American Institute of Mathematical Sciences

Advanced
April  2005, 13(3): 683-700. doi: 10.3934/dcds.2005.13.683

Properties of blow-up solutions to a parabolic system with nonlinear localized terms

Huiling Li 1, and  Mingxin Wang 2,

1. 

Department of Mathematics, Southeast University, Nanjing 210018, China
2. 

Department of Mathematics, Xuzhou Normal University, Xuzhou 221116, Department of Mathematics, Southeast University, Nanjing 210018, China

Received  September 2004 Revised  February 2005 Published  May 2005

This paper deals with blow-up properties of the solution to a semi-linear parabolic system with nonlinear localized sources involved in a product with local terms, subject to the null Dirichlet boundary condition. We investigate the influence of localized sources and local terms on blow-up properties for this system. It will be proved that: (i) when $m, q\leq 1$ this system possesses uniform blow-up profiles. In other words, the localized terms play a leading role in the blow-up profile for this case. (ii) when $m, q>1$, this system presents single point blow-up patterns, or say that, in this time, local terms dominate localized terms in the blow-up profile. Moreover, the blow-up rate estimates in time and space are obtained, respectively.
Keywords: blow-up rate estimate., uniform blow-up profile, nonlinear local terms, single point blow-up pattern, nonlinear localized sources, Parabolic system.
Mathematics Subject Classification: Primary: 35K15, 35K6.
Citation: Huiling Li, Mingxin Wang. Properties of blow-up solutions to a parabolic system with nonlinear localized terms. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 683-700. doi: 10.3934/dcds.2005.13.683
[1]

Victor A. Galaktionov, Juan-Luis Vázquez. The problem Of blow-up in nonlinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 399-433. doi: 10.3934/dcds.2002.8.399
[2]

Pavol Quittner, Philippe Souplet. Blow-up rate of solutions of parabolic poblems with nonlinear boundary conditions. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 671-681. doi: 10.3934/dcdss.2012.5.671
[3]

Evgeny Galakhov, Olga Salieva. Blow-up for nonlinear inequalities with gradient terms and singularities on unbounded sets. Conference Publications, 2015, 2015 (special) : 489-494. doi: 10.3934/proc.2015.0489
[4]

Jorge A. Esquivel-Avila. Blow-up in damped abstract nonlinear equations. Electronic Research Archive, 2020, 28 (1) : 347-367. doi: 10.3934/era.2020020
[5]

Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71
[6]

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
[7]

Vural Bayrak, Emil Novruzov, Ibrahim Ozkol. Local-in-space blow-up criteria for two-component nonlinear dispersive wave system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 6023-6037. doi: 10.3934/dcds.2019263
[8]

Nejib Mahmoudi. Single-point blow-up for a multi-component reaction-diffusion system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 209-230. doi: 10.3934/dcds.2018010
[9]

Satyanad Kichenassamy. Control of blow-up singularities for nonlinear wave equations. Evolution Equations & Control Theory, 2013, 2 (4) : 669-677. doi: 10.3934/eect.2013.2.669
[10]

Dapeng Du, Yifei Wu, Kaijun Zhang. On blow-up criterion for the nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3639-3650. doi: 10.3934/dcds.2016.36.3639
[11]

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
[12]

Filippo Gazzola, Paschalis Karageorgis. Refined blow-up results for nonlinear fourth order differential equations. Communications on Pure & Applied Analysis, 2015, 14 (2) : 677-693. doi: 10.3934/cpaa.2015.14.677
[13]

Lili Du, Zheng-An Yao. Localization of blow-up points for a nonlinear nonlocal porous medium equation. Communications on Pure & Applied Analysis, 2007, 6 (1) : 183-190. doi: 10.3934/cpaa.2007.6.183
[14]

Türker Özsarı. Blow-up of solutions of nonlinear Schrödinger equations with oscillating nonlinearities. Communications on Pure & Applied Analysis, 2019, 18 (1) : 539-558. doi: 10.3934/cpaa.2019027
[15]

Zhijun Zhang. Boundary blow-up for elliptic problems involving exponential nonlinearities with nonlinear gradient terms and singular weights. Communications on Pure & Applied Analysis, 2007, 6 (2) : 521-529. doi: 10.3934/cpaa.2007.6.521
[16]

Jian Zhang, Shihui Zhu, Xiaoguang Li. Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential. Mathematical Control & Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119
[17]

Nguyen Thanh Long, Hoang Hai Ha, Le Thi Phuong Ngoc, Nguyen Anh Triet. Existence, blow-up and exponential decay estimates for a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (1) : 455-492. doi: 10.3934/cpaa.2020023
[18]

Zhiqing Liu, Zhong Bo Fang. Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3619-3635. doi: 10.3934/dcdsb.2016113
[19]

Monica Marras, Stella Vernier Piro. On global existence and bounds for blow-up time in nonlinear parabolic problems with time dependent coefficients. Conference Publications, 2013, 2013 (special) : 535-544. doi: 10.3934/proc.2013.2013.535
[20]

Monica Marras, Stella Vernier-Piro, Giuseppe Viglialoro. Blow-up phenomena for nonlinear pseudo-parabolic equations with gradient term. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2291-2300. doi: 10.3934/dcdsb.2017096

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (41)
  • HTML views (0)
  • Cited by (20)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences