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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. [2] T. Aiki and E. Minchev, A prey-predator model with hysteresis effect, SIAM Journal on Mathematical Analysis, 36 (2005), 2020-2032. [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. [4] H. Brézis, Opérateurs Maximaux Monotone et Semi-Groupes de Contractions dans les Espaces Hilbert, North-Holland Math. Studies 5,1973. [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. [6] P. Colli, N. Kenmochi and M. Kubo, A phase field model with temperature dependent constraint, J. Math. Anal. Appl., 256 (2001), 668-685. [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. [8] N. Kenmochi, T. Koyama and G. H. Meyer, Parabolic PDEs with hysteresis and quasivariational inequalities, Nonlinear Analysis, 34 (1998), 665-686. [9] N. Kenmochi, E. Minchev and T. Okazaki, Ordinary differential systems describing hysteresis effects and numerical simulations, Abstr. Appl. Anal., 7 (2002), 563-583. [10] J.-P. Kernevez, G. Joly, M.-C. Duban, B. Bunow and D. Thomas, Hysteresis, oscillations, and pattern formation in realistic immobilized enzyme systems, J. Math. Biol., 7 (1979), 41-56. [11] M. A. Krasnosel'skii and A. V. Pokrovskii, Systems with Hysteresis, Springer, Heidelberg, 1989. [12] M. Kubo, A filtration model with hysteresis, J. Differential Equations, 201 (2004), 75-98. [13] M. Landau, P. Lorente, J. 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. [14] J. W. Macki, P. Nistri and P. Zecca, Mathematical models for hysteresis, SIAM Review, 35 (1993), 94-123. [15] I. D. Mayergoyz, Mathematical Models for Hysteresis, Springer, New York, 1991. [16] E. Minchev, A diffusion-convection prey-predator model with hysteresis, Math. J. Toyama Univ., 27 (2004), 51-69. [17] U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Advances in Math., 3 (1969), 510-585. [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. [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. [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. [21] M. Ôtani, Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators, Cauchy problems, Journal of Differential Equations, 46 (1982), 268-299. [22] M. Ôtani and Y. Sugiyama, Lipschitz continuous solutions of some doubly nonlinear parabolic equations, Discrete and Continuous Dynamical Systems, 8 (2002), 647-670. [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. [24] A. Visintin, Differential Models of Hysteresis, Springer, Berlin, 1994.

show all references

##### References:
 [1] T. Aiki, One-dimensional shape memory alloy problems, Funkcial. Ekvac., 46 (2003), 441-469. [2] T. Aiki and E. Minchev, A prey-predator model with hysteresis effect, SIAM Journal on Mathematical Analysis, 36 (2005), 2020-2032. [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. [4] H. Brézis, Opérateurs Maximaux Monotone et Semi-Groupes de Contractions dans les Espaces Hilbert, North-Holland Math. Studies 5,1973. [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. [6] P. Colli, N. Kenmochi and M. Kubo, A phase field model with temperature dependent constraint, J. Math. Anal. Appl., 256 (2001), 668-685. [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. [8] N. Kenmochi, T. Koyama and G. H. Meyer, Parabolic PDEs with hysteresis and quasivariational inequalities, Nonlinear Analysis, 34 (1998), 665-686. [9] N. Kenmochi, E. Minchev and T. Okazaki, Ordinary differential systems describing hysteresis effects and numerical simulations, Abstr. Appl. Anal., 7 (2002), 563-583. [10] J.-P. Kernevez, G. Joly, M.-C. Duban, B. Bunow and D. Thomas, Hysteresis, oscillations, and pattern formation in realistic immobilized enzyme systems, J. Math. Biol., 7 (1979), 41-56. [11] M. A. Krasnosel'skii and A. V. Pokrovskii, Systems with Hysteresis, Springer, Heidelberg, 1989. [12] M. Kubo, A filtration model with hysteresis, J. Differential Equations, 201 (2004), 75-98. [13] M. Landau, P. Lorente, J. 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. [14] J. W. Macki, P. Nistri and P. Zecca, Mathematical models for hysteresis, SIAM Review, 35 (1993), 94-123. [15] I. D. Mayergoyz, Mathematical Models for Hysteresis, Springer, New York, 1991. [16] E. Minchev, A diffusion-convection prey-predator model with hysteresis, Math. J. Toyama Univ., 27 (2004), 51-69. [17] U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Advances in Math., 3 (1969), 510-585. [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. [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. [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. [21] M. Ôtani, Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators, Cauchy problems, Journal of Differential Equations, 46 (1982), 268-299. [22] M. Ôtani and Y. Sugiyama, Lipschitz continuous solutions of some doubly nonlinear parabolic equations, Discrete and Continuous Dynamical Systems, 8 (2002), 647-670. [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. [24] A. Visintin, Differential Models of Hysteresis, Springer, Berlin, 1994.
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