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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

 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.
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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