Blow-up for parabolic equations and systems with nonnegative potential (Q652086)

The following parabolic system is considered: \[ \begin{aligned} &u_t=\Delta u+\mu(x)f(x, t, u),\quad x\in \Omega, t>0,\\ &u(x, 0)=u_0(x),\quad x\in \Omega,\\ &u(x, t)=0,\quad x\in \partial\Omega, t>0, \end{aligned} \] where \(u=(u_1, \ldots, u_m)\) is an unknown function, \(m=1, 2\), \(\Omega\) is a bounded smooth domain, \(\mu\) is Hölder continuous in \(\Omega\), \(\mu(x)\geq 0\), \(\mu(x)\) is not identically equal to zero and all zeros of \(\mu(x)\) are included in \(\Omega\), \(u_0(x)=(u_{01}(x),\ldots, u_{0m}(x))\), \(u_{0i}\geq 0\), \(u_{0i}\) are not identically equal to zero, \(u_{0i}\) are smooth functions for which \({u_{0i}}_{|_{\partial\Omega}}=0\). When \(m=1\), \(f(x, t, u_1)=f_1(x, t, u_1)=\int_0^t u_1^p(x, s) ds\), \(p>1\), and the solution \(u_1(x, t)\) blows up at \(T<\infty\), and there exists a constant \(C_1>0\) so that \(||u_1(\cdot, t)||_{L^{\infty}(\Omega)}\leq C_1(T-t)^{-{2\over {p-1}}}\) for every \(t\in (0, T)\), the authors prove that any zero of \(\mu(x)\) is not a blow-up point. When \(m=2\), \(f(x, t, u_1, u_2)=(u_2^p, u_1^q)\), \(p>1\), \(q>1\), the authors prove that the solution \(u=(u_1, u_2)\), for which \(u_{it}(x, 0)\geq 0\), \(x\in \Omega\), \(i=1, 2\), blows up in a finite time \(T\) and \(||u_1(\cdot, t)||_{L^{\infty}(\Omega)}\leq C_2(T-t)^{-{{p+1}\over {pq-1}}}\), \(||u_2(\cdot, t)||_{L^{\infty}(\Omega)}\leq C_2(T-t)^{-{{q+1}\over {pq-1}}}\) for every \(t\in (0, T)\) and for some positive constant \(C_2\), and any zero point of \(\mu(x)\) is not a blow-up point.