# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021222
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## 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 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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021222
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