Blow-up results of the positive solution for a weakly coupled quasilinear parabolic system

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August 2022, 27(8): 4173-4183. doi: 10.3934/dcdsb.2021222

Blow-up results of the positive solution for a weakly coupled quasilinear parabolic system

Juntang Ding ^, and Chenyu Dong

School of Mathematical Sciences, Shanxi University, Taiyuan 030006, China

^*Corresponding author: Juntang Ding

Received May 2021 Revised June 2021 Published August 2022 Early access September 2021

Fund Project: This work was supported by the National Natural Science Foundation of China (No. 61473180)

The main purpose of the present paper is to study the blow-up problem of a weakly coupled quasilinear parabolic system as follows:

$ \left\{ \begin{array}{ll} u_{t} = \nabla\cdot\left(r(u)\nabla u\right)+f(u,v,x,t), & \\ v_{t} = \nabla\cdot\left(s(v)\nabla v\right)+g(u,v,x,t) &{\rm in} \ \Omega\times(0,t^{*}), \\ \frac{\partial u}{\partial\nu} = h(u), \ \frac{\partial v}{\partial\nu} = k(v) &{\rm on} \ \partial\Omega\times(0,t^{*}), \\ u(x,0) = u_{0}(x), \ v(x,0) = v_{0}(x) &{\rm in} \ \overline{\Omega}. \end{array} \right. $

Here

$ \Omega $

is a spatial bounded region in

$ \mathbb{R}^{n} \ (n\geq2) $

and the boundary

$ \partial\Omega $

of the spatial region

$ \Omega $

is smooth. We give a sufficient condition to guarantee that the positive solution

$ (u,v) $

of the above problem must be a blow-up solution with a finite blow-up time

$ t^* $

. In addition, an upper bound on

$ t^* $

and an upper estimate of the blow-up rate on

$ (u,v) $

are obtained.

Keywords: Blow-up, quasilinear parabolic system, nonlinear boundary condition.

Mathematics Subject Classification: 35K51, 35B44.

Citation: Juntang Ding, Chenyu Dong. Blow-up results of the positive solution for a weakly coupled quasilinear parabolic system. Discrete and Continuous Dynamical Systems - B, 2022, 27 (8) : 4173-4183. doi: 10.3934/dcdsb.2021222

References:

[1]	X. L. Bai, Finite time blow-up for a reaction-diffusion system in bounded domain, Z. Angew. Math. Phys., 65 (2014), 135-138. doi: 10.1007/s00033-013-0330-4.
[2]	J. T. Ding, Blow-up analysis for parabolic p-Laplacian equations with a gradient source term, J. Inequal. Appl., 2020 (2020), 1-11. doi: 10.1186/s13660-020-02481-y.
[3]	J. T. Ding and W. Kou, Blow-up solutions for reaction diffusion equations with nonlocal boundary conditions, J. Math. Anal. Appl., 470 (2019), 1-15.
[4]	J. T. Ding and X. H. Shen, Blow-up time estimates in porous medium equations with nonlinear boundary conditions, Z. Angew. Math. Phys., 69 (2018), 1-13. doi: 10.1007/s00033-018-0993-y.
[5]	L. L. Du, Blow-up for a degenerate reaction-diffusion system with nonlinear nonlocal sources, J. Comput. Appl. Math., 202 (2007), 237-247. doi: 10.1016/j.cam.2006.02.028.
[6]	Y. L. Du and B. C. Liu, Time-weighted blow-up profiles in a nonlinear parabolic system with Fujita exponent, Comput. Math. Appl., 76 (2018), 1034-1055. doi: 10.1016/j.camwa.2018.05.039.
[7]	C. Enache, Blow-up, global existence and exponential decay estimates for a class of quasilinear parabolic problems, Nonlinear Anal. TMA, 69 (2008), 2864-2874. doi: 10.1016/j.na.2007.08.063.
[8]	A. Friedman, Partial Differential Equation of Parabolic Type, , Prentice-Hall, Englewood Cliffs, N. J., 1964.
[9]	S. C. Fu and J. S. Guo, Blow-up for a semilinear reaction-diffusion system coupled in both equations and boundary conditions, J. Math. Anal. Anal., 276 (2002), 458-475. doi: 10.1016/S0022-247X(02)00506-1.
[10]	W. Guo, W. J. Gao and B. Guo, Global existence and blowing-up of solutions to a class of coupled reaction-convection-diffusion systems, Appl. Math. Lett., 28 (2014), 72-76. doi: 10.1016/j.aml.2013.10.003.
[11]	W. Kou and J. T. Ding, Blow-up phenomena for p-Laplacian parabolic equations under nonlocal boundary conditions, Appl. Anal., 2020. doi: 10.1080/00036811.2020.1716972.
[12]	F. J. Li and B. C. Liu, Critical exponents for non-simultaneous blow-up in a localized parabolic system, Nonlinear Anal. TMA, 70 (2009), 3452-3460. doi: 10.1016/j.na.2008.07.002.
[13]	G. Li, P. Fan and J. Zhu, Blow-up estimates for a semilinear coupled parabolic system, Appl. Math. Lett., 22 (2009), 1297-1302. doi: 10.1016/j.aml.2009.01.046.
[14]	F. Liang, Global existence and blow-up for a degenerate reaction-diffusion system with nonlinear localized sources and nonlocal boundary conditions, J. Korean Math. Soc., 53 (2016), 27-43. doi: 10.4134/JKMS.2016.53.1.027.
[15]	H. H. Lu, Global existence and blow-up analysis for some degenerate and quasilinear parabolic systems, Electron. J. Qual. Theory Differ. Equ., 49 (2009), 1-14. doi: 10.14232/ejqtde.2009.1.49.
[16]	N. Mahmoudi, P. Souplet and S. Tayachi, Improved conditions for single-point blow-up in reaction-diffusion systems, J. Differential Equations, 259 (2015), 1898-1932. doi: 10.1016/j.jde.2015.03.024.
[17]	M. Marras, Bounds for blow-up time in nonlinear parabolic systems under various boundary conditions, Numer. Funct. Anal. Optim., 32 (2011), 453-468. doi: 10.1080/01630563.2011.554949.
[18]	M. Marras and S. Vernier-Piro, Finite time collapse in chemotaxis systems with logistic-type superlinear source, Math. Methods Appl. Sci., 43 (2020), 10027-10040. doi: 10.1002/mma.6676.
[19]	L. E. Payne and G. A. Philippin, Blow-up phenomena for a class of parabolic systems with time dependent coefficients, Appl. Math., 3 (2012), 325-330. doi: 10.4236/am.2012.34049.
[20]	L. E. Payne and P. W. Schaefer, Blow-up phenomena for some nonlinear parabolic systems, Int. J. Pure Appl. Math., 48 (2008), 193-202.
[21]	J. D. Rossi and P. Souplet, Coexistence of simultaneous and nonsimultaneous blow-up in a semilinear parabolic system, Differ. Integral Equ., 18 (2005), 405-418.
[22]	X. H. Shen and J. T. Ding, Blow-up phenomena in porous medium equation systems with nonlinear boundary conditions, Comput. Math. Appl., 77 (2019), 3250-3263. doi: 10.1016/j.camwa.2019.02.007.
[23]	P. Souplet and S. Tayachi, Optimal condition for non-simultaneous blow-up in a reaction-diffusion system, J. Math. Soc. Japan, 56 (2004), 571-584. doi: 10.2969/jmsj/1191418646.
[24]	R. P. Sperb, Maximum Principles and Their Applications,, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981.
[25]	N. Umeda, Large time behavior and uniqueness of solutions of a weakly coupled system of reaction-diffusion equations, Tokyo J. Math., 26 (2003), 347-372. doi: 10.3836/tjm/1244208595.
[26]	J. Z. Zhang and F. S. Li, Global existence and blow-up phenomena for divergence form parabolic equation with time-dependent coefficient in multidimensional space, Z. Angew. Math. Phys., 70 (2019), 1-16. doi: 10.1007/s00033-019-1195-y.
[27]	L. L. Zhang, H. Wang and X. Q. Wang, Global and blow-up analysis for a class of nonlinear reaction diffusion model with Dirichlet boundary conditions, Math. Methods Appl. Sci., 41 (2018), 7789-7803. doi: 10.1002/mma.5241.
[28]	H. H. Zou, Blow-up rates for semi-linear reaction-diffusion systems, J. Differential Equations, 257 (2014), 843-867. doi: 10.1016/j.jde.2014.04.019.

show all references

References:

[1]	X. L. Bai, Finite time blow-up for a reaction-diffusion system in bounded domain, Z. Angew. Math. Phys., 65 (2014), 135-138. doi: 10.1007/s00033-013-0330-4.
[2]	J. T. Ding, Blow-up analysis for parabolic p-Laplacian equations with a gradient source term, J. Inequal. Appl., 2020 (2020), 1-11. doi: 10.1186/s13660-020-02481-y.
[3]	J. T. Ding and W. Kou, Blow-up solutions for reaction diffusion equations with nonlocal boundary conditions, J. Math. Anal. Appl., 470 (2019), 1-15.
[4]	J. T. Ding and X. H. Shen, Blow-up time estimates in porous medium equations with nonlinear boundary conditions, Z. Angew. Math. Phys., 69 (2018), 1-13. doi: 10.1007/s00033-018-0993-y.
[5]	L. L. Du, Blow-up for a degenerate reaction-diffusion system with nonlinear nonlocal sources, J. Comput. Appl. Math., 202 (2007), 237-247. doi: 10.1016/j.cam.2006.02.028.
[6]	Y. L. Du and B. C. Liu, Time-weighted blow-up profiles in a nonlinear parabolic system with Fujita exponent, Comput. Math. Appl., 76 (2018), 1034-1055. doi: 10.1016/j.camwa.2018.05.039.
[7]	C. Enache, Blow-up, global existence and exponential decay estimates for a class of quasilinear parabolic problems, Nonlinear Anal. TMA, 69 (2008), 2864-2874. doi: 10.1016/j.na.2007.08.063.
[8]	A. Friedman, Partial Differential Equation of Parabolic Type, , Prentice-Hall, Englewood Cliffs, N. J., 1964.
[9]	S. C. Fu and J. S. Guo, Blow-up for a semilinear reaction-diffusion system coupled in both equations and boundary conditions, J. Math. Anal. Anal., 276 (2002), 458-475. doi: 10.1016/S0022-247X(02)00506-1.
[10]	W. Guo, W. J. Gao and B. Guo, Global existence and blowing-up of solutions to a class of coupled reaction-convection-diffusion systems, Appl. Math. Lett., 28 (2014), 72-76. doi: 10.1016/j.aml.2013.10.003.
[11]	W. Kou and J. T. Ding, Blow-up phenomena for p-Laplacian parabolic equations under nonlocal boundary conditions, Appl. Anal., 2020. doi: 10.1080/00036811.2020.1716972.
[12]	F. J. Li and B. C. Liu, Critical exponents for non-simultaneous blow-up in a localized parabolic system, Nonlinear Anal. TMA, 70 (2009), 3452-3460. doi: 10.1016/j.na.2008.07.002.
[13]	G. Li, P. Fan and J. Zhu, Blow-up estimates for a semilinear coupled parabolic system, Appl. Math. Lett., 22 (2009), 1297-1302. doi: 10.1016/j.aml.2009.01.046.
[14]	F. Liang, Global existence and blow-up for a degenerate reaction-diffusion system with nonlinear localized sources and nonlocal boundary conditions, J. Korean Math. Soc., 53 (2016), 27-43. doi: 10.4134/JKMS.2016.53.1.027.
[15]	H. H. Lu, Global existence and blow-up analysis for some degenerate and quasilinear parabolic systems, Electron. J. Qual. Theory Differ. Equ., 49 (2009), 1-14. doi: 10.14232/ejqtde.2009.1.49.
[16]	N. Mahmoudi, P. Souplet and S. Tayachi, Improved conditions for single-point blow-up in reaction-diffusion systems, J. Differential Equations, 259 (2015), 1898-1932. doi: 10.1016/j.jde.2015.03.024.
[17]	M. Marras, Bounds for blow-up time in nonlinear parabolic systems under various boundary conditions, Numer. Funct. Anal. Optim., 32 (2011), 453-468. doi: 10.1080/01630563.2011.554949.
[18]	M. Marras and S. Vernier-Piro, Finite time collapse in chemotaxis systems with logistic-type superlinear source, Math. Methods Appl. Sci., 43 (2020), 10027-10040. doi: 10.1002/mma.6676.
[19]	L. E. Payne and G. A. Philippin, Blow-up phenomena for a class of parabolic systems with time dependent coefficients, Appl. Math., 3 (2012), 325-330. doi: 10.4236/am.2012.34049.
[20]	L. E. Payne and P. W. Schaefer, Blow-up phenomena for some nonlinear parabolic systems, Int. J. Pure Appl. Math., 48 (2008), 193-202.
[21]	J. D. Rossi and P. Souplet, Coexistence of simultaneous and nonsimultaneous blow-up in a semilinear parabolic system, Differ. Integral Equ., 18 (2005), 405-418.
[22]	X. H. Shen and J. T. Ding, Blow-up phenomena in porous medium equation systems with nonlinear boundary conditions, Comput. Math. Appl., 77 (2019), 3250-3263. doi: 10.1016/j.camwa.2019.02.007.
[23]	P. Souplet and S. Tayachi, Optimal condition for non-simultaneous blow-up in a reaction-diffusion system, J. Math. Soc. Japan, 56 (2004), 571-584. doi: 10.2969/jmsj/1191418646.
[24]	R. P. Sperb, Maximum Principles and Their Applications,, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981.
[25]	N. Umeda, Large time behavior and uniqueness of solutions of a weakly coupled system of reaction-diffusion equations, Tokyo J. Math., 26 (2003), 347-372. doi: 10.3836/tjm/1244208595.
[26]	J. Z. Zhang and F. S. Li, Global existence and blow-up phenomena for divergence form parabolic equation with time-dependent coefficient in multidimensional space, Z. Angew. Math. Phys., 70 (2019), 1-16. doi: 10.1007/s00033-019-1195-y.
[27]	L. L. Zhang, H. Wang and X. Q. Wang, Global and blow-up analysis for a class of nonlinear reaction diffusion model with Dirichlet boundary conditions, Math. Methods Appl. Sci., 41 (2018), 7789-7803. doi: 10.1002/mma.5241.
[28]	H. H. Zou, Blow-up rates for semi-linear reaction-diffusion systems, J. Differential Equations, 257 (2014), 843-867. doi: 10.1016/j.jde.2014.04.019.

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