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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

Aníbal Rodríguez-Bernal 1, and  Alejandro Vidal-López 2,

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.
Keywords: asymptotic behavior., Singular initial data, global existence, critical exponents, strong solutions, reaction--diffusion.
Mathematics Subject Classification: Primary: 35A01, 35B33, 35D35, 35K57, 35K20; Secondary: 35B40, 35B4.
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 &amp; 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), 515-554. 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), 370-373. 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), 277-304. 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, New York, 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), 1995-2037. doi: 10.1142/S0218202509004029.  Google Scholar

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D. Henry, "Geometric Theory of Semilinear Parabolic Equations,'' Volume 840 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 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), 1445-1499. 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), 289-337. 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), 165-196. 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), 2983-3030. 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), 515-554. 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), 370-373. 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), 277-304. 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, New York, 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), 1995-2037. doi: 10.1142/S0218202509004029.  Google Scholar

[6]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,'' Volume 840 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 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), 1445-1499. 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), 289-337. 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), 165-196. 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), 2983-3030. doi: 10.1016/j.jde.2008.02.046.  Google Scholar

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