• Previous Article
    Infinitely many homoclinic solutions for damped vibration problems with subquadratic potentials
  • CPAA Home
  • This Issue
  • Next Article
    Boundedness of solutions for a class of impact oscillators with time-denpendent polynomial potentials
March  2014, 13(2): 635-644. doi: 10.3934/cpaa.2014.13.635

A note on the existence of global solutions for reaction-diffusion equations with almost-monotonic nonlinearities

1. 

Departamento de Matemática Aplicada, Universidad Complutense de Madrid, Madrid 28040

2. 

Mathematics Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom

Received  January 2013 Revised  September 2013 Published  October 2013

We show existence and uniqueness of global solutions for reaction-diffusion equations with almost-monotonic nonlinear terms in $L^q(\Omega)$ for each $1\leq q < \infty$. In particular, we do not assume restriction on the growth of the nonlinearites required by the standar local existence theory.
Citation: Aníbal Rodríguez-Bernal, Alejandro Vidal-López. A note on the existence of global solutions for reaction-diffusion equations with almost-monotonic nonlinearities. Communications on Pure & Applied Analysis, 2014, 13 (2) : 635-644. doi: 10.3934/cpaa.2014.13.635
References:
[1]

J. M. Arrieta, J. W. Cholewa, T. Dlotko and A. Rodríguez-Bernal, Asymptotic behavior and attractors for reaction diffusion equations in unbounded domains,, Nonlinear Anal., 56 (2004), 515. doi: 10.1016/j.na.2003.09.023. Google Scholar

[2]

J. M. Ball, Strongly continuous semigroups, weak solutions, and the variation of constants formula,, Proc. Amer. Math. Soc., 63 (1977), 370. doi: 10.2307/2041821. Google Scholar

[3]

H. Brezis and T. Cazenave, A nonlinear heat equation with singular initial data,, J. Anal. Math., 68 (1996), 277. doi: 10.1007/BF02790212. Google Scholar

[4]

A. Carvalho, J. A. Langa and J. Robinson., "Attractors for Infinite-dimensional Non-autonomous Dynamical Systems,'' volume 182 of Applied Mathematical Sciences,, Springer, (2012). doi: 10.1007/978-1-4614-4581-4_1. Google Scholar

[5]

J. W. Cholewa and A. Rodríguez-Bernal, Extremal equilibria for dissipative parabolic equations in locally uniform spaces,, Math. Models Methods Appl. Sci., 19 (2009), 1995. doi: 10.1142/S0218202509004029. Google Scholar

[6]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,'' Volume 840 of Lecture Notes in Mathematics,, Springer-Verlag, (1981). Google Scholar

[7]

M. Marcus and L. Véron, Initial trace of positive solutions of some nonlinear parabolic equations,, Comm. Partial Differential Equations, 24 (1999), 1445. doi: 10.1080/03605309908821471. Google Scholar

[8]

J. C. Robinson, A. Rodríguez-Bernal and A. Vidal-López, Pullback attractors and extremal complete trajectories for non-autonomous reaction-diffusion problems,, J. Differ. Equations, 238 (2007), 289. doi: 10.1016/j.jde.2007.03.028. Google Scholar

[9]

A. Rodríguez-Bernal, Attractors for parabolic equations with nonlinear boundary conditions, critical exponents, and singular initial data,, J. Differ. Equations, 181 (2002), 165. doi: 10.1006/jdeq.2001.4072. Google Scholar

[10]

A. Rodríguez-Bernal and A. Vidal-López, Extremal equilibria for reaction-diffusion equations in bounded domains and applications,, Journal of Differential Equations, 244 (2008), 2983. doi: 10.1016/j.jde.2008.02.046. Google Scholar

show all references

References:
[1]

J. M. Arrieta, J. W. Cholewa, T. Dlotko and A. Rodríguez-Bernal, Asymptotic behavior and attractors for reaction diffusion equations in unbounded domains,, Nonlinear Anal., 56 (2004), 515. doi: 10.1016/j.na.2003.09.023. Google Scholar

[2]

J. M. Ball, Strongly continuous semigroups, weak solutions, and the variation of constants formula,, Proc. Amer. Math. Soc., 63 (1977), 370. doi: 10.2307/2041821. Google Scholar

[3]

H. Brezis and T. Cazenave, A nonlinear heat equation with singular initial data,, J. Anal. Math., 68 (1996), 277. doi: 10.1007/BF02790212. Google Scholar

[4]

A. Carvalho, J. A. Langa and J. Robinson., "Attractors for Infinite-dimensional Non-autonomous Dynamical Systems,'' volume 182 of Applied Mathematical Sciences,, Springer, (2012). doi: 10.1007/978-1-4614-4581-4_1. Google Scholar

[5]

J. W. Cholewa and A. Rodríguez-Bernal, Extremal equilibria for dissipative parabolic equations in locally uniform spaces,, Math. Models Methods Appl. Sci., 19 (2009), 1995. doi: 10.1142/S0218202509004029. Google Scholar

[6]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,'' Volume 840 of Lecture Notes in Mathematics,, Springer-Verlag, (1981). Google Scholar

[7]

M. Marcus and L. Véron, Initial trace of positive solutions of some nonlinear parabolic equations,, Comm. Partial Differential Equations, 24 (1999), 1445. doi: 10.1080/03605309908821471. Google Scholar

[8]

J. C. Robinson, A. Rodríguez-Bernal and A. Vidal-López, Pullback attractors and extremal complete trajectories for non-autonomous reaction-diffusion problems,, J. Differ. Equations, 238 (2007), 289. doi: 10.1016/j.jde.2007.03.028. Google Scholar

[9]

A. Rodríguez-Bernal, Attractors for parabolic equations with nonlinear boundary conditions, critical exponents, and singular initial data,, J. Differ. Equations, 181 (2002), 165. doi: 10.1006/jdeq.2001.4072. Google Scholar

[10]

A. Rodríguez-Bernal and A. Vidal-López, Extremal equilibria for reaction-diffusion equations in bounded domains and applications,, Journal of Differential Equations, 244 (2008), 2983. doi: 10.1016/j.jde.2008.02.046. Google Scholar

[1]

Honglv Ma, Jin Zhang, Chengkui Zhong. Global existence and asymptotic behavior of global smooth solutions to the Kirchhoff equations with strong nonlinear damping. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4721-4737. doi: 10.3934/dcdsb.2019027

[2]

Kazuo Yamazaki, Xueying Wang. Global well-posedness and asymptotic behavior of solutions to a reaction-convection-diffusion cholera epidemic model. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1297-1316. doi: 10.3934/dcdsb.2016.21.1297

[3]

Guangyu Xu, Jun Zhou. Global existence and blow-up of solutions to a singular Non-Newton polytropic filtration equation with critical and supercritical initial energy. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1805-1820. doi: 10.3934/cpaa.2018086

[4]

Zhoude Shao. Existence and continuity of strong solutions of partly dissipative reaction diffusion systems. Conference Publications, 2011, 2011 (Special) : 1319-1328. doi: 10.3934/proc.2011.2011.1319

[5]

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

[6]

Jishan Fan, Shuxiang Huang, Fucai Li. Global strong solutions to the planar compressible magnetohydrodynamic equations with large initial data and vacuum. Kinetic & Related Models, 2017, 10 (4) : 1035-1053. doi: 10.3934/krm.2017041

[7]

Kosuke Ono. Global existence and asymptotic behavior of small solutions for semilinear dissipative wave equations. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 651-662. doi: 10.3934/dcds.2003.9.651

[8]

P. Blue, J. Colliander. Global well-posedness in Sobolev space implies global existence for weighted $L^2$ initial data for $L^2$-critical NLS. Communications on Pure & Applied Analysis, 2006, 5 (4) : 691-708. doi: 10.3934/cpaa.2006.5.691

[9]

Bingkang Huang, Lan Zhang. A global existence of classical solutions to the two-dimensional Vlasov-Fokker-Planck and magnetohydrodynamics equations with large initial data. Kinetic & Related Models, 2019, 12 (2) : 357-396. doi: 10.3934/krm.2019016

[10]

Michele Campiti, Giovanni P. Galdi, Matthias Hieber. Global existence of strong solutions for $2$-dimensional Navier-Stokes equations on exterior domains with growing data at infinity. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1613-1627. doi: 10.3934/cpaa.2014.13.1613

[11]

Vo Van Au, Mokhtar Kirane, Nguyen Huy Tuan. Determination of initial data for a reaction-diffusion system with variable coefficients. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 771-801. doi: 10.3934/dcds.2019032

[12]

Joana Terra, Noemi Wolanski. Large time behavior for a nonlocal diffusion equation with absorption and bounded initial data. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 581-605. doi: 10.3934/dcds.2011.31.581

[13]

Zongming Guo, Juncheng Wei. Asymptotic behavior of touch-down solutions and global bifurcations for an elliptic problem with a singular nonlinearity. Communications on Pure & Applied Analysis, 2008, 7 (4) : 765-786. doi: 10.3934/cpaa.2008.7.765

[14]

Shota Sato, Eiji Yanagida. Asymptotic behavior of singular solutions for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 4027-4043. doi: 10.3934/dcds.2012.32.4027

[15]

Marco Di Francesco, Alexander Lorz, Peter A. Markowich. Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1437-1453. doi: 10.3934/dcds.2010.28.1437

[16]

Congming Li, Eric S. Wright. Global existence of solutions to a reaction diffusion system based upon carbonate reaction kinetics. Communications on Pure & Applied Analysis, 2002, 1 (1) : 77-84. doi: 10.3934/cpaa.2002.1.77

[17]

Huajun Gong, Jinkai Li. Global existence of strong solutions to incompressible MHD. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1553-1561. doi: 10.3934/cpaa.2014.13.1553

[18]

Huajun Gong, Jinkai Li. Global existence of strong solutions to incompressible MHD. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1337-1345. doi: 10.3934/cpaa.2014.13.1337

[19]

Angelo Favini, Atsushi Yagi. Global existence for Laplace reaction-diffusion equations. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-21. doi: 10.3934/dcdss.2020083

[20]

Young-Pil Choi, Seung-Yeal Ha, Seok-Bae Yun. Global existence and asymptotic behavior of measure valued solutions to the kinetic Kuramoto--Daido model with inertia. Networks & Heterogeneous Media, 2013, 8 (4) : 943-968. doi: 10.3934/nhm.2013.8.943

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (1)

[Back to Top]