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 and 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.

[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.

[3]

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

[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.

[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.,

[6]

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

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[27]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[35]

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

[36]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[3]

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

[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.

[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.,

[6]

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

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[27]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[35]

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

[36]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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