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.
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