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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

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
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