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 and 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), 47-83. 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.

show all references

References:
[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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