Blow-up phenomena in a class of coupled reaction-diffusion system with
nonlocal boundary conditions
Abstract
The paper deals with blow-up phenomena for the following coupled
reaction-diffusion system with nonlocal boundary conditions:
\begin{equation*}
\left\{\begin{aligned}
&u_t=\nabla\cdot\big(\rho_1(u)\nabla
u\big)+a_1(x)f_1(v),~~v_t=\nabla\cdot\big(\rho_2(v)\nabla
v\big)+a_2(x)f_2(u),&(x,t)\in
D\times(0,T),~~\\
&\frac{\partial
u}{\partial
\nu}=k_1(t)\int_D g_1(u)
{\rm d
}x,~~\frac{\partial
v}{\partial
\nu}=k_2(t)\int_D g_2(v)
{\rm d
}x,&(x,t)\in\partial
D\times(0,T),\\
&u(x,0)=u_0(x),~~v(x,0)=v_0(x),\quad
&x\in
\overline{D}.~~~~~~~~~~~~~~~~~
\end{aligned}\right.
\end{equation*} Based some differential inequalities
and Sobolev inequality, we establish conditions on the data to guarantee
the occurrence of the blow-up. Moreover, when the blow-up occurs,
explicit lower and upper bounds on blow-up time are obtained. At last,
an example is presented to illustrate our main results.