35010 · doi:10.1142/9789812702937_0041 [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. · Zbl 0970.35073 · doi:10.1006/jdeq.2000.3769 [28] Pierre, M.: Weak solutions and supersolutions in \(L^ 1\) for reaction-diffusion systems. Dedicated to Philippe Bénilan. J. Evol. Eq. 3 (2003), no. 1, 153-168. · Zbl 1026.35047 · doi:10.1007/s000280300007 [29] Pierre, M., Schmitt, D.: Blowup in reaction-diffusion systems with dissipation of mass. SIAM Rev. 42 (2000), no. 1, 93-106 (electronic). JSTOR: · Zbl 0942.35033 · doi:10.1137/S0036144599359735 [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. · Zbl 0546.35003 [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. · Zbl 0944.35066 · doi:10.1007/s002200050631 [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. · Zbl 1034.82032 · doi:10.1023/A:1018623930325 [35] Villani, C.: Cercignani’s conjecture is sometimes true and always almost true. Comm. Math. Phys. 234 (2003), no. 3, 455-490. · Zbl 1041.82018 · doi:10.1007/s00220-002-0777-1 [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. · Zbl 0555.92014 [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. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.