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doi: 10.3934/dcdss.2020341

Solution of contrast structure type for a reaction-diffusion equation with discontinuous reactive term

Xiao Wu 1, and  Mingkang Ni 2,,

1. 

School of Mathematical Sciences, East China Normal University, Shanghai 200000, P. R. China
2. 

School of Mathematical Sciences, East China Normal University, Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, Shanghai 200000, P. R. China

* Corresponding author: Mingkang Ni

Received  June 2019 Revised  October 2019 Published  April 2020

Fund Project: The second author is supported by the National Natural Science Foundation of China (No. 11871217) and the Science and Technology Commission of Shanghai Municipality (No. 18dz2271000)

In this paper, we consider the Dirichlet boundary value problem for a singularly perturbed reaction-diffusion equation with discontinuous reactive term. We use the asymptotic analysis to construct the formal asymptotic approximation of the solution with internal and boundary layers. The internal layer is located in the vicinity of a curve of the discontinuous reactive term. By using sufficiently precise lower and upper solutions, we prove the existence of a periodic solution and estimate the accuracy of its asymptotic approximation.

Keywords: discontinuous reactive term, periodic solution, asymptotic approximation, lower and upper solutions, small parameter.
Mathematics Subject Classification: Primary: 34D15, 34E15; Secondary: 34K13.
Citation: Xiao Wu, Mingkang Ni. Solution of contrast structure type for a reaction-diffusion equation with discontinuous reactive term. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020341
References:
[1]

F. M. Arscott, Periodic-parabolic boundary value problems and positivity, Bulletin of the London Mathematical Society, 24 (1991). doi: 10.1112/blms/24.6.619.  Google Scholar

[2]

C. De Coster, F. Obersnel and P. Omari, A qualitative analysis, via lower and upper solutions, of first order periodic evolutionary equations with lack of uniqueness, Handbook of Differential Equations: Ordinary Differential Equations, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 3 (2006), 203-339.  doi: 10.1016/S1874-5725(06)80007-6.  Google Scholar

[3]

Z. J. Du and Z. S. Feng, Existence and asymptotic behaviors of traveling waves of a modified vector-disease model, Commun. Pure Appl. Anal., 17 (2018), 1899-1920.  doi: 10.3934/cpaa.2018090.  Google Scholar

[4]

Z. J. Du, J. Li and X. W. Li, The existence of solitary wave solutions of delayed Camassa-Holm equation via a geometric approach, J. Funct. Anal., 275 (2018), 988-1007.  doi: 10.1016/j.jfa.2018.05.005.  Google Scholar

[5]

G. Hek, Geometric singular perturbation theory in biological pratice, J. Math. Biol., 60 (2010), 347-386.  doi: 10.1007/s00285-009-0266-7.  Google Scholar

[6]

N. T. Levashova, O. A. Nikolaeva and A. D. Pashkin, Simulation of the temperature distribution at the water-air interface using the theory of contrast structures, Vestnik Moskov. Univ. Ser. III Fiz. Astronom., (2015), 12–16.  Google Scholar

[7]

N. T. Levashova, Y. V. Mukhartova and W. A. Davydova, Application of the theory of constract structures to the description of the wind velocity field in the space-inhomogeneous vegetable cover, Moscow University Physics Bulletin, 3 (2015), 3-10.   Google Scholar

[8]

N. T. Levashova and O. A. Nikolaeva, The heat equation solution near the interface between two media, Lomonosov Moscow State University, 24 (2017), 339-352.  doi: 10.18255/1818-1015-2017-3-339-352.  Google Scholar

[9]

N. T. Levashova, N. N. Nefedov and A. O. Orlov, Time-independent reaction-diffusion equation with a discontinuous reactive term, Computational Mathematics and Mathematical Physics, 57 (2017), 854-866.  doi: 10.1134/S0965542517050062.  Google Scholar

[10]

N. T. Levashova, N. N. Nefedov, O. A. Nikolaeva, A. O. Orlov and A. A. Panin, The solution with internal transition layer of the reaction-diffusion equation in case of discontinuous reactive and diffusive terms, Mathematical Methods in the Applied Sciences, 41 (2018), 9203-9217.  doi: 10.1002/mma.5134.  Google Scholar

[11]

N. T. Levashova, N. N. Nefedov and A. O. Orlov, Asymptotic stability of a stationary solution of a multidimensional reaction-diffusion equation with a discontinuous source, Comput. Math. Math. Phys., 59 (2019), 573-582.  doi: 10.1134/S0965542519040109.  Google Scholar

[12]

E. A. Mikhailov, Wavefronts of the magnetic field in galaxies: Asymptotic and numerical approaches, Magnetohydrodynamics, 52 (2016), 117-125.   Google Scholar

[13]

N. N. Nefedov and M. K. Ni, Internal layers in the one-dimensional reaction-diffusion equation with a discontinuous reactive term, Computational Mathematics and Mathematical Physics, 55 (2015), 2001-2007.  doi: 10.1134/S096554251512012X.  Google Scholar

[14]

N. N. Nefedov, The method of differential inequalities for some singularly perturbed partial differential equations, Differential Equations, 31 (1995), 668-671.   Google Scholar

[15]

N. N. Nefedov and M. A. Davydova, Contrast structures in multidimensional singularly perturbed reaction-diffusion-advection problems, Differential Equations, 48 (2012), 745-755.  doi: 10.1134/S0012266112050138.  Google Scholar

[16]

N. N. Nefedov and M. A. Davydova, Contrast structures in singularly perturbed quasilinear reaction-diffusion-advection equations, Differential Equations, 49 (2013), 688-706.  doi: 10.1134/S0012266113060049.  Google Scholar

[17]

N. N. Nefedov, An asymptotic method of differential inequalities for the investigation of periodic contrast structures: Existence, asymptotics and stability, Differential Equations, 36 (2000), 298-305.  doi: 10.1007/BF02754216.  Google Scholar

[18]

A. Orlov, N. Levashova and T. Burbaev, The use of asymptotic methods for modelling of the carriers wave functions in the Si/SiGe heterostructures with quantum-confined layers, Journal of Physics: Conference Series, 586 (2014). doi: 10.1088/1742-6596/586/1/012003.  Google Scholar

[19]

A. O. Orlov, N. T. Levashova and N. N. Nefedov, Solution of contrast structure type for a parabolic reaction-diffusion problem in a medium with discontinuous characteristics, Differential Equations, 54 (2018), 669-686.  doi: 10.1134/S0012266118050105.  Google Scholar

[20]

C. V. Pao, Periodic solutions of parabolic systems with nonlinear boundary conditions, Journal of Mathematical Analysis and Applications, 234 (1999), 695-716.  doi: 10.1006/jmaa.1999.6412.  Google Scholar

[21] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.   Google Scholar
[22]

S. I. Pohozaev, On equations of the form $ \Delta u = f(x, u, Du)$, Mathematics of the USSR-Sbornik, 41 (1982), 269-280.  doi: 10.1070/SM1982v041n02ABEH002233.  Google Scholar

[23]

V. N. Pavlenko and O. V. Ul'Yanova, The method of upper and lower solutions for elliptic-type equations with discontinuous nonlinearities, Izv. Vyssh. Uchebn. Zaved. Mat., 42 (1998), 69-76.   Google Scholar

[24]

V. N. Pavlenko, Strong solutions of periodic parabolic problems with discontinuous nonlinearities, Differential Equations, 52 (2016), 505-516.  doi: 10.1134/S0012266116040108.  Google Scholar

[25]

Y. Pang, M. K. Ni, N. T. Levashova and O. A. Nikolaeva, Internal layers for a singualrly perturbed second-order quasilinear differential equation with discontinuous right-hand side, Differential Equations, 53 (2017), 1567-1577.  doi: 10.1134/S0012266117120059.  Google Scholar

[26]

Y. F. Pang, M. K. Ni and M. A. Davaydova, Contrast structures in problems for a stationary equation of reaction-diffusion-advection type with discontinuous nonlinearity, Mathematical Notes, 104 (2018), 735-744.  doi: 10.4213/mzm11699.  Google Scholar

[27]

Y. F. Pang, M. K. Ni and N. T. Levashova, Internal layer for a system of singularly perturbed equations with discontinuous right-hand side, Differential Equations, 54 (2018), 1583-1594.  doi: 10.1134/S0012266118120054.  Google Scholar

[28]

X. H. Shang and Z. J. Du, Existence of traveling waves in a generalized nonlinear dispersive-dissipative equation, Math. Methods Appl. Sci., 39 (2016), 3035-3042.  doi: 10.1002/mma.3750.  Google Scholar

[29]

A. B. Vasil'Yeva, Step-Like Contrasting Structures for a System of Singularly Perturbed Equations, 1994. Google Scholar

[30]

A. B. Vasil'Yeva and V. F. Butuzov, Asymptitic Expansions of Solutions to Singularly Perturbed Equations, 1973. Google Scholar

[31]

A. B. Vasil'eva, V. F. Butuzov and N. N. Nefedov, Singularly perturbed problems with boundary and internal layers, Proceedings of the Steklov Institute of Mathematics, 268 (2010), 258-273.  doi: 10.1134/S0081543810010189.  Google Scholar

[32]

V. T. Volkov and N. N. Nefëdov, Development of the asymptotic method of differential inequalities for investigation of periodic constrast structures in reaction-diffusion equations, Computational Mathematics and Mathematical Physics, 46 (2006), 585-593.  doi: 10.1134/s0965542506040075.  Google Scholar

[33]

V. T. Volkov and N. N. Nefëdov, Periodic solutions with boundary layers of a singularly perturbed reaction-diffusion model, Computational Mathematics and Mathematical Physics, 34 (1994), 1133-1140.   Google Scholar

[34]

C. Wang and X. Zhang, Stability loss delay and smoothness of the return map in slow-fast systems, SIAM J. Appl. Dyn. Syst., 17 (2018), 788-822.  doi: 10.1137/17M1130010.  Google Scholar

[35]

Y. Xu, Z. J. Du and L. Wei, Geometric singular perturbation method to the existence and asymptotic behavior of traveling waves for a generalized burgers-kdv equation, Nolinear Dynamics, 83 (2016), 65-73.  doi: 10.1007/s11071-015-2309-5.  Google Scholar

show all references

References:
[1]

F. M. Arscott, Periodic-parabolic boundary value problems and positivity, Bulletin of the London Mathematical Society, 24 (1991). doi: 10.1112/blms/24.6.619.  Google Scholar

[2]

C. De Coster, F. Obersnel and P. Omari, A qualitative analysis, via lower and upper solutions, of first order periodic evolutionary equations with lack of uniqueness, Handbook of Differential Equations: Ordinary Differential Equations, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 3 (2006), 203-339.  doi: 10.1016/S1874-5725(06)80007-6.  Google Scholar

[3]

Z. J. Du and Z. S. Feng, Existence and asymptotic behaviors of traveling waves of a modified vector-disease model, Commun. Pure Appl. Anal., 17 (2018), 1899-1920.  doi: 10.3934/cpaa.2018090.  Google Scholar

[4]

Z. J. Du, J. Li and X. W. Li, The existence of solitary wave solutions of delayed Camassa-Holm equation via a geometric approach, J. Funct. Anal., 275 (2018), 988-1007.  doi: 10.1016/j.jfa.2018.05.005.  Google Scholar

[5]

G. Hek, Geometric singular perturbation theory in biological pratice, J. Math. Biol., 60 (2010), 347-386.  doi: 10.1007/s00285-009-0266-7.  Google Scholar

[6]

N. T. Levashova, O. A. Nikolaeva and A. D. Pashkin, Simulation of the temperature distribution at the water-air interface using the theory of contrast structures, Vestnik Moskov. Univ. Ser. III Fiz. Astronom., (2015), 12–16.  Google Scholar

[7]

N. T. Levashova, Y. V. Mukhartova and W. A. Davydova, Application of the theory of constract structures to the description of the wind velocity field in the space-inhomogeneous vegetable cover, Moscow University Physics Bulletin, 3 (2015), 3-10.   Google Scholar

[8]

N. T. Levashova and O. A. Nikolaeva, The heat equation solution near the interface between two media, Lomonosov Moscow State University, 24 (2017), 339-352.  doi: 10.18255/1818-1015-2017-3-339-352.  Google Scholar

[9]

N. T. Levashova, N. N. Nefedov and A. O. Orlov, Time-independent reaction-diffusion equation with a discontinuous reactive term, Computational Mathematics and Mathematical Physics, 57 (2017), 854-866.  doi: 10.1134/S0965542517050062.  Google Scholar

[10]

N. T. Levashova, N. N. Nefedov, O. A. Nikolaeva, A. O. Orlov and A. A. Panin, The solution with internal transition layer of the reaction-diffusion equation in case of discontinuous reactive and diffusive terms, Mathematical Methods in the Applied Sciences, 41 (2018), 9203-9217.  doi: 10.1002/mma.5134.  Google Scholar

[11]

N. T. Levashova, N. N. Nefedov and A. O. Orlov, Asymptotic stability of a stationary solution of a multidimensional reaction-diffusion equation with a discontinuous source, Comput. Math. Math. Phys., 59 (2019), 573-582.  doi: 10.1134/S0965542519040109.  Google Scholar

[12]

E. A. Mikhailov, Wavefronts of the magnetic field in galaxies: Asymptotic and numerical approaches, Magnetohydrodynamics, 52 (2016), 117-125.   Google Scholar

[13]

N. N. Nefedov and M. K. Ni, Internal layers in the one-dimensional reaction-diffusion equation with a discontinuous reactive term, Computational Mathematics and Mathematical Physics, 55 (2015), 2001-2007.  doi: 10.1134/S096554251512012X.  Google Scholar

[14]

N. N. Nefedov, The method of differential inequalities for some singularly perturbed partial differential equations, Differential Equations, 31 (1995), 668-671.   Google Scholar

[15]

N. N. Nefedov and M. A. Davydova, Contrast structures in multidimensional singularly perturbed reaction-diffusion-advection problems, Differential Equations, 48 (2012), 745-755.  doi: 10.1134/S0012266112050138.  Google Scholar

[16]

N. N. Nefedov and M. A. Davydova, Contrast structures in singularly perturbed quasilinear reaction-diffusion-advection equations, Differential Equations, 49 (2013), 688-706.  doi: 10.1134/S0012266113060049.  Google Scholar

[17]

N. N. Nefedov, An asymptotic method of differential inequalities for the investigation of periodic contrast structures: Existence, asymptotics and stability, Differential Equations, 36 (2000), 298-305.  doi: 10.1007/BF02754216.  Google Scholar

[18]

A. Orlov, N. Levashova and T. Burbaev, The use of asymptotic methods for modelling of the carriers wave functions in the Si/SiGe heterostructures with quantum-confined layers, Journal of Physics: Conference Series, 586 (2014). doi: 10.1088/1742-6596/586/1/012003.  Google Scholar

[19]

A. O. Orlov, N. T. Levashova and N. N. Nefedov, Solution of contrast structure type for a parabolic reaction-diffusion problem in a medium with discontinuous characteristics, Differential Equations, 54 (2018), 669-686.  doi: 10.1134/S0012266118050105.  Google Scholar

[20]

C. V. Pao, Periodic solutions of parabolic systems with nonlinear boundary conditions, Journal of Mathematical Analysis and Applications, 234 (1999), 695-716.  doi: 10.1006/jmaa.1999.6412.  Google Scholar

[21] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.   Google Scholar
[22]

S. I. Pohozaev, On equations of the form $ \Delta u = f(x, u, Du)$, Mathematics of the USSR-Sbornik, 41 (1982), 269-280.  doi: 10.1070/SM1982v041n02ABEH002233.  Google Scholar

[23]

V. N. Pavlenko and O. V. Ul'Yanova, The method of upper and lower solutions for elliptic-type equations with discontinuous nonlinearities, Izv. Vyssh. Uchebn. Zaved. Mat., 42 (1998), 69-76.   Google Scholar

[24]

V. N. Pavlenko, Strong solutions of periodic parabolic problems with discontinuous nonlinearities, Differential Equations, 52 (2016), 505-516.  doi: 10.1134/S0012266116040108.  Google Scholar

[25]

Y. Pang, M. K. Ni, N. T. Levashova and O. A. Nikolaeva, Internal layers for a singualrly perturbed second-order quasilinear differential equation with discontinuous right-hand side, Differential Equations, 53 (2017), 1567-1577.  doi: 10.1134/S0012266117120059.  Google Scholar

[26]

Y. F. Pang, M. K. Ni and M. A. Davaydova, Contrast structures in problems for a stationary equation of reaction-diffusion-advection type with discontinuous nonlinearity, Mathematical Notes, 104 (2018), 735-744.  doi: 10.4213/mzm11699.  Google Scholar

[27]

Y. F. Pang, M. K. Ni and N. T. Levashova, Internal layer for a system of singularly perturbed equations with discontinuous right-hand side, Differential Equations, 54 (2018), 1583-1594.  doi: 10.1134/S0012266118120054.  Google Scholar

[28]

X. H. Shang and Z. J. Du, Existence of traveling waves in a generalized nonlinear dispersive-dissipative equation, Math. Methods Appl. Sci., 39 (2016), 3035-3042.  doi: 10.1002/mma.3750.  Google Scholar

[29]

A. B. Vasil'Yeva, Step-Like Contrasting Structures for a System of Singularly Perturbed Equations, 1994. Google Scholar

[30]

A. B. Vasil'Yeva and V. F. Butuzov, Asymptitic Expansions of Solutions to Singularly Perturbed Equations, 1973. Google Scholar

[31]

A. B. Vasil'eva, V. F. Butuzov and N. N. Nefedov, Singularly perturbed problems with boundary and internal layers, Proceedings of the Steklov Institute of Mathematics, 268 (2010), 258-273.  doi: 10.1134/S0081543810010189.  Google Scholar

[32]

V. T. Volkov and N. N. Nefëdov, Development of the asymptotic method of differential inequalities for investigation of periodic constrast structures in reaction-diffusion equations, Computational Mathematics and Mathematical Physics, 46 (2006), 585-593.  doi: 10.1134/s0965542506040075.  Google Scholar

[33]

V. T. Volkov and N. N. Nefëdov, Periodic solutions with boundary layers of a singularly perturbed reaction-diffusion model, Computational Mathematics and Mathematical Physics, 34 (1994), 1133-1140.   Google Scholar

[34]

C. Wang and X. Zhang, Stability loss delay and smoothness of the return map in slow-fast systems, SIAM J. Appl. Dyn. Syst., 17 (2018), 788-822.  doi: 10.1137/17M1130010.  Google Scholar

[35]

Y. Xu, Z. J. Du and L. Wei, Geometric singular perturbation method to the existence and asymptotic behavior of traveling waves for a generalized burgers-kdv equation, Nolinear Dynamics, 83 (2016), 65-73.  doi: 10.1007/s11071-015-2309-5.  Google Scholar

Figure 1.  the picture for the zero approximation $ U_{0}(x,t,\epsilon) $ for the solution $ u(x,t,\epsilon) $ of (50)
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