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July  2018, 17(4): 1613-1632. doi: 10.3934/cpaa.2018077

$L^∞$-energy method for a parabolic system with convection and hysteresis effect

1. 

Hikari Ltd, P.O. Box 85 Ruse 7000, Bulgaria

2. 

Department of Applied Physics, School of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan

* Corresponding author

Received  March 2017 Revised  October 2017 Published  April 2018

Fund Project: The second author is supported by the Grant-in-Aid for Scientific Research #15K13451, the Ministry of Education, Culture, Sports, Science, and Technology, Japan.

The $L^∞$-energy method is developed so as to handle nonlinear parabolic systems with convection and hysteresis effect. The system under consideration originates from a biological model where the hysteresis and convective effects are taken into account in the evolution of species. Some results for the existence of local and global solutions as well as the uniqueness of solution are presented.

Citation: Emil Minchev, Mitsuharu Ôtani. $L^∞$-energy method for a parabolic system with convection and hysteresis effect. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1613-1632. doi: 10.3934/cpaa.2018077
References:
[1]

T. Aiki, One-dimensional shape memory alloy problems, Funkcial. Ekvac., 46 (2003), 441-469.   Google Scholar

[2]

T. Aiki and E. Minchev, A prey-predator model with hysteresis effect, SIAM Journal on Mathematical Analysis, 36 (2005), 2020-2032.   Google Scholar

[3]

H. Brézis, Monotonicity methods in Hilbert spaces and some applications to non-linear partial differential equations, Contributions to Nonlinear Functional Analysis (ed. E. Zarantonello), Academic Press, New York/London, (1971), 101-156. Google Scholar

[4]

H. Brézis, Opérateurs Maximaux Monotone et Semi-Groupes de Contractions dans les Espaces Hilbert, North-Holland Math. Studies 5,1973. Google Scholar

[5]

P. Colli and K. H. Hoffmann, A nonlinear evolution problem describing multi-component phase changes with dissipation, Numer. Funct. Anal. Optim., 14 (1993), 275-297.   Google Scholar

[6]

P. ColliN. Kenmochi and M. Kubo, A phase field model with temperature dependent constraint, J. Math. Anal. Appl., 256 (2001), 668-685.   Google Scholar

[7]

F. C. Hoppensteadt, W. Jäger and C. Pöppe, A hysteresis model for bacterial growth patterns, in Modelling of Patterns in Space and Time (W. Jäger and J. D. Murray eds.), Lecture Notes in Biomath., 55, Springer-Verlag, Berlin, (1984), 123-134. Google Scholar

[8]

N. KenmochiT. Koyama and G. H. Meyer, Parabolic PDEs with hysteresis and quasivariational inequalities, Nonlinear Analysis, 34 (1998), 665-686.   Google Scholar

[9]

N. KenmochiE. Minchev and T. Okazaki, Ordinary differential systems describing hysteresis effects and numerical simulations, Abstr. Appl. Anal., 7 (2002), 563-583.   Google Scholar

[10]

J.-P. KernevezG. JolyM.-C. DubanB. Bunow and D. Thomas, Hysteresis, oscillations, and pattern formation in realistic immobilized enzyme systems, J. Math. Biol., 7 (1979), 41-56.   Google Scholar

[11]

M. A. Krasnosel'skii and A. V. Pokrovskii, Systems with Hysteresis, Springer, Heidelberg, 1989. Google Scholar

[12]

M. Kubo, A filtration model with hysteresis, J. Differential Equations, 201 (2004), 75-98.   Google Scholar

[13]

M. LandauP. LorenteJ. Henry and S. Canu, Hysteresis phenomena between periodic and stationary solutions in a model of pacemaker and nonpacemaker coupled cardiac cells, J. Math. Biol., 25 (1987), 491-509.   Google Scholar

[14]

J. W. MackiP. Nistri and P. Zecca, Mathematical models for hysteresis, SIAM Review, 35 (1993), 94-123.   Google Scholar

[15]

I. D. Mayergoyz, Mathematical Models for Hysteresis, Springer, New York, 1991. Google Scholar

[16]

E. Minchev, A diffusion-convection prey-predator model with hysteresis, Math. J. Toyama Univ., 27 (2004), 51-69.   Google Scholar

[17]

U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Advances in Math., 3 (1969), 510-585.   Google Scholar

[18]

M. Ôtani, L-energy method and its applications, Mathematical Analysis and Applications, Gakuto International Series, Nonlinear Partial Differential Equations and Their Applications, 20 (2004), 506-516. Google Scholar

[19]

M. Ôtani, L-energy method and its applications to some nonlinear parabolic systems, Mathematical Analysis and Applications, Gakuto International Series, Mathematical Sciences and Applications, Gakkotosho, Tokyo, 22 (2005), 233-244. Google Scholar

[20]

M. Ôtani, L-energy method, basic tools and usage, Differential Equations, Chaos and Variational Problems, Progress in Nonlinear Differential Equations and Their Applications, 75 (V. Staicu ed.), Birkhauser, (2007), 357-376. Google Scholar

[21]

M. Ôtani, Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators, Cauchy problems, Journal of Differential Equations, 46 (1982), 268-299.   Google Scholar

[22]

M. Ôtani and Y. Sugiyama, Lipschitz continuous solutions of some doubly nonlinear parabolic equations, Discrete and Continuous Dynamical Systems, 8 (2002), 647-670.   Google Scholar

[23]

M. Ôtani and Y. Sugiyama, A method of energy estimates in L and its applications to porous medium equations, Journal of Mathematical Society of Japan, 53 (2001), 746-789.   Google Scholar

[24]

A. Visintin, Differential Models of Hysteresis, Springer, Berlin, 1994. Google Scholar

show all references

References:
[1]

T. Aiki, One-dimensional shape memory alloy problems, Funkcial. Ekvac., 46 (2003), 441-469.   Google Scholar

[2]

T. Aiki and E. Minchev, A prey-predator model with hysteresis effect, SIAM Journal on Mathematical Analysis, 36 (2005), 2020-2032.   Google Scholar

[3]

H. Brézis, Monotonicity methods in Hilbert spaces and some applications to non-linear partial differential equations, Contributions to Nonlinear Functional Analysis (ed. E. Zarantonello), Academic Press, New York/London, (1971), 101-156. Google Scholar

[4]

H. Brézis, Opérateurs Maximaux Monotone et Semi-Groupes de Contractions dans les Espaces Hilbert, North-Holland Math. Studies 5,1973. Google Scholar

[5]

P. Colli and K. H. Hoffmann, A nonlinear evolution problem describing multi-component phase changes with dissipation, Numer. Funct. Anal. Optim., 14 (1993), 275-297.   Google Scholar

[6]

P. ColliN. Kenmochi and M. Kubo, A phase field model with temperature dependent constraint, J. Math. Anal. Appl., 256 (2001), 668-685.   Google Scholar

[7]

F. C. Hoppensteadt, W. Jäger and C. Pöppe, A hysteresis model for bacterial growth patterns, in Modelling of Patterns in Space and Time (W. Jäger and J. D. Murray eds.), Lecture Notes in Biomath., 55, Springer-Verlag, Berlin, (1984), 123-134. Google Scholar

[8]

N. KenmochiT. Koyama and G. H. Meyer, Parabolic PDEs with hysteresis and quasivariational inequalities, Nonlinear Analysis, 34 (1998), 665-686.   Google Scholar

[9]

N. KenmochiE. Minchev and T. Okazaki, Ordinary differential systems describing hysteresis effects and numerical simulations, Abstr. Appl. Anal., 7 (2002), 563-583.   Google Scholar

[10]

J.-P. KernevezG. JolyM.-C. DubanB. Bunow and D. Thomas, Hysteresis, oscillations, and pattern formation in realistic immobilized enzyme systems, J. Math. Biol., 7 (1979), 41-56.   Google Scholar

[11]

M. A. Krasnosel'skii and A. V. Pokrovskii, Systems with Hysteresis, Springer, Heidelberg, 1989. Google Scholar

[12]

M. Kubo, A filtration model with hysteresis, J. Differential Equations, 201 (2004), 75-98.   Google Scholar

[13]

M. LandauP. LorenteJ. Henry and S. Canu, Hysteresis phenomena between periodic and stationary solutions in a model of pacemaker and nonpacemaker coupled cardiac cells, J. Math. Biol., 25 (1987), 491-509.   Google Scholar

[14]

J. W. MackiP. Nistri and P. Zecca, Mathematical models for hysteresis, SIAM Review, 35 (1993), 94-123.   Google Scholar

[15]

I. D. Mayergoyz, Mathematical Models for Hysteresis, Springer, New York, 1991. Google Scholar

[16]

E. Minchev, A diffusion-convection prey-predator model with hysteresis, Math. J. Toyama Univ., 27 (2004), 51-69.   Google Scholar

[17]

U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Advances in Math., 3 (1969), 510-585.   Google Scholar

[18]

M. Ôtani, L-energy method and its applications, Mathematical Analysis and Applications, Gakuto International Series, Nonlinear Partial Differential Equations and Their Applications, 20 (2004), 506-516. Google Scholar

[19]

M. Ôtani, L-energy method and its applications to some nonlinear parabolic systems, Mathematical Analysis and Applications, Gakuto International Series, Mathematical Sciences and Applications, Gakkotosho, Tokyo, 22 (2005), 233-244. Google Scholar

[20]

M. Ôtani, L-energy method, basic tools and usage, Differential Equations, Chaos and Variational Problems, Progress in Nonlinear Differential Equations and Their Applications, 75 (V. Staicu ed.), Birkhauser, (2007), 357-376. Google Scholar

[21]

M. Ôtani, Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators, Cauchy problems, Journal of Differential Equations, 46 (1982), 268-299.   Google Scholar

[22]

M. Ôtani and Y. Sugiyama, Lipschitz continuous solutions of some doubly nonlinear parabolic equations, Discrete and Continuous Dynamical Systems, 8 (2002), 647-670.   Google Scholar

[23]

M. Ôtani and Y. Sugiyama, A method of energy estimates in L and its applications to porous medium equations, Journal of Mathematical Society of Japan, 53 (2001), 746-789.   Google Scholar

[24]

A. Visintin, Differential Models of Hysteresis, Springer, Berlin, 1994. Google Scholar

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