American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Synchronization for stochastic differential equations with nonlinear multiplicative noise in the mean square sense
  • DCDS-B Home
  • This Issue
  • Next Article
    Limiting behavior of trajectory attractors of perturbed reaction-diffusion equations
October  2019, 24(10): 5695-5707. doi: 10.3934/dcdsb.2019102

Existence and uniqueness of solutions to a family of semi-linear parabolic systems using coupled upper-lower solutions

Anne Mund 1, , Christina Kuttler 1, and  Judith Pérez-Velázquez 1,2,,

1. 

Zentrum Mathematik, Technische Universität Müenchen, Boltzmannstr. 3, 85748 Garching, Germany
2. 

Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas (IIMAS), Universidad Nacional Autónoma de México (UNAM), Circuito Escolar s/n, Ciudad Universitaria, 04510 Cd. de México

* Corresponding author: perez-velazquez@mym.iimas.unam.mx

Received  April 2018 Revised  November 2018 Published  June 2019

We prove existence and uniqueness of weak and classical solutions to certain semi-linear parabolic systems with Robin boundary conditions using the coupled upper-lower solution approach. Our interest lies in cross-dependencies on the gradient parts of the reaction term, which prevents the straight-forward application of standard theorems. Such cross-dependencies emerge e.g. in a model describing evolution of bacterial quorum sensing, but are interesting also in a more general context. We show the existence and uniqueness of solutions for this example.

Keywords: Parabolic system, coupled upper-lower solutions, existence, uniqueness, semi-linear equation.
Mathematics Subject Classification: Primary: 35K51, 35B51; Secondary: 35K58.
Citation: Anne Mund, Christina Kuttler, Judith Pérez-Velázquez. Existence and uniqueness of solutions to a family of semi-linear parabolic systems using coupled upper-lower solutions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5695-5707. doi: 10.3934/dcdsb.2019102
References:
[1]

M. Abudiab, I. Ahn and L. Li, Upper–lower solutions for nonlinear parabolic systems and their applications, Journal of Mathematical Analysis and Applications, 378 (2011), 620-633.  doi: 10.1016/j.jmaa.2011.01.003.  Google Scholar

[2]

D. Bothe, A. Fischer, M. Pierre and G. Rolland, Global well-posedness for a class of reaction–advection–anisotropic-diffusion systems, Journal of Evolution Equations, 17 (2017), 101-130.  doi: 10.1007/s00028-016-0348-0.  Google Scholar

[3]

K. J. Brown, P. C. Dunne and R. A. Gardner, A semilinear parabolic system arising in the theory of superconductivity, Journal of Differential Equations, 40 (1981), 232-252.  doi: 10.1016/0022-0396(81)90020-6.  Google Scholar

[4]

A. Friedmann, Partial Differential Equations of Parabolic Type, Prentice Hall, 1964.  Google Scholar

[5]

N. Kronik and Y. Cohen, Evolutionary games in space, Mathematical Modelling of Natural Phenomena, 4 (2009), 54-90.  doi: 10.1051/mmnp/20094602.  Google Scholar

[6]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and quasilinear equations of parabolic type (Russian), (Russian) Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R.I. 1968  Google Scholar

[7]

J. Morgan, Global existence for semilinear parabolic systems, SIAM Journal on Mathematical Analysis, 20 (1989), 1128-1144.  doi: 10.1137/0520075.  Google Scholar

[8]

I. G. Petrovskii, On the Cauchy problem for systems of linear partial differential equations in the domain of non-analytic functions, Bull. MGU, Sect. A, 1938. Google Scholar

[9]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, volume 258 of Comprehensive Studies in Mathematics, Springer Verlag, 1983.  Google Scholar

[10]

D. Werner, Funktionalanalysis, Third, revised and extended edition. Springer-Verlag, Berlin, 2000.  Google Scholar

show all references

References:
[1]

M. Abudiab, I. Ahn and L. Li, Upper–lower solutions for nonlinear parabolic systems and their applications, Journal of Mathematical Analysis and Applications, 378 (2011), 620-633.  doi: 10.1016/j.jmaa.2011.01.003.  Google Scholar

[2]

D. Bothe, A. Fischer, M. Pierre and G. Rolland, Global well-posedness for a class of reaction–advection–anisotropic-diffusion systems, Journal of Evolution Equations, 17 (2017), 101-130.  doi: 10.1007/s00028-016-0348-0.  Google Scholar

[3]

K. J. Brown, P. C. Dunne and R. A. Gardner, A semilinear parabolic system arising in the theory of superconductivity, Journal of Differential Equations, 40 (1981), 232-252.  doi: 10.1016/0022-0396(81)90020-6.  Google Scholar

[4]

A. Friedmann, Partial Differential Equations of Parabolic Type, Prentice Hall, 1964.  Google Scholar

[5]

N. Kronik and Y. Cohen, Evolutionary games in space, Mathematical Modelling of Natural Phenomena, 4 (2009), 54-90.  doi: 10.1051/mmnp/20094602.  Google Scholar

[6]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and quasilinear equations of parabolic type (Russian), (Russian) Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R.I. 1968  Google Scholar

[7]

J. Morgan, Global existence for semilinear parabolic systems, SIAM Journal on Mathematical Analysis, 20 (1989), 1128-1144.  doi: 10.1137/0520075.  Google Scholar

[8]

I. G. Petrovskii, On the Cauchy problem for systems of linear partial differential equations in the domain of non-analytic functions, Bull. MGU, Sect. A, 1938. Google Scholar

[9]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, volume 258 of Comprehensive Studies in Mathematics, Springer Verlag, 1983.  Google Scholar

[10]

D. Werner, Funktionalanalysis, Third, revised and extended edition. Springer-Verlag, Berlin, 2000.  Google Scholar

[1]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825
[2]

M. Grasselli, V. Pata. Asymptotic behavior of a parabolic-hyperbolic system. Communications on Pure & Applied Analysis, 2004, 3 (4) : 849-881. doi: 10.3934/cpaa.2004.3.849
[3]

Haiyan Wang. Existence and nonexistence of positive radial solutions for quasilinear systems. Conference Publications, 2009, 2009 (Special) : 810-817. doi: 10.3934/proc.2009.2009.810
[4]

Shu-Yu Hsu. Existence and properties of ancient solutions of the Yamabe flow. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 91-129. doi: 10.3934/dcds.2018005
[5]

Nizami A. Gasilov. Solving a system of linear differential equations with interval coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2739-2747. doi: 10.3934/dcdsb.2020203
[6]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437
[7]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399
[8]

Arunima Bhattacharya, Micah Warren. $ C^{2, \alpha} $ estimates for solutions to almost Linear elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021024
[9]

Lucas C. F. Ferreira, Jhean E. Pérez-López, Élder J. Villamizar-Roa. On the product in Besov-Lorentz-Morrey spaces and existence of solutions for the stationary Boussinesq equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2423-2439. doi: 10.3934/cpaa.2018115
[10]

Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267
[11]

Misha Bialy, Andrey E. Mironov. Rich quasi-linear system for integrable geodesic flows on 2-torus. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 81-90. doi: 10.3934/dcds.2011.29.81
[12]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309
[13]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258
[14]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1649-1672. doi: 10.3934/dcdss.2020448
[15]

Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1779-1799. doi: 10.3934/dcdss.2020454
[16]

Wenmin Gong, Guangcun Lu. On coupled Dirac systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4329-4346. doi: 10.3934/dcds.2017185
[17]

Ying Yang. Global classical solutions to two-dimensional chemotaxis-shallow water system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2625-2643. doi: 10.3934/dcdsb.2020198
[18]

Prasanta Kumar Barik, Ankik Kumar Giri, Rajesh Kumar. Mass-conserving weak solutions to the coagulation and collisional breakage equation with singular rates. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021009
[19]

Junichi Minagawa. On the uniqueness of Nash equilibrium in strategic-form games. Journal of Dynamics & Games, 2020, 7 (2) : 97-104. doi: 10.3934/jdg.2020006
[20]

Qian Liu. The lower bounds on the second-order nonlinearity of three classes of Boolean functions. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020136

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (129)
  • HTML views (253)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences