January  2018, 38(1): 209-230. doi: 10.3934/dcds.2018010

Single-point blow-up for a multi-component reaction-diffusion system

Université de Tunis El Manar, Faculté des Sciences de Tunis, Département de Mathématiques, Laboratoire Équations aux Dérivées Partielles LR03ES04, Tunis, 2092, Tunisie

Received  December 2016 Revised  July 2017 Published  September 2017

In this work, we prove single-point blow-up for any positive, radially decreasing, classical and blowing-up solution of a system of $m≥q3$ heat equations in a ball of $\mathbb{R}^n$, which are coupled cyclically by superlinear monomial reaction terms. We also obtain lower pointwise estimates for the blow-up profiles.

Citation: Nejib Mahmoudi. Single-point blow-up for a multi-component reaction-diffusion system. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 209-230. doi: 10.3934/dcds.2018010
References:
[1]

X. Y. Chen and H. Matano, Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations, J. Differ. Equations, 78 (1989), 160-190.  doi: 10.1016/0022-0396(89)90081-8.

[2]

F. Fila and P. Quittner, The blow-up rate for a semilinear parabolic system, J. Math. Anal. Appl., 238 (1999), 468-476.  doi: 10.1006/jmaa.1999.6525.

[3]

A. Friedman and Y. Giga, A single point blow-up for solutions of semilinear parabolic systems, J. Fac. Sci. Univ. Tokyo Sec. IA. Math., 34 (1987), 65-79. 

[4]

A. Friedman and B. Mcleod, Blow-up of positive solution of semilinear heat equations, Indiana Univ. Math. J., 34 (1985), 425-447.  doi: 10.1512/iumj.1985.34.34025.

[5]

Y. Giga and R. V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math., 38 (1985), 297-319.  doi: 10.1002/cpa.3160380304.

[6]

M. A. Herrero and J. J. A. Velázquez, A blow up result for semilinear heat equations in the supercritical case, Preprint.

[7]

N. Mahmoudi, Single-point blow-up for a semilinear reaction-diffusion system, Differ. Equ. Appl., 6 (2014), 563-591.  doi: 10.7153/dea-06-33.

[8]

N. MahmoudiPh. Souplet and S. Tayachi, Improved conditions for single-point blow-up in reaction-diffusion systems, J. Differ. Equations, 259 (2015), 1898-1932.  doi: 10.1016/j.jde.2015.03.024.

[9]

H. Matano and F. Merle, On nonexistence of type Ⅱ blowup for a supercritical nonlinear heat equation, Comm. Pure Appl. Math., 57 (2004), 1494-1541.  doi: 10.1002/cpa.20044.

[10]

C. E. Mueller and F. B. Weissler, Single point blow-up for a general semilinear heat equation, Indiana Univ. Math. J., 34 (1985), 881-913.  doi: 10.1512/iumj.1985.34.34049.

[11]

P. Quittner and Ph. Souplet, Superlinear Parabolic Problems Blow-Up, Global Existence and Steady States, Birkhäuser Verlag AG, Basel Boston Berlin, 2007.

[12]

J. Renclawowicz, Blow-up, global existence and growth rate estimates in nonlinear parabolic systems, Colloq. Math., 86 (2000), 43-66. 

[13]

J. Renclawowicz, Global existence and blow-up of solutions for a completely coupled Fujita type system, Appl. Math., 27 (2000), 203-218. 

[14]

Ph. Souplet, Single-point blow-up for a semilinear parabolic system, J. Eur. Math. Soc., 11 (2009), 169-188.  doi: 10.4171/JEMS/145.

[15]

M. Wang, Blow-up rate for a semilinear reaction diffusion system, Comput. Math. Appl., 44 (2002), 573-585.  doi: 10.1016/S0898-1221(02)00172-4.

[16]

F. B. Weissler, Single point blow-up for a semilinear initial value problem, J. Differ. Equations, 55 (1984), 204-224.  doi: 10.1016/0022-0396(84)90081-0.

show all references

References:
[1]

X. Y. Chen and H. Matano, Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations, J. Differ. Equations, 78 (1989), 160-190.  doi: 10.1016/0022-0396(89)90081-8.

[2]

F. Fila and P. Quittner, The blow-up rate for a semilinear parabolic system, J. Math. Anal. Appl., 238 (1999), 468-476.  doi: 10.1006/jmaa.1999.6525.

[3]

A. Friedman and Y. Giga, A single point blow-up for solutions of semilinear parabolic systems, J. Fac. Sci. Univ. Tokyo Sec. IA. Math., 34 (1987), 65-79. 

[4]

A. Friedman and B. Mcleod, Blow-up of positive solution of semilinear heat equations, Indiana Univ. Math. J., 34 (1985), 425-447.  doi: 10.1512/iumj.1985.34.34025.

[5]

Y. Giga and R. V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math., 38 (1985), 297-319.  doi: 10.1002/cpa.3160380304.

[6]

M. A. Herrero and J. J. A. Velázquez, A blow up result for semilinear heat equations in the supercritical case, Preprint.

[7]

N. Mahmoudi, Single-point blow-up for a semilinear reaction-diffusion system, Differ. Equ. Appl., 6 (2014), 563-591.  doi: 10.7153/dea-06-33.

[8]

N. MahmoudiPh. Souplet and S. Tayachi, Improved conditions for single-point blow-up in reaction-diffusion systems, J. Differ. Equations, 259 (2015), 1898-1932.  doi: 10.1016/j.jde.2015.03.024.

[9]

H. Matano and F. Merle, On nonexistence of type Ⅱ blowup for a supercritical nonlinear heat equation, Comm. Pure Appl. Math., 57 (2004), 1494-1541.  doi: 10.1002/cpa.20044.

[10]

C. E. Mueller and F. B. Weissler, Single point blow-up for a general semilinear heat equation, Indiana Univ. Math. J., 34 (1985), 881-913.  doi: 10.1512/iumj.1985.34.34049.

[11]

P. Quittner and Ph. Souplet, Superlinear Parabolic Problems Blow-Up, Global Existence and Steady States, Birkhäuser Verlag AG, Basel Boston Berlin, 2007.

[12]

J. Renclawowicz, Blow-up, global existence and growth rate estimates in nonlinear parabolic systems, Colloq. Math., 86 (2000), 43-66. 

[13]

J. Renclawowicz, Global existence and blow-up of solutions for a completely coupled Fujita type system, Appl. Math., 27 (2000), 203-218. 

[14]

Ph. Souplet, Single-point blow-up for a semilinear parabolic system, J. Eur. Math. Soc., 11 (2009), 169-188.  doi: 10.4171/JEMS/145.

[15]

M. Wang, Blow-up rate for a semilinear reaction diffusion system, Comput. Math. Appl., 44 (2002), 573-585.  doi: 10.1016/S0898-1221(02)00172-4.

[16]

F. B. Weissler, Single point blow-up for a semilinear initial value problem, J. Differ. Equations, 55 (1984), 204-224.  doi: 10.1016/0022-0396(84)90081-0.

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