April  2020, 19(4): 2347-2368. doi: 10.3934/cpaa.2020102

Convergence of nonautonomous multivalued problems with large diffusion to ordinary differential inclusions

1. 

Instituto de Matemática e Computacão, Universidade Federal de Itajubá, Av. BPS n. 1303, Bairro Pinheirinho, 37500-903, Itajubá - MG - Brazil

2. 

Centro de Investigación Operativa, Universidad Miguel Hernández de Elche, 03202-Elche, Alicante, Spain

Dedicated to Professor Tomás Caraballo on occasion of his 60th Birthday

Received  November 2018 Revised  October 2019 Published  January 2020

In this work we consider a family of nonautonomous partial differential inclusions governed by $ p $-laplacian operators with variable exponents and large diffusion and driven by a forcing nonlinear term of Heaviside type. We prove first that this problem generates a sequence of multivalued nonautonomous dynamical systems possessing a pullback attractor. The main result of this paper states that the solutions of the family of partial differential inclusions converge to the solutions of a limit ordinary differential inclusion for large diffusion and when the exponents go to $ 2 $. After that we prove the upper semicontinuity of the pullback attractors.

Citation: Jacson Simsen, Mariza Stefanello Simsen, José Valero. Convergence of nonautonomous multivalued problems with large diffusion to ordinary differential inclusions. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2347-2368. doi: 10.3934/cpaa.2020102
References:
[1]

J. M. ArrietaA. N. Carvalho and A. Rodríguez-Bernal, Upper semicontinuity for attractors of parabolic problems with localized large diffusion and nonlinear boundary conditions, J. Differential Equations, 168 (2000), 33-59.  doi: 10.1006/jdeq.2000.3876.  Google Scholar

[2]

J. M. ArrietaA. Rodríguez-Bernal and J. Valero, Dynamics of a reaction-diffusion equation with a discontinuous nonlinearity, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 16 (2006), 2695-2984.  doi: 10.1142/S0218127406016586.  Google Scholar

[3]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei, Bucuresti, 1976.  Google Scholar

[4]

S. Bensid and J. I. Díaz, Stability results for discontinuous nonlinear elliptic and parabolic problems with a s-shaped bifurcation branch of stationary solutions, Discrete Contin. Dyn. Syst., Ser. B, 22 (2017), 1757-1778.  doi: 10.3934/dcdsb.2017105.  Google Scholar

[5]

S. Bensid and J. I. Díaz, On the exact number of monotone solutions of a simplified Budyko climate model and their different stability, Discrete Contin. Dyn. Syst., Ser. B, 24 (2019), 1033–1047.,  Google Scholar

[6]

H. Brézis, Analyse Fonctionalle, Paris, Masson Editeur, 1983. Google Scholar

[7]

M. I. Budyko, The effects of solar radiation variations on the climate of the Earth, Tellus, 21 (1969), 611–619. Google Scholar

[8]

T. CaraballoP. E. Kloeden and P. Marín-Rubio, Weak pullback attractors of setvalued processes, J. Math. Anal. Appl., 288 (2003), 692-707.  doi: 10.1016/j.jmaa.2003.09.039.  Google Scholar

[9]

T. CaraballoJ. A. LangaV. S. Melnik and J. Valero, Pullback attractors for nonautonomous and stochastic multivalued dynamical systems, Set-Valued Anal., 11 (2003), 153-201.  doi: 10.1023/A:1022902802385.  Google Scholar

[10]

T. CaraballoJ. A. Langa and J. Valero, Structure of the pullback attractor for a non-autonomous scalar differential inclusion, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 979-994.  doi: 10.3934/dcdss.2016037.  Google Scholar

[11]

V. L. CarboneA. N. Carvalho and K. Schiabel-Silva, Continuity of attractors for parabolic problems with localized large diffusion, Nonlinear Anal., 68 (2008), 515-535.  doi: 10.1016/j.na.2006.11.017.  Google Scholar

[12]

V. L. CarboneC. Gentile and K. Schiabel-Silva, Asymptotic properties in parabolic problems dominated by a p-Laplacian operator with localized large diffusion, Nonlinear Anal., 74 (2011), 4002-4011.  doi: 10.1016/j.na.2011.03.028.  Google Scholar

[13]

A. N. Carvalho, Infinite dimensional dynamics described by ordinary differential equations, J. Differential Equations, 116 (1995), 338-404.  doi: 10.1006/jdeq.1995.1039.  Google Scholar

[14]

A. N. Carvalho and J. K. Hale, Large diffusion with dispersion, Nonlinear Anal., 17 (1991), 1139-1151.  doi: 10.1016/0362-546X(91)90233-Q.  Google Scholar

[15]

A. N. Carvalho and A. L. Pereira, A scalar parabolic equation whose asymptotic behavior is dictated by a system of ordinary differential equations, J. Differential Equations, 112 (1994), 81-130.  doi: 10.1006/jdeq.1994.1096.  Google Scholar

[16]

A. N. Carvalho and S. Piskarev, A general approximation scheme for attractors of abstract parabolic problems, Numer. Funct. Anal. Optim., 27 (2006), 785-829.  doi: 10.1080/01630560600882723.  Google Scholar

[17]

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.  Google Scholar

[18]

J. I. Díaz and I. I. Vrabie, Existence for reaction diffusion systems. A compactness method approach, J. Math. Anal. Appl., 188 (1994) 521–540. doi: 10.1006/jmaa.1994.1443.  Google Scholar

[19]

J. I. DíazJ. Hernández and L. Tello, Some results about multiplicity and bifurcation of stationary solutions of a reaction diffusion climatological model, Rev. R. Acad. Cien. Serie A. Mat., 96 (2002), 357-366.   Google Scholar

[20]

L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Springer-Verlag, Berlin, Heidelberg, 2011. doi: 10.1007/978-3-642-18363-8.  Google Scholar

[21]

X. L. Fan and Q. H. Zhang, Existence of solutions for $p(x)$-laplacian Dirichlet problems, Nonlinear Anal., 52 (2003), 1843-1852.  doi: 10.1016/S0362-546X(02)00150-5.  Google Scholar

[22]

E. Feireisl and J. Norbury, Some existence and nonuniqueness theorems for solutions of parabolic equations with discontinuous nonlinearities, Proc. Roy. Soc. Edinburgh, 119A (1991), 1-17.  doi: 10.1017/S0308210500028262.  Google Scholar

[23]

A. C. Fernandes, M. C. Gonçalves and J. Simsen, Non-autonomous reaction-diffusion equations with variable exponents and large diffusion, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1485–1510.  Google Scholar

[24]

G. Fusco, On the explicit construction of an ODE which has the same dynamics as a scalar parabolic PDE, J. Differential Equations, 69 (1987), 85-110.  doi: 10.1016/0022-0396(87)90104-5.  Google Scholar

[25]

Z. GuoQ. LiuJ. Sun and B. Wu, Reaction-diffusion systems with $p(x)$-growth for image denoising, Nonlinear Anal. Real World Appl., 12 (2011), 2904-2918.  doi: 10.1016/j.nonrwa.2011.04.015.  Google Scholar

[26]

J. K. Hale, Large diffusivity and asymptotic behavior in parabolic systems, J. Math. Anal. Appl., 118 (1986), 455-466.  doi: 10.1016/0022-247X(86)90273-8.  Google Scholar

[27]

J. K. Hale and C. Rocha, Varying boundary conditions with large diffusivity, J. Math. Pures Appl., 66 (1987), 139–158.  Google Scholar

[28]

J. K. Hale and C. Rocha, Interaction of diffusion and boundary conditions, Nonlinear Anal., 11 (1987), 633-64.  doi: 10.1016/0362-546X(87)90078-2.  Google Scholar

[29]

O. V. Kapustyan, V. S. Melnik, J. Valero and V. V. Yasinski, Global Attractors of Multi-Valued Dynamical Systems and Evolution Equations Without Uniqueness, National Academy of Sciences of Ukraine, Naukova Dumka, 2008. Google Scholar

[30]

P. E. Kloeden and J. Simsen, Pullback attractors for non-autonomous evolution equations with spatially variable exponents, Commun. Pure Appl. Anal., 13 (2014), 2543-2557.  doi: 10.3934/cpaa.2014.13.2543.  Google Scholar

[31]

P. E. KloedenJ. Simsen and M. S. Simsen, Asymptotically autonomous multivalued Cauchy problems with spatially variable exponents, J. Math. Anal. Appl., 445 (2017), 513-531.  doi: 10.1016/j.jmaa.2016.08.004.  Google Scholar

[32]

V. S. Melnik and J. Valero, On attractors of multi-valued semi-flows and differential inclusions, Set-Valued Anal., 6 (1998), 83-111.  doi: 10.1023/A:1008608431399.  Google Scholar

[33]

V. S. Melnik and J. Valero, On global attractors of multivalued semiprocesses and nonautonomous evolution inclusions, Set-Valued Anal., 8 (2000), 375-403.  doi: 10.1023/A:1026514727329.  Google Scholar

[34]

J. SimsenM. S. Simsen and M. R. T. Primo, Reaction-diffusion equations with spatially variable exponents and large diffusion, Commun. Pure Appl. Anal., 15 (2016), 495-506.  doi: 10.3934/cpaa.2016.15.495.  Google Scholar

[35]

J. Simsen, Partial differential inclusions with spatially variable exponents and large diffusion, Mathematics in Enginnering, Science and Aerospace MESA, 7 (2016), 479–489. Google Scholar

[36]

J. Simsen, Weak upper semicontinuity of pullback attractors for nonautonomous reaction-diffusion equations, Electron. J. Qual. Theory Differ. Equ., 68 (2019), 1–14.  Google Scholar

[37]

J. Simsen and C. B. Gentile, Well-posed p-laplacian problems with large diffusion, Nonlinear Anal., 71 (2009), 4609-4617.  doi: 10.1016/j.na.2009.03.041.  Google Scholar

[38]

J. Simsen and C. B. Gentile, Systems of p-Laplacian differential inclusions with large diffusion, J. Math. Anal. Appl., 368 (2010), 525-537.  doi: 10.1016/j.jmaa.2010.02.006.  Google Scholar

[39]

J. Simsen, M. S. Simsen and F. B. Rocha, Existence of solutions for some classes of parabolic problems involving variable exponents, Nonlinear Stud., 21 (2014), 113–128.  Google Scholar

[40]

J. SimsenM. S. Simsen and A. Zimmermann, Study of ODE limit problems for reaction-diffusion equations, Opuscula Math., 38 (2018), 117-131.  doi: 10.7494/opmath.2018.38.1.117.  Google Scholar

[41]

J. Simsen and J. Valero, Characterization of pullback attractors for multivalued nonautonomous dynamical systems, in, Advances in Dynamical Systems and Control (V. A. Sadovnichiy and Z. Zgurovsky eds.), Studies in Systems, Decision and Control 659, Springer (2016), pp. 179–195.  Google Scholar

[42]

D. Terman, A free boundary problem arising from a bistable reaction-diffusion equation, SIAM J. Math. Anal., 14 (1983), 1107-1129.  doi: 10.1137/0514086.  Google Scholar

[43]

D. Terman, A free boundary arising from a model for nerve conduction, J. Differential Equations, 58 (1985), 345-363.  doi: 10.1016/0022-0396(85)90004-X.  Google Scholar

[44]

R. Willie, Large diffusivity stability of attractors in the $C$-topology for a semilinear reaction and diffusion system of equations, Dynamics of PDE, 3 (2006), 173-197.  doi: 10.4310/DPDE.2006.v3.n3.a1.  Google Scholar

show all references

References:
[1]

J. M. ArrietaA. N. Carvalho and A. Rodríguez-Bernal, Upper semicontinuity for attractors of parabolic problems with localized large diffusion and nonlinear boundary conditions, J. Differential Equations, 168 (2000), 33-59.  doi: 10.1006/jdeq.2000.3876.  Google Scholar

[2]

J. M. ArrietaA. Rodríguez-Bernal and J. Valero, Dynamics of a reaction-diffusion equation with a discontinuous nonlinearity, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 16 (2006), 2695-2984.  doi: 10.1142/S0218127406016586.  Google Scholar

[3]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei, Bucuresti, 1976.  Google Scholar

[4]

S. Bensid and J. I. Díaz, Stability results for discontinuous nonlinear elliptic and parabolic problems with a s-shaped bifurcation branch of stationary solutions, Discrete Contin. Dyn. Syst., Ser. B, 22 (2017), 1757-1778.  doi: 10.3934/dcdsb.2017105.  Google Scholar

[5]

S. Bensid and J. I. Díaz, On the exact number of monotone solutions of a simplified Budyko climate model and their different stability, Discrete Contin. Dyn. Syst., Ser. B, 24 (2019), 1033–1047.,  Google Scholar

[6]

H. Brézis, Analyse Fonctionalle, Paris, Masson Editeur, 1983. Google Scholar

[7]

M. I. Budyko, The effects of solar radiation variations on the climate of the Earth, Tellus, 21 (1969), 611–619. Google Scholar

[8]

T. CaraballoP. E. Kloeden and P. Marín-Rubio, Weak pullback attractors of setvalued processes, J. Math. Anal. Appl., 288 (2003), 692-707.  doi: 10.1016/j.jmaa.2003.09.039.  Google Scholar

[9]

T. CaraballoJ. A. LangaV. S. Melnik and J. Valero, Pullback attractors for nonautonomous and stochastic multivalued dynamical systems, Set-Valued Anal., 11 (2003), 153-201.  doi: 10.1023/A:1022902802385.  Google Scholar

[10]

T. CaraballoJ. A. Langa and J. Valero, Structure of the pullback attractor for a non-autonomous scalar differential inclusion, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 979-994.  doi: 10.3934/dcdss.2016037.  Google Scholar

[11]

V. L. CarboneA. N. Carvalho and K. Schiabel-Silva, Continuity of attractors for parabolic problems with localized large diffusion, Nonlinear Anal., 68 (2008), 515-535.  doi: 10.1016/j.na.2006.11.017.  Google Scholar

[12]

V. L. CarboneC. Gentile and K. Schiabel-Silva, Asymptotic properties in parabolic problems dominated by a p-Laplacian operator with localized large diffusion, Nonlinear Anal., 74 (2011), 4002-4011.  doi: 10.1016/j.na.2011.03.028.  Google Scholar

[13]

A. N. Carvalho, Infinite dimensional dynamics described by ordinary differential equations, J. Differential Equations, 116 (1995), 338-404.  doi: 10.1006/jdeq.1995.1039.  Google Scholar

[14]

A. N. Carvalho and J. K. Hale, Large diffusion with dispersion, Nonlinear Anal., 17 (1991), 1139-1151.  doi: 10.1016/0362-546X(91)90233-Q.  Google Scholar

[15]

A. N. Carvalho and A. L. Pereira, A scalar parabolic equation whose asymptotic behavior is dictated by a system of ordinary differential equations, J. Differential Equations, 112 (1994), 81-130.  doi: 10.1006/jdeq.1994.1096.  Google Scholar

[16]

A. N. Carvalho and S. Piskarev, A general approximation scheme for attractors of abstract parabolic problems, Numer. Funct. Anal. Optim., 27 (2006), 785-829.  doi: 10.1080/01630560600882723.  Google Scholar

[17]

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.  Google Scholar

[18]

J. I. Díaz and I. I. Vrabie, Existence for reaction diffusion systems. A compactness method approach, J. Math. Anal. Appl., 188 (1994) 521–540. doi: 10.1006/jmaa.1994.1443.  Google Scholar

[19]

J. I. DíazJ. Hernández and L. Tello, Some results about multiplicity and bifurcation of stationary solutions of a reaction diffusion climatological model, Rev. R. Acad. Cien. Serie A. Mat., 96 (2002), 357-366.   Google Scholar

[20]

L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Springer-Verlag, Berlin, Heidelberg, 2011. doi: 10.1007/978-3-642-18363-8.  Google Scholar

[21]

X. L. Fan and Q. H. Zhang, Existence of solutions for $p(x)$-laplacian Dirichlet problems, Nonlinear Anal., 52 (2003), 1843-1852.  doi: 10.1016/S0362-546X(02)00150-5.  Google Scholar

[22]

E. Feireisl and J. Norbury, Some existence and nonuniqueness theorems for solutions of parabolic equations with discontinuous nonlinearities, Proc. Roy. Soc. Edinburgh, 119A (1991), 1-17.  doi: 10.1017/S0308210500028262.  Google Scholar

[23]

A. C. Fernandes, M. C. Gonçalves and J. Simsen, Non-autonomous reaction-diffusion equations with variable exponents and large diffusion, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1485–1510.  Google Scholar

[24]

G. Fusco, On the explicit construction of an ODE which has the same dynamics as a scalar parabolic PDE, J. Differential Equations, 69 (1987), 85-110.  doi: 10.1016/0022-0396(87)90104-5.  Google Scholar

[25]

Z. GuoQ. LiuJ. Sun and B. Wu, Reaction-diffusion systems with $p(x)$-growth for image denoising, Nonlinear Anal. Real World Appl., 12 (2011), 2904-2918.  doi: 10.1016/j.nonrwa.2011.04.015.  Google Scholar

[26]

J. K. Hale, Large diffusivity and asymptotic behavior in parabolic systems, J. Math. Anal. Appl., 118 (1986), 455-466.  doi: 10.1016/0022-247X(86)90273-8.  Google Scholar

[27]

J. K. Hale and C. Rocha, Varying boundary conditions with large diffusivity, J. Math. Pures Appl., 66 (1987), 139–158.  Google Scholar

[28]

J. K. Hale and C. Rocha, Interaction of diffusion and boundary conditions, Nonlinear Anal., 11 (1987), 633-64.  doi: 10.1016/0362-546X(87)90078-2.  Google Scholar

[29]

O. V. Kapustyan, V. S. Melnik, J. Valero and V. V. Yasinski, Global Attractors of Multi-Valued Dynamical Systems and Evolution Equations Without Uniqueness, National Academy of Sciences of Ukraine, Naukova Dumka, 2008. Google Scholar

[30]

P. E. Kloeden and J. Simsen, Pullback attractors for non-autonomous evolution equations with spatially variable exponents, Commun. Pure Appl. Anal., 13 (2014), 2543-2557.  doi: 10.3934/cpaa.2014.13.2543.  Google Scholar

[31]

P. E. KloedenJ. Simsen and M. S. Simsen, Asymptotically autonomous multivalued Cauchy problems with spatially variable exponents, J. Math. Anal. Appl., 445 (2017), 513-531.  doi: 10.1016/j.jmaa.2016.08.004.  Google Scholar

[32]

V. S. Melnik and J. Valero, On attractors of multi-valued semi-flows and differential inclusions, Set-Valued Anal., 6 (1998), 83-111.  doi: 10.1023/A:1008608431399.  Google Scholar

[33]

V. S. Melnik and J. Valero, On global attractors of multivalued semiprocesses and nonautonomous evolution inclusions, Set-Valued Anal., 8 (2000), 375-403.  doi: 10.1023/A:1026514727329.  Google Scholar

[34]

J. SimsenM. S. Simsen and M. R. T. Primo, Reaction-diffusion equations with spatially variable exponents and large diffusion, Commun. Pure Appl. Anal., 15 (2016), 495-506.  doi: 10.3934/cpaa.2016.15.495.  Google Scholar

[35]

J. Simsen, Partial differential inclusions with spatially variable exponents and large diffusion, Mathematics in Enginnering, Science and Aerospace MESA, 7 (2016), 479–489. Google Scholar

[36]

J. Simsen, Weak upper semicontinuity of pullback attractors for nonautonomous reaction-diffusion equations, Electron. J. Qual. Theory Differ. Equ., 68 (2019), 1–14.  Google Scholar

[37]

J. Simsen and C. B. Gentile, Well-posed p-laplacian problems with large diffusion, Nonlinear Anal., 71 (2009), 4609-4617.  doi: 10.1016/j.na.2009.03.041.  Google Scholar

[38]

J. Simsen and C. B. Gentile, Systems of p-Laplacian differential inclusions with large diffusion, J. Math. Anal. Appl., 368 (2010), 525-537.  doi: 10.1016/j.jmaa.2010.02.006.  Google Scholar

[39]

J. Simsen, M. S. Simsen and F. B. Rocha, Existence of solutions for some classes of parabolic problems involving variable exponents, Nonlinear Stud., 21 (2014), 113–128.  Google Scholar

[40]

J. SimsenM. S. Simsen and A. Zimmermann, Study of ODE limit problems for reaction-diffusion equations, Opuscula Math., 38 (2018), 117-131.  doi: 10.7494/opmath.2018.38.1.117.  Google Scholar

[41]

J. Simsen and J. Valero, Characterization of pullback attractors for multivalued nonautonomous dynamical systems, in, Advances in Dynamical Systems and Control (V. A. Sadovnichiy and Z. Zgurovsky eds.), Studies in Systems, Decision and Control 659, Springer (2016), pp. 179–195.  Google Scholar

[42]

D. Terman, A free boundary problem arising from a bistable reaction-diffusion equation, SIAM J. Math. Anal., 14 (1983), 1107-1129.  doi: 10.1137/0514086.  Google Scholar

[43]

D. Terman, A free boundary arising from a model for nerve conduction, J. Differential Equations, 58 (1985), 345-363.  doi: 10.1016/0022-0396(85)90004-X.  Google Scholar

[44]

R. Willie, Large diffusivity stability of attractors in the $C$-topology for a semilinear reaction and diffusion system of equations, Dynamics of PDE, 3 (2006), 173-197.  doi: 10.4310/DPDE.2006.v3.n3.a1.  Google Scholar

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