December  2016, 21(10): 3441-3462. doi: 10.3934/dcdsb.2016106

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

1. 

Institut für Mathematik und Wissenschaftliches Rechnen, Heinrichstraße 36, 8010 Graz, Austria, Austria

2. 

Graduate School of Engineering Science, Department of Systems Innovation, Division of Mathematical Science, Osaka University, Japan

Received  January 2016 Revised  March 2016 Published  November 2016

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.
Citation: Klemens Fellner, Evangelos Latos, Takashi Suzuki. Global classical solutions for mass-conserving, (super)-quadratic reaction-diffusion systems in three and higher space dimensions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3441-3462. doi: 10.3934/dcdsb.2016106
References:
[1]

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

[2]

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

[3]

M. Bisi, F. Conforto and L. Desvillettes, Quasi-steady-state approximation for reaction-diffusion equations,, Bull. Inst. Math., 2 (2007), 823.   Google Scholar

[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.  doi: 10.1016/j.jmaa.2010.02.044.  Google Scholar

[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.  doi: 10.1080/03605302.2013.829500.  Google Scholar

[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.  doi: 10.1080/03605300903089867.  Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[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.  doi: 10.1515/ans-2007-0309.  Google Scholar

[13]

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

[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.  doi: 10.1216/rmjm/1181072348.  Google Scholar

[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.  doi: 10.3934/dcds.2013.33.2885.  Google Scholar

[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.  doi: 10.1002/mma.2524.  Google Scholar

[17]

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

[18]

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

[19]

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

[20]

F. Rothe, Global Solutions of Reaction-Diffusion Equations,, Lecture Notes in Mathematics, (1984).   Google Scholar

[21]

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

show all references

References:
[1]

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

[2]

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

[3]

M. Bisi, F. Conforto and L. Desvillettes, Quasi-steady-state approximation for reaction-diffusion equations,, Bull. Inst. Math., 2 (2007), 823.   Google Scholar

[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.  doi: 10.1016/j.jmaa.2010.02.044.  Google Scholar

[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.  doi: 10.1080/03605302.2013.829500.  Google Scholar

[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.  doi: 10.1080/03605300903089867.  Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[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.  doi: 10.1515/ans-2007-0309.  Google Scholar

[13]

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

[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.  doi: 10.1216/rmjm/1181072348.  Google Scholar

[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.  doi: 10.3934/dcds.2013.33.2885.  Google Scholar

[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.  doi: 10.1002/mma.2524.  Google Scholar

[17]

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

[18]

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

[19]

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

[20]

F. Rothe, Global Solutions of Reaction-Diffusion Equations,, Lecture Notes in Mathematics, (1984).   Google Scholar

[21]

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

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