Boundedness in a two-species chemotaxis parabolic system with two chemicals

Previous Article
Diffusive heat transport in Budyko's energy balance climate model with a dynamic ice line
DCDS-B Home
This Issue
Next Article
Adaptive methods for stochastic differential equations via natural embeddings and rejection sampling with memory

September 2017, 22(7): 2717-2729. doi: 10.3934/dcdsb.2017132

Boundedness in a two-species chemotaxis parabolic system with two chemicals

1.	College of Mathematic & Information, China West Normal University, Nanchong, 637002, China
2.	School of Sciences, Southwest Petroleum University, Chengdu, 610500, China

* Corresponding author: Xie Li

Received August 2016 Revised February 2017 Published April 2017

This paper is devoted to the chemotaxis system

$\left\{\begin{aligned}&u_t=Δ u-χ\nabla·(u\nabla v), &x∈Ω,\,t>0,\\& τ v_t=Δ v-v+w, &x∈Ω,\,t>0,\\&w_t=Δ w-ξ\nabla·(w\nabla z), &x∈Ω,\,t>0,\\& τ z_t=Δ z-z+u, &x∈Ω,\,t>0,\end{aligned}\right.$

which models the interaction between two species in presence of two chemicals, where

$χ, \, ξ∈\mathbb{R}$

$Ω\subset\mathbb{R}^2$

are bounded domains with smooth boundary. It is shown that under the homogeneous Neumann boundary conditions the system possesses a unique global classical solution which is bounded whenever both

$\int_Ω u_0dx$

and

$\int_Ω w_0dx$

are appropriately small. In particular, we extend the recent results obtained by Tao and Winkler (2015, Disc. Cont. Dyn. Syst. B) to the fully parabolic case, i.e., the case of

$τ=1$

Keywords: Chemotaxis, attraction, repulsion, blow-up.

Mathematics Subject Classification: Primary:35A01, 35B44, 35K57, 35Q92;Secondary:92C17.

Citation: Xie Li, Yilong Wang. Boundedness in a two-species chemotaxis parabolic system with two chemicals. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2717-2729. doi: 10.3934/dcdsb.2017132

References:

[1]	N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868. doi: 10.1080/03605307908820113.
[2]	P. Biler, W. Hebisch and T. Nadzieja, The Debye system: Existence and large time behavior of solutions, Nonlinear Anal., 23 (1994), 1189-1209. doi: 10.1016/0362-546X(94)90101-5.
[3]	P. Biler, E. E. Espejo and I. Guerra, Blowup in higher dimensional two species chemotactic systems, Commun. Pure Appl. Anal., 12 (2013), 89-98. doi: 10.3934/cpaa.2013.12.89.
[4]	V. Calvez and J. A. Carrillo, Volume effects in the Keller-Segel model: Energy estimates preventing blow-up, J. Math. Pure Appl., 86 (2006), 155-175. doi: 10.1016/j.matpur.2006.04.002.
[5]	T. Cieslak, P. LaurenÇot and C. Morales-Rodrigo, Global existence and convergence to steady-states in a chemorepulsion system, Banach Center Publications, 81 (2008), 105-117. doi: 10.4064/bc81-0-7.
[6]	A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.
[7]	M. A. Gates, V. M. Coupe, E. M. Torres, R. A. Fricker-Gares and S. B. Dunnett, Spatially and temporally restricted chemoattractant and chemorepulsive cues direct the formation of the nigro-striatal circuit, Euro. J. Neurosci., 19 (2004), 831-844.
[8]	M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Normale Superiore, 24 (1997), 633-683.
[9]	M. E. Hibbing, C. Fuqua, M. R. Parsek and S. B. Peterson, Bacterial competition: surviving and thriving in the microbial jungle, Nature Reviews Microbiology., 8 (2010), 15-25. doi: 10.1038/nrmicro2259.
[10]	D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177. doi: 10.1017/S0956792501004363.
[11]	D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, Ⅰ, Jahresber. Deutsch. Math.-Verien, 105 (2003), 103-165.
[12]	D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107. doi: 10.1016/j.jde.2004.10.022.
[13]	H. Jin and Z. A. Wang, Asymptotic dynamics of the one-dimensional attraction-repulsion Keller-Segel model, Math. Meth. Appl. Sci., 38 (2015), 444-457. doi: 10.1002/mma.3080.
[14]	H. Jin and Z. A. Wang, Boundedness, blowup and critical mass phenomenon in competing chemotaxis, J. Differential Equations, 206 (2016), 162-196. doi: 10.1016/j.jde.2015.08.040.
[15]	H. Jin, Boundedness of the attraction-repulsion Keller-Segel system, J. Math. Anal. Appl., 422 (2015), 1463-1478. doi: 10.1016/j.jmaa.2014.09.049.
[16]	E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.
[17]	R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398. doi: 10.1016/j.jmaa.2008.01.005.
[18]	M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner, Chemotactic signalling, microglia, and Alzheimers disease senile plague: Is there a connection?, Bull.Math. Biol., 65 (2003), 673-730.
[19]	X. Li, Boundedness in a two-dimensional attraction-repulsion system with nonlinear diffusion, Math. Methods Appl. Sci., 39 (2016), 289-301. doi: 10.1002/mma.3477.
[20]	X. Li and Z. Xiang, Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source, Disc. Cont. Dyn. System -A, 35 (2015), 3503-3531. doi: 10.3934/dcds.2015.35.3503.
[21]	X. Li and Z. Xiang, On an attraction-repulsion chemotaxis system with logistic source, IMA J. of Appl. Math., 81 (2016), 165-198. doi: 10.1093/imamat/hxv033.
[22]	K. Lin, C. Mu and L. Wang, Large-time behavior of an attraction-repulsion chemotaxis system, J. Math. Anal. Appl., 426 (2015), 105-124. doi: 10.1016/j.jmaa.2014.12.052.
[23]	T. Nagai, Blow-up of nonradial solutions to parabolic-elliptic systems modelling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55. doi: 10.1155/S1025583401000042.
[24]	M. Negreanu and J. I. Tello, On a two species chemotaxis model with slow chemical diffusion, SIAM J. Math. Anal., 46 (2014), 3761-3781. doi: 10.1137/140971853.
[25]	K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal. Ser. A: Theory Methods and Applications, 51 (2002), 119-144. doi: 10.1016/S0362-546X(01)00815-X.
[26]	K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Canad. Appl. Math. Quart., 10 (2002), 501-543.
[27]	K. J. Painter and J. A. Sherratt, Modelling the movement of interacting cell populations, Journal of Theoretical Biology, 225 (2003), 327-339. doi: 10.1016/S0022-5193(03)00258-3.
[28]	C. Stinner, J. I. Tello and M. Winkler, Competitive exclusion in a two-species chemotaxis model, J. Math. Biology, 68 (2014), 1607-1626. doi: 10.1007/s00285-013-0681-7.
[29]	Y. Tao and Z. A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36. doi: 10.1142/S0218202512500443.
[30]	Y. Tao, Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity, Discr. Cont. Dyn. Syst. B, 18 (2013), 2705-2722. doi: 10.3934/dcdsb.2013.18.2705.
[31]	Y. Tao and M. Winkler, A chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-704. doi: 10.1137/100802943.
[32]	Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715. doi: 10.1016/j.jde.2011.08.019.
[33]	Y. Tao and M. Winkler, Energy-type estimates and global solvability in a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, J. Differential Equations, 257 (2014), 784-815. doi: 10.1016/j.jde.2014.04.014.
[34]	Y. Tao and M. Winkler, Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discr. Cont. Dyn. Syst. B, 20 (2015), 3165-3183. doi: 10.3934/dcdsb.2015.20.3165.
[35]	J. I. Tello and M. Winkler, Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 1413-1425. doi: 10.1088/0951-7715/25/5/1413.
[36]	Y. Wang, A quasilinear attraction-repulsion chemotaxis system of parabolic-elliptic type with logistic source, J. Math. Anal. Appl., 441 (2016), 259-292. doi: 10.1016/j.jmaa.2016.03.061.
[37]	Y. Wang, Global bounded weak solutions to a degenerate quasilinear attraction-repulsion chemotaxis system with rotation, Comput. Math. Appl., 72 (2016), 2226-2240. doi: 10.1016/j.camwa.2016.08.024.
[38]	Y. Wang, Global existence and boundedness in a quasilinear attraction-repulsion chemotaxis system of parabolic-elliptic type, Bound. Value Probl., 2016 (2016), 1-22. doi: 10.1186/s13661-016-0518-6.
[39]	Y. Wang and Z. Xiang, Boundedness in a quasilinear 2D parabolic-parabolic attraction-repulsion chemotaxis system, Discr. Cont. Dyn. Syst. B, 21 (2016), 1953-1973. doi: 10.3934/dcdsb.2016031.
[40]	M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767. doi: 10.1016/j.matpur.2013.01.020.
[41]	M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24. doi: 10.1002/mma.1146.
[42]	M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Commun. Partial Differential Equations, 35 (2010), 1516-1537. doi: 10.1080/03605300903473426.
[43]	Q. Zhang and Y. Li, Global boundedness of solutions to a two-species chemotaxis system, Z. Angew. Math. Phys., 66 (2015), 83-93. doi: 10.1007/s00033-013-0383-4.

show all references

References:

[1]	N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868. doi: 10.1080/03605307908820113.
[2]	P. Biler, W. Hebisch and T. Nadzieja, The Debye system: Existence and large time behavior of solutions, Nonlinear Anal., 23 (1994), 1189-1209. doi: 10.1016/0362-546X(94)90101-5.
[3]	P. Biler, E. E. Espejo and I. Guerra, Blowup in higher dimensional two species chemotactic systems, Commun. Pure Appl. Anal., 12 (2013), 89-98. doi: 10.3934/cpaa.2013.12.89.
[4]	V. Calvez and J. A. Carrillo, Volume effects in the Keller-Segel model: Energy estimates preventing blow-up, J. Math. Pure Appl., 86 (2006), 155-175. doi: 10.1016/j.matpur.2006.04.002.
[5]	T. Cieslak, P. LaurenÇot and C. Morales-Rodrigo, Global existence and convergence to steady-states in a chemorepulsion system, Banach Center Publications, 81 (2008), 105-117. doi: 10.4064/bc81-0-7.
[6]	A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.
[7]	M. A. Gates, V. M. Coupe, E. M. Torres, R. A. Fricker-Gares and S. B. Dunnett, Spatially and temporally restricted chemoattractant and chemorepulsive cues direct the formation of the nigro-striatal circuit, Euro. J. Neurosci., 19 (2004), 831-844.
[8]	M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Normale Superiore, 24 (1997), 633-683.
[9]	M. E. Hibbing, C. Fuqua, M. R. Parsek and S. B. Peterson, Bacterial competition: surviving and thriving in the microbial jungle, Nature Reviews Microbiology., 8 (2010), 15-25. doi: 10.1038/nrmicro2259.
[10]	D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177. doi: 10.1017/S0956792501004363.
[11]	D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, Ⅰ, Jahresber. Deutsch. Math.-Verien, 105 (2003), 103-165.
[12]	D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107. doi: 10.1016/j.jde.2004.10.022.
[13]	H. Jin and Z. A. Wang, Asymptotic dynamics of the one-dimensional attraction-repulsion Keller-Segel model, Math. Meth. Appl. Sci., 38 (2015), 444-457. doi: 10.1002/mma.3080.
[14]	H. Jin and Z. A. Wang, Boundedness, blowup and critical mass phenomenon in competing chemotaxis, J. Differential Equations, 206 (2016), 162-196. doi: 10.1016/j.jde.2015.08.040.
[15]	H. Jin, Boundedness of the attraction-repulsion Keller-Segel system, J. Math. Anal. Appl., 422 (2015), 1463-1478. doi: 10.1016/j.jmaa.2014.09.049.
[16]	E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.
[17]	R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398. doi: 10.1016/j.jmaa.2008.01.005.
[18]	M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner, Chemotactic signalling, microglia, and Alzheimers disease senile plague: Is there a connection?, Bull.Math. Biol., 65 (2003), 673-730.
[19]	X. Li, Boundedness in a two-dimensional attraction-repulsion system with nonlinear diffusion, Math. Methods Appl. Sci., 39 (2016), 289-301. doi: 10.1002/mma.3477.
[20]	X. Li and Z. Xiang, Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source, Disc. Cont. Dyn. System -A, 35 (2015), 3503-3531. doi: 10.3934/dcds.2015.35.3503.
[21]	X. Li and Z. Xiang, On an attraction-repulsion chemotaxis system with logistic source, IMA J. of Appl. Math., 81 (2016), 165-198. doi: 10.1093/imamat/hxv033.
[22]	K. Lin, C. Mu and L. Wang, Large-time behavior of an attraction-repulsion chemotaxis system, J. Math. Anal. Appl., 426 (2015), 105-124. doi: 10.1016/j.jmaa.2014.12.052.
[23]	T. Nagai, Blow-up of nonradial solutions to parabolic-elliptic systems modelling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55. doi: 10.1155/S1025583401000042.
[24]	M. Negreanu and J. I. Tello, On a two species chemotaxis model with slow chemical diffusion, SIAM J. Math. Anal., 46 (2014), 3761-3781. doi: 10.1137/140971853.
[25]	K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal. Ser. A: Theory Methods and Applications, 51 (2002), 119-144. doi: 10.1016/S0362-546X(01)00815-X.
[26]	K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Canad. Appl. Math. Quart., 10 (2002), 501-543.
[27]	K. J. Painter and J. A. Sherratt, Modelling the movement of interacting cell populations, Journal of Theoretical Biology, 225 (2003), 327-339. doi: 10.1016/S0022-5193(03)00258-3.
[28]	C. Stinner, J. I. Tello and M. Winkler, Competitive exclusion in a two-species chemotaxis model, J. Math. Biology, 68 (2014), 1607-1626. doi: 10.1007/s00285-013-0681-7.
[29]	Y. Tao and Z. A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36. doi: 10.1142/S0218202512500443.
[30]	Y. Tao, Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity, Discr. Cont. Dyn. Syst. B, 18 (2013), 2705-2722. doi: 10.3934/dcdsb.2013.18.2705.
[31]	Y. Tao and M. Winkler, A chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-704. doi: 10.1137/100802943.
[32]	Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715. doi: 10.1016/j.jde.2011.08.019.
[33]	Y. Tao and M. Winkler, Energy-type estimates and global solvability in a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, J. Differential Equations, 257 (2014), 784-815. doi: 10.1016/j.jde.2014.04.014.
[34]	Y. Tao and M. Winkler, Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discr. Cont. Dyn. Syst. B, 20 (2015), 3165-3183. doi: 10.3934/dcdsb.2015.20.3165.
[35]	J. I. Tello and M. Winkler, Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 1413-1425. doi: 10.1088/0951-7715/25/5/1413.
[36]	Y. Wang, A quasilinear attraction-repulsion chemotaxis system of parabolic-elliptic type with logistic source, J. Math. Anal. Appl., 441 (2016), 259-292. doi: 10.1016/j.jmaa.2016.03.061.
[37]	Y. Wang, Global bounded weak solutions to a degenerate quasilinear attraction-repulsion chemotaxis system with rotation, Comput. Math. Appl., 72 (2016), 2226-2240. doi: 10.1016/j.camwa.2016.08.024.
[38]	Y. Wang, Global existence and boundedness in a quasilinear attraction-repulsion chemotaxis system of parabolic-elliptic type, Bound. Value Probl., 2016 (2016), 1-22. doi: 10.1186/s13661-016-0518-6.
[39]	Y. Wang and Z. Xiang, Boundedness in a quasilinear 2D parabolic-parabolic attraction-repulsion chemotaxis system, Discr. Cont. Dyn. Syst. B, 21 (2016), 1953-1973. doi: 10.3934/dcdsb.2016031.
[40]	M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767. doi: 10.1016/j.matpur.2013.01.020.
[41]	M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24. doi: 10.1002/mma.1146.
[42]	M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Commun. Partial Differential Equations, 35 (2010), 1516-1537. doi: 10.1080/03605300903473426.
[43]	Q. Zhang and Y. Li, Global boundedness of solutions to a two-species chemotaxis system, Z. Angew. Math. Phys., 66 (2015), 83-93. doi: 10.1007/s00033-013-0383-4.

[1]	Maria Antonietta Farina, Monica Marras, Giuseppe Viglialoro. On explicit lower bounds and blow-up times in a model of chemotaxis. Conference Publications, 2015, 2015 (special) : 409-417. doi: 10.3934/proc.2015.0409
[2]	Sainan Wu, Junping Shi, Boying Wu. Global existence of solutions to an attraction-repulsion chemotaxis model with growth. Communications on Pure & Applied Analysis, 2017, 16 (3) : 1037-1058. doi: 10.3934/cpaa.2017050
[3]	Abelardo Duarte-Rodríguez, Lucas C. F. Ferreira, Élder J. Villamizar-Roa. Global existence for an attraction-repulsion chemotaxis fluid model with logistic source. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 423-447. doi: 10.3934/dcdsb.2018180
[4]	Hai-Yang Jin, Tian Xiang. Repulsion effects on boundedness in a quasilinear attraction-repulsion chemotaxis model in higher dimensions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3071-3085. doi: 10.3934/dcdsb.2017197
[5]	Pan Zheng, Chunlai Mu, Xuegang Hu. Boundedness and blow-up for a chemotaxis system with generalized volume-filling effect and logistic source. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2299-2323. doi: 10.3934/dcds.2015.35.2299
[6]	Yan Li. Emergence of large densities and simultaneous blow-up in a two-species chemotaxis system with competitive kinetics. Discrete & Continuous Dynamical Systems - B, 2017, 12 (11) : 1-20. doi: 10.3934/dcdsb.2019066
[7]	Yilong Wang, Zhaoyin Xiang. Boundedness in a quasilinear 2D parabolic-parabolic attraction-repulsion chemotaxis system. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1953-1973. doi: 10.3934/dcdsb.2016031
[8]	Shijie Shi, Zhengrong Liu, Hai-Yang Jin. Boundedness and large time behavior of an attraction-repulsion chemotaxis model with logistic source. Kinetic & Related Models, 2017, 10 (3) : 855-878. doi: 10.3934/krm.2017034
[9]	C. Y. Chan. Recent advances in quenching and blow-up of solutions. Conference Publications, 2001, 2001 (Special) : 88-95. doi: 10.3934/proc.2001.2001.88
[10]	Marina Chugunova, Chiu-Yen Kao, Sarun Seepun. On the Benilov-Vynnycky blow-up problem. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1443-1460. doi: 10.3934/dcdsb.2015.20.1443
[11]	Alberto Bressan, Massimo Fonte. On the blow-up for a discrete Boltzmann equation in the plane. Discrete & Continuous Dynamical Systems - A, 2005, 13 (1) : 1-12. doi: 10.3934/dcds.2005.13.1
[12]	Marek Fila, Hiroshi Matano. Connecting equilibria by blow-up solutions. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 155-164. doi: 10.3934/dcds.2000.6.155
[13]	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
[14]	W. Edward Olmstead, Colleen M. Kirk, Catherine A. Roberts. Blow-up in a subdiffusive medium with advection. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1655-1667. doi: 10.3934/dcds.2010.28.1655
[15]	Yukihiro Seki. A remark on blow-up at space infinity. Conference Publications, 2009, 2009 (Special) : 691-696. doi: 10.3934/proc.2009.2009.691
[16]	Petri Juutinen. Convexity of solutions to boundary blow-up problems. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2267-2275. doi: 10.3934/cpaa.2013.12.2267
[17]	José M. Arrieta, Raúl Ferreira, Arturo de Pablo, Julio D. Rossi. Stability of the blow-up time and the blow-up set under perturbations. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 43-61. doi: 10.3934/dcds.2010.26.43
[18]	Mohamed-Ali Hamza, Hatem Zaag. Blow-up results for semilinear wave equations in the superconformal case. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2315-2329. doi: 10.3934/dcdsb.2013.18.2315
[19]	Huyuan Chen, Hichem Hajaiej, Ying Wang. Boundary blow-up solutions to fractional elliptic equations in a measure framework. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1881-1903. doi: 10.3934/dcds.2016.36.1881
[20]	Yoshikazu Giga. Interior derivative blow-up for quasilinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 1995, 1 (3) : 449-461. doi: 10.3934/dcds.1995.1.449

American Institute of Mathematical Sciences