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November  2012, 32(11): 4001-4014. doi: 10.3934/dcds.2012.32.4001

Blow-up phenomena in reaction-diffusion systems

Monica Marras 1, and  Stella Vernier Piro 1,

1. 

Dipartimento di Matematica e Informatica, Università di Cagliari, 09123, Italy, Italy

Received  December 2010 Revised  September 2011 Published  June 2012

In this paper we deal with the blow-up phenomena of solutions to two different classes of reaction-diffusion systems coupled through nonlinearities with nonlinear boundary conditions. By using a differential inequality technique, we derive upper and lower bounds for the blow-up time, if blow-up occurs. Moreover by introducing suitable auxiliary functions, we give sufficient conditions on data in order to obtain global existence.
Keywords: Parabolic problems, global existence., blow-up.
Mathematics Subject Classification: Primary: 35K55, 35K60; Secondary: 35K4.
Citation: Monica Marras, Stella Vernier Piro. Blow-up phenomena in reaction-diffusion systems. Discrete & Continuous Dynamical Systems, 2012, 32 (11) : 4001-4014. doi: 10.3934/dcds.2012.32.4001
References:
[1]

J. M. Ball, Remarks on blow-up and nonexistence theorems for nonlinearevolution equations, Quart. J. Math. Oxford, 28 (1977), 473-486. doi: 10.1093/qmath/28.4.473.  Google Scholar

[2]

C. Bandle and H. Brunner, Blow-up in diffusion equations, A survey, J. Comput. Appl. Math., 97 (1998), 3-22.  Google Scholar

[3]

M. Chipot, M. Fila and P. Quittner, Stationary solutions, blow-up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions, Acta Math.Univ. Comenian. (N.S.), LX (1991), 35103.  Google Scholar

[4]

A. A. Lacey, Diffusion models with blow-up, J.Comput. Appl. Math., 97 (1998), 39-49. doi: 10.1016/S0377-0427(98)00105-8.  Google Scholar

[5]

J. López-Gómez, V. Márquez and N. Wolanski, Blow up results and localization of blow up points for the heat equation with a nonlinear boundary condition, J. Diff. Equ., 92 (1991), 384-401. doi: 10.1016/0022-0396(91)90056-F.  Google Scholar

[6]

J. López-Gómez, V. Márquez and N. Wolanski, "Global Behaviour of Positive Solutions to a Semilinear Equation with a Nonlinear Flux Condition," IMA Preprint Series, 810 , University of Minnesota, May 1991. Google Scholar

[7]

H. Kielhöfer, Halbgruppen und semilineare Anfangs-randwert-probleme, Manuscripta Math. 12 (1974), 121-152.  Google Scholar

[8]

M. Marras, Bounds for blow-up time in nonlinear parabolic systems under various boundary conditions, Numer. Funct. Anal. Optim., (2010), 32 (2011), 453-468.  Google Scholar

[9]

L. E. Payne, G. A. Philippin and S. Vernier-Piro, Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition I, Z. Angew. Math. Phys., 61 (2010), 971-978. doi: 10.1007/s00033-010-0071-6.  Google Scholar

[10]

L. E. Payne, G. A. Philippin and S. Vernier-Piro, Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition II, Nonlinear Analysis, 73 (2010), 971-978. doi: 10.1016/j.na.2010.04.023.  Google Scholar

[11]

L. E. Payne and P. W. Schaefer, Blow-up phenomena for some nonlinear parabolic systems, Int. J. Pure Appl. Math., 48 (2008), 193-202.  Google Scholar

[12]

G. A. Philippin and V. Proytcheva, Some remarks on the asymptotic behaviour of the solutions of a class of parabolic problems, Math.Meth. Appl. Sci., 29 (2006), 297-307. doi: 10.1002/mma.679.  Google Scholar

[13]

P. Quittner, On global existence and stationary solutions of two classes of semilinear parabolic equations, Comm.Math. Univ.Carolinae, 34 (1993), 105-124.  Google Scholar

[14]

P. Quittner and P. Souplet, "Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States," Birkhäuser Advanced Texts, Basel, 2007.  Google Scholar

[15]

B. Straughan, "Explosive Instabilities in Mechanics," Springer, Berlin, 1998.  Google Scholar

[16]

J. L. Vázquez, The problems of blow-up for nonlinear heat equations. Complete blow-up and avalanche formation, Rend. Mat. Acc. Lincei s. IX, 15 (2004), 281-300.  Google Scholar

[17]

F. B. Weissler, Local existence and nonexistence for semilinear parabolic equations in $L^p$, Indiana Univ. Math. J., 29 (1980), 79-102. doi: 10.1512/iumj.1980.29.29007.  Google Scholar

[18]

F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math., 38 (1981), 29-40. doi: 10.1007/BF02761845.  Google Scholar

show all references

References:
[1]

J. M. Ball, Remarks on blow-up and nonexistence theorems for nonlinearevolution equations, Quart. J. Math. Oxford, 28 (1977), 473-486. doi: 10.1093/qmath/28.4.473.  Google Scholar

[2]

C. Bandle and H. Brunner, Blow-up in diffusion equations, A survey, J. Comput. Appl. Math., 97 (1998), 3-22.  Google Scholar

[3]

M. Chipot, M. Fila and P. Quittner, Stationary solutions, blow-up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions, Acta Math.Univ. Comenian. (N.S.), LX (1991), 35103.  Google Scholar

[4]

A. A. Lacey, Diffusion models with blow-up, J.Comput. Appl. Math., 97 (1998), 39-49. doi: 10.1016/S0377-0427(98)00105-8.  Google Scholar

[5]

J. López-Gómez, V. Márquez and N. Wolanski, Blow up results and localization of blow up points for the heat equation with a nonlinear boundary condition, J. Diff. Equ., 92 (1991), 384-401. doi: 10.1016/0022-0396(91)90056-F.  Google Scholar

[6]

J. López-Gómez, V. Márquez and N. Wolanski, "Global Behaviour of Positive Solutions to a Semilinear Equation with a Nonlinear Flux Condition," IMA Preprint Series, 810 , University of Minnesota, May 1991. Google Scholar

[7]

H. Kielhöfer, Halbgruppen und semilineare Anfangs-randwert-probleme, Manuscripta Math. 12 (1974), 121-152.  Google Scholar

[8]

M. Marras, Bounds for blow-up time in nonlinear parabolic systems under various boundary conditions, Numer. Funct. Anal. Optim., (2010), 32 (2011), 453-468.  Google Scholar

[9]

L. E. Payne, G. A. Philippin and S. Vernier-Piro, Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition I, Z. Angew. Math. Phys., 61 (2010), 971-978. doi: 10.1007/s00033-010-0071-6.  Google Scholar

[10]

L. E. Payne, G. A. Philippin and S. Vernier-Piro, Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition II, Nonlinear Analysis, 73 (2010), 971-978. doi: 10.1016/j.na.2010.04.023.  Google Scholar

[11]

L. E. Payne and P. W. Schaefer, Blow-up phenomena for some nonlinear parabolic systems, Int. J. Pure Appl. Math., 48 (2008), 193-202.  Google Scholar

[12]

G. A. Philippin and V. Proytcheva, Some remarks on the asymptotic behaviour of the solutions of a class of parabolic problems, Math.Meth. Appl. Sci., 29 (2006), 297-307. doi: 10.1002/mma.679.  Google Scholar

[13]

P. Quittner, On global existence and stationary solutions of two classes of semilinear parabolic equations, Comm.Math. Univ.Carolinae, 34 (1993), 105-124.  Google Scholar

[14]

P. Quittner and P. Souplet, "Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States," Birkhäuser Advanced Texts, Basel, 2007.  Google Scholar

[15]

B. Straughan, "Explosive Instabilities in Mechanics," Springer, Berlin, 1998.  Google Scholar

[16]

J. L. Vázquez, The problems of blow-up for nonlinear heat equations. Complete blow-up and avalanche formation, Rend. Mat. Acc. Lincei s. IX, 15 (2004), 281-300.  Google Scholar

[17]

F. B. Weissler, Local existence and nonexistence for semilinear parabolic equations in $L^p$, Indiana Univ. Math. J., 29 (1980), 79-102. doi: 10.1512/iumj.1980.29.29007.  Google Scholar

[18]

F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math., 38 (1981), 29-40. doi: 10.1007/BF02761845.  Google Scholar

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