May 2017, 37(5): 2431-2453. doi: 10.3934/dcds.2017105

Parabolic reaction-diffusion systems with nonlocal coupled diffusivity terms

1. 

EEIMVR -VCE -Universidade Federal Fluminense, Volta Redonda, RJ, Brazil

2. 

FCT -Universidade do Algarve, Faro, Portugal

3. 

CMAFCIO -Universidade de Lisboa, Portugal

* Corresponding author: H.B. de Oliveira, holivei@ualg.pt

Received  June 2016 Revised  December 2016 Published  February 2017

Fund Project: The first author was partially supported by the Research Project CAPES -Grant BEX 2478-12-8 and MEC/MCTI/CAPES/CNPq/FAPs no. 71/2013, Grant 88881.030388/2013-01, Brazil. The second author was partially supported by Fundação para a Ciência e a Tecnologia, UID/MAT/04561/2013-2015, Portugal

In this work we study a system of parabolic reaction-diffusion equations which are coupled not only through the reaction terms but also by way of nonlocal diffusivity functions. For the associated initial problem, endowed with homogeneous Dirichlet or Neumann boundary conditions, we prove the existence of global solutions. We also prove the existence of local solutions but with less assumptions on the boundedness of the nonlocal terms. The uniqueness result is established next and then we find the conditions under which the existence of strong solutions is assured. We establish several blow-up results for the strong solutions to our problem and we give a criterium for the convergence of these solutions towards a homogeneous state.

Citation: Jorge Ferreira, Hermenegildo Borges de Oliveira. Parabolic reaction-diffusion systems with nonlocal coupled diffusivity terms. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2431-2453. doi: 10.3934/dcds.2017105
References:
[1]

A. S. Ackleh and L. Ke, Existence-uniqueness and long time behavior for a class of nonlocal nonlinear parabolic evolution equations, Proc. Am. Math. Soc., 128 (2000), 3483-3492.

[2]

N.-H. Chang and M. Chipot, Nonlinear nonlocal evolution problems, RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., 97 (2003), 423-445.

[3]

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

[4]

M. Chipot and B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal. Theory Methods Appl., 30 (1997), 4619-4627.

[5]

M. Chipot and B. Lovat, On the asymptotic behaviour of some nonlocal problems, Positivity, 3 (1999), 65-81.

[6]

M. Chipot and B. Lovat, Existence and uniqueness results for a class of nonlocal elliptic and parabolic problems, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 8 (2001), 35-51.

[7]

M. Chipot and L. Molinet, Asymptotic behaviour of some nonlocal diffusion problems, Appl. Anal., 80 (2001), 279-315.

[8]

M. Chipot and J. F. Rodrigues, On a class of nonlocal nonlinear elliptic problems, RAIRO Modél. Math. Anal. Numér., 26 (1992), 447-467.

[9]

M. Chipot and T. Savitska, Nonlocal p-Laplace equations depending on the $L^p$ norm of the gradient, Adv. Differential Equations, 19 (2014), 997-1020.

[10]

M. ChipotV. Valente and G. Vergara Caffarelli, Remarks on a nonlocal problem involving the Dirichtlet energy, Rend. Sem. Mat. Univ. Padova., 110 (2003), 199-220.

[11]

M. Chlebík and M. Fila, From critical exponents to blow-up rates for parabolic problems, Rend. Mat. Appl. Ser Ⅶ, 19 (1999), 449-470.

[12]

E. ConwayD. Hoff and J. Smoller, Large time behavior of solutions of systems of nonlinear reaction-diffusion equations, SIAM J. Appl. Math., 35 (1978), 1-16. doi: 10.1137/0135001.

[13]

F. J. S. A. CorrêaS. D. B. Menezes and J. Ferreira, On a class of problems involving a nonlocal operator, Appl. Math. Comput., 147 (2004), 475-489.

[14]

M. Escobedo and M. Herrero, Boundedness and blow up for a semilinear reaction diffusion system, J. Differential Equations, 89 (1991), 176-202. doi: 10.1016/0022-0396(91)90118-S.

[15]

L. C. Evans, Partial Differential Equations Graduate Studies in Math. 19 American Mathematical Society, Providence, RI, 1998. doi: 10.1090/gsm/019.

[16]

K. IchikawaM. Rouzimaimaiti and T. Suzuki, Reaction diffusion equation with non-local term arises as a mean field limit of the master equation, Discrete Contin. Dyn. Syst. S(Special Issue), 5 (2012), 115-126.

[17]

J. -L. Lions, On some questions in boundary value problems of mathematical physics, In Proc. Internat. Sympos. , Inst. Mat. , Univ. Fed. Rio de Janeiro, North-Holland Math. Stud. 30, 284-346, North-Holland, Amsterdam-New York, 1978. doi: 10.1016/S0304-0208(08)70870-3.

[18]

B. Lovat, Etudes de Quelques Problemes Paraboliques non Locaux Thése présentée pour l'obtention du doctorat en Mathématiques, Université de Metz, 1995. Available from: http://docnum.univ-lorraine.fr/public/UPV-M/Theses/1995/Lovat.Bruno.SMZ9536.pdf

[19]

M. Negreanu and J. I. Tello, On a competitive system under chemotactic effects with non-local terms, Nonlinearity, 26 (2013), 1083-1103. doi: 10.1088/0951-7715/26/4/1083.

[20]

J. Smoller, Shock Waves and Reaction-Diffusion Equations Second edition. Grundlehren der Mathematischen Wissenschaften, 258. Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.

[21]

Ph. Souplet and S. Tayachi, Optimal condition for non-simultaneous blow-up in a reaction-diffusion system, J. Math. Soc. Japan, 56 (2004), 571-584. doi: 10.2969/jmsj/1191418646.

[22]

I. Vrabie, Compactness Methods for Nonlinear Evolutions Second edition. Pitman Monographs and Surveys in Pure and Applied Mathematics, 75. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc. , New York, 1995.

[23]

S. Zheng and M. Chipot, Asymptotic behavior of solutions to nonlinear parabolic equations with nonlocal terms, Asymptot. Anal., 45 (2005), 301-312.

show all references

References:
[1]

A. S. Ackleh and L. Ke, Existence-uniqueness and long time behavior for a class of nonlocal nonlinear parabolic evolution equations, Proc. Am. Math. Soc., 128 (2000), 3483-3492.

[2]

N.-H. Chang and M. Chipot, Nonlinear nonlocal evolution problems, RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., 97 (2003), 423-445.

[3]

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

[4]

M. Chipot and B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal. Theory Methods Appl., 30 (1997), 4619-4627.

[5]

M. Chipot and B. Lovat, On the asymptotic behaviour of some nonlocal problems, Positivity, 3 (1999), 65-81.

[6]

M. Chipot and B. Lovat, Existence and uniqueness results for a class of nonlocal elliptic and parabolic problems, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 8 (2001), 35-51.

[7]

M. Chipot and L. Molinet, Asymptotic behaviour of some nonlocal diffusion problems, Appl. Anal., 80 (2001), 279-315.

[8]

M. Chipot and J. F. Rodrigues, On a class of nonlocal nonlinear elliptic problems, RAIRO Modél. Math. Anal. Numér., 26 (1992), 447-467.

[9]

M. Chipot and T. Savitska, Nonlocal p-Laplace equations depending on the $L^p$ norm of the gradient, Adv. Differential Equations, 19 (2014), 997-1020.

[10]

M. ChipotV. Valente and G. Vergara Caffarelli, Remarks on a nonlocal problem involving the Dirichtlet energy, Rend. Sem. Mat. Univ. Padova., 110 (2003), 199-220.

[11]

M. Chlebík and M. Fila, From critical exponents to blow-up rates for parabolic problems, Rend. Mat. Appl. Ser Ⅶ, 19 (1999), 449-470.

[12]

E. ConwayD. Hoff and J. Smoller, Large time behavior of solutions of systems of nonlinear reaction-diffusion equations, SIAM J. Appl. Math., 35 (1978), 1-16. doi: 10.1137/0135001.

[13]

F. J. S. A. CorrêaS. D. B. Menezes and J. Ferreira, On a class of problems involving a nonlocal operator, Appl. Math. Comput., 147 (2004), 475-489.

[14]

M. Escobedo and M. Herrero, Boundedness and blow up for a semilinear reaction diffusion system, J. Differential Equations, 89 (1991), 176-202. doi: 10.1016/0022-0396(91)90118-S.

[15]

L. C. Evans, Partial Differential Equations Graduate Studies in Math. 19 American Mathematical Society, Providence, RI, 1998. doi: 10.1090/gsm/019.

[16]

K. IchikawaM. Rouzimaimaiti and T. Suzuki, Reaction diffusion equation with non-local term arises as a mean field limit of the master equation, Discrete Contin. Dyn. Syst. S(Special Issue), 5 (2012), 115-126.

[17]

J. -L. Lions, On some questions in boundary value problems of mathematical physics, In Proc. Internat. Sympos. , Inst. Mat. , Univ. Fed. Rio de Janeiro, North-Holland Math. Stud. 30, 284-346, North-Holland, Amsterdam-New York, 1978. doi: 10.1016/S0304-0208(08)70870-3.

[18]

B. Lovat, Etudes de Quelques Problemes Paraboliques non Locaux Thése présentée pour l'obtention du doctorat en Mathématiques, Université de Metz, 1995. Available from: http://docnum.univ-lorraine.fr/public/UPV-M/Theses/1995/Lovat.Bruno.SMZ9536.pdf

[19]

M. Negreanu and J. I. Tello, On a competitive system under chemotactic effects with non-local terms, Nonlinearity, 26 (2013), 1083-1103. doi: 10.1088/0951-7715/26/4/1083.

[20]

J. Smoller, Shock Waves and Reaction-Diffusion Equations Second edition. Grundlehren der Mathematischen Wissenschaften, 258. Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.

[21]

Ph. Souplet and S. Tayachi, Optimal condition for non-simultaneous blow-up in a reaction-diffusion system, J. Math. Soc. Japan, 56 (2004), 571-584. doi: 10.2969/jmsj/1191418646.

[22]

I. Vrabie, Compactness Methods for Nonlinear Evolutions Second edition. Pitman Monographs and Surveys in Pure and Applied Mathematics, 75. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc. , New York, 1995.

[23]

S. Zheng and M. Chipot, Asymptotic behavior of solutions to nonlinear parabolic equations with nonlocal terms, Asymptot. Anal., 45 (2005), 301-312.

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