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Global classical solutions for mass-conserving, (super)-quadratic reaction-diffusion systems in three and higher space dimensions

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  • This paper considers quadratic and super-quadratic reaction-diffusion systems, for which all species satisfy uniform-in-time $L^1$ a-priori estimates, for instance, as a consequence of suitable mass conservation laws. A new result on the global existence of classical solutions is proved in three and higher space dimensions by combining regularity and interpolation arguments in Bochner spaces, a bootstrap scheme and a weak comparison argument. Moreover, provided that the considered system allows for entropy entropy-dissipation estimates proving exponential convergence to equilibrium, we are also able to prove that solutions are bounded uniformly-in-time.
    Mathematics Subject Classification: 35K61, 35A01, 35K57, 35B40.

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  • [1]

    H. Amann, Global existence for semilinear parabolic systems, J. Reine Angew. Math., 360 (1985), 47-83.doi: 10.1515/crll.1985.360.47.

    [2]

    M. Bisi and L. Desvillettes, From reactive Boltzmann equations to reaction-diffusion systems, J. Stat. Phys., 125 (2006), 249-280.doi: 10.1007/s10955-005-8075-x.

    [3]

    M. Bisi, F. Conforto and L. Desvillettes, Quasi-steady-state approximation for reaction-diffusion equations, Bull. Inst. Math., Acad. Sin. (N.S.), 2 (2007), 823-850.

    [4]

    D. Bothe and M. Pierre, Quasi-steady-state approximation for a reaction-diffusion system with fast intermediate, J. Math. Anal. Appl., 368 (2010), 120-132.doi: 10.1016/j.jmaa.2010.02.044.

    [5]

    J. A. Cañizo, L. Desvillettes and K. Fellner, Improved duality estimates and applications to reaction-diffusion equations, Comm. Partial Differential Equations, 39 (2014), 1185-1204.doi: 10.1080/03605302.2013.829500.

    [6]

    C. Caputo and A. Vasseur, Global regularity of solutions to systems of reaction-diffusion with sub-quadratic growth in any dimension, Commun. Partial Differential Equations, 34 (2009), 1228-1250.doi: 10.1080/03605300903089867.

    [7]

    M. Chipot, Elements of Nonlinear Analysis, Birkhäuser, 2000.doi: 10.1007/978-3-0348-8428-0.

    [8]

    A. De Masi and E. Presutti, Mathematical Methods for Hydrodynamic Limits, Springer-Verlag, Berlin, 1991.doi: 10.1007/BFb0086457.

    [9]

    L. Desvillettes and K. Fellner, Exponential decay toward equilibrium via entropy methods for reaction-diffusion equations, J. Math. Anal. Appl., 319 (2006), 157-176.doi: 10.1016/j.jmaa.2005.07.003.

    [10]

    L. Desvillettes and K. Fellner, Entropy methods for reaction-diffusion equations: Slowly growing a priori bounds, Revista Matemática Iberoamericana, 24 (2008), 407-431.doi: 10.4171/RMI/541.

    [11]

    L. Desvillettes and K. Fellner, Exponential convergence to equilibrium for a nonlinear reaction-diffusion systems arising in reversible chemistry, System Modelling and Optimization, IFIP AICT, 443 (2014), 96-104.doi: 10.1007/978-3-662-45504-3_9.

    [12]

    L. Desvillettes, K. Fellner, M. Pierre and J. Vovelle, About global existence for quadratic systems of reaction-diffusion, J. Advanced Nonlinear Studies, 7 (2007), 491-511.doi: 10.1515/ans-2007-0309.

    [13]

    T. Goudon and A. Vasseur, Regularity analysis for systems of reaction-diffusion equations, Ann. Sci. Ec. Norm. Super., (4) 43 (2010), 117-142.

    [14]

    S. L. Hollis and J. J. Morgan, On the blow-up of solution to some semilinear and quasilinear reaction-diffusion systems, Rocky Mountain Journal of Mathematics, 24 (1994), 1447-1465.doi: 10.1216/rmjm/1181072348.

    [15]

    G. Karali and T. Suzuki, Global-in-time behavior of the solution to a Gierer-Meinhardt system, Discrete and Continuous Dynamical Systems, 33 (2013), 2885-2900.doi: 10.3934/dcds.2013.33.2885.

    [16]

    E. Latos, T. Suzuki and Y. Yamada, Transient and asymptotic dynamics of a prey-predator system with diffusion, Math. Meth. Appl. Sci., 35 (2012), 1101-1109.doi: 10.1002/mma.2524.

    [17]

    M. Pierre, Weak solutions and supersolutions in $L^1$ for reaction-diffusion systems, J. Evol. Equ., 3 (2003), 153-168.doi: 10.1007/s000280300007.

    [18]

    M. Pierre, Global existence in reaction-diffusion systems with dissipation of mass: A Survey, Milan J. Math., 78 (2010), 417-455.doi: 10.1007/s00032-010-0133-4.

    [19]

    M. Pierre and D. Schmitt, Blowup in reaction-diffusion systems with dissipation of mass, SIAM Review, 42 (2000), 93-106.doi: 10.1137/S0036144599359735.

    [20]

    F. Rothe, Global Solutions of Reaction-Diffusion Equations, Lecture Notes in Mathematics, Springer-Verlag, 1984.

    [21]

    T. Suzuki and Y. Yamada, Global-in-time behavior of Lotka-Volterra system with diffusion, Indiana Univ. Math., 64 (2015), 181-216.doi: 10.1512/iumj.2015.64.5460.

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