A parabolic system with nonlocal boundary conditions and nonlocal sources (Q2904307)

The paper deals with blow-up properties of solutions to the following porous medium system with nonlocal boundary conditions and nonlocal sources NEWLINE\[NEWLINE \begin{cases} u_t=\Delta u^m+a \int_\Omega v^pdx, & x\in\Omega,\;t>0,\\ v_t=\Delta v^m+b \int_\Omega u^pdx, & x\in\Omega,\;t>0,\\ u(x,t)=\int_\Omega k_1(x,y) u(y,t)dy, & x\in\partial\Omega,\;t>0,\\ v(x,t)=\int_\Omega k_2(x,y) v(y,t)dy, & x\in\partial\Omega,\;t>0,\\ u(x,0)=u_0(x),\;v(x,0)=v_0(x), & x\in\Omega, \end{cases} NEWLINE\]NEWLINE where \(m,n>1\), \(a,b,p,q>0\) are constants, \(\Omega\subset\mathbb R^N\) is a bounded and smooth domain, \(k_i(x,y)\not\equiv0\) are nonnegative continuous functions and \(u_0\) and \(v_0\) are positive, continuous and satisfy suitable compatibility conditions on \(\partial\Omega.\)NEWLINENEWLINEThe authors provide criteria for the existence of global or blow-up solutions to the Cauchy problem. Moreover, the global blow-up property and precise blow-up rate estimates are obtained as well.