# American Institute of Mathematical Sciences

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. Mahmoudi, Ph. 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. Mahmoudi, Ph. 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.
 [1] Bouthaina Abdelhedi, Hatem Zaag. Single point blow-up and final profile for a perturbed nonlinear heat equation with a gradient and a non-local term. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 2607-2623. doi: 10.3934/dcdss.2021032 [2] Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete and Continuous Dynamical Systems, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71 [3] Victor A. Galaktionov, Juan-Luis Vázquez. The problem Of blow-up in nonlinear parabolic equations. Discrete and Continuous Dynamical Systems, 2002, 8 (2) : 399-433. doi: 10.3934/dcds.2002.8.399 [4] Pavol Quittner, Philippe Souplet. Blow-up rate of solutions of parabolic poblems with nonlinear boundary conditions. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 671-681. doi: 10.3934/dcdss.2012.5.671 [5] Marek Fila, Hirokazu Ninomiya, Juan-Luis Vázquez. Dirichlet boundary conditions can prevent blow-up in reaction-diffusion equations and systems. Discrete and Continuous Dynamical Systems, 2006, 14 (1) : 63-74. doi: 10.3934/dcds.2006.14.63 [6] Monica Marras, Stella Vernier Piro. Blow-up phenomena in reaction-diffusion systems. Discrete and Continuous Dynamical Systems, 2012, 32 (11) : 4001-4014. doi: 10.3934/dcds.2012.32.4001 [7] Hongwei Chen. Blow-up estimates of positive solutions of a reaction-diffusion system. Conference Publications, 2003, 2003 (Special) : 182-188. doi: 10.3934/proc.2003.2003.182 [8] Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021 [9] Lili Du, Chunlai Mu, Zhaoyin Xiang. Global existence and blow-up to a reaction-diffusion system with nonlinear memory. Communications on Pure and Applied Analysis, 2005, 4 (4) : 721-733. doi: 10.3934/cpaa.2005.4.721 [10] Monica Marras, Stella Vernier Piro. Bounds for blow-up time in nonlinear parabolic systems. Conference Publications, 2011, 2011 (Special) : 1025-1031. doi: 10.3934/proc.2011.2011.1025 [11] Petri Juutinen. Convexity of solutions to boundary blow-up problems. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2267-2275. doi: 10.3934/cpaa.2013.12.2267 [12] Huiling Li, Mingxin Wang. Properties of blow-up solutions to a parabolic system with nonlinear localized terms. Discrete and Continuous Dynamical Systems, 2005, 13 (3) : 683-700. doi: 10.3934/dcds.2005.13.683 [13] Jorge A. Esquivel-Avila. Blow-up in damped abstract nonlinear equations. Electronic Research Archive, 2020, 28 (1) : 347-367. doi: 10.3934/era.2020020 [14] Antonio Vitolo, Maria E. Amendola, Giulio Galise. On the uniqueness of blow-up solutions of fully nonlinear elliptic equations. Conference Publications, 2013, 2013 (special) : 771-780. doi: 10.3934/proc.2013.2013.771 [15] Türker Özsarı. Blow-up of solutions of nonlinear Schrödinger equations with oscillating nonlinearities. Communications on Pure and Applied Analysis, 2019, 18 (1) : 539-558. doi: 10.3934/cpaa.2019027 [16] Frank Merle, Hatem Zaag. O.D.E. type behavior of blow-up solutions of nonlinear heat equations. Discrete and Continuous Dynamical Systems, 2002, 8 (2) : 435-450. doi: 10.3934/dcds.2002.8.435 [17] Qiong Chen, Chunlai Mu, Zhaoyin Xiang. Blow-up and asymptotic behavior of solutions to a semilinear integrodifferential system. Communications on Pure and Applied Analysis, 2006, 5 (3) : 435-446. doi: 10.3934/cpaa.2006.5.435 [18] Shu-Xiang Huang, Fu-Cai Li, Chun-Hong Xie. Global existence and blow-up of solutions to a nonlocal reaction-diffusion system. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1519-1532. doi: 10.3934/dcds.2003.9.1519 [19] Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure and Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264 [20] Zhiqing Liu, Zhong Bo Fang. Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3619-3635. doi: 10.3934/dcdsb.2016113

2021 Impact Factor: 1.588