Double thresholds for blowup and global existence of the solution to a system of parabolic equations

We considered the following parabolic system:

$ \begin{equation*} \left\{ \begin{array}{llll} &u_t = d_1\Delta u-a(x)\cdot \nabla u+f(u,v),\quad x\in\Omega,\ t>0,\\ &v_t = d_2\Delta v-b(x) \cdot \nabla v+g(u,v),\quad x\in\Omega,\ t>0,\\ &u(x,0) = u_0(x),\quad v(x,0) = v_0(x),\quad x\in\Omega, \end{array}\right. \end{equation*} $

subject to Dirichlet (or Neumann) boundary conditions. Here $ \Omega\subset \mathbb{R}^N(N\geq 1) $ is a bounded smooth domain. In addition to some results on blowup and global existence of the solution, we found some more interesting results as follows: (1) There exists double thresholds for blowup and global existence of the solution. Under certain conditions, if $ f(u, v) = f_1(v)g_1(u) $ and $ g(u, v) = f_2(v)g_2(u) $, then the first watershed is

$ \int^{+\infty}_{c_1}\frac{du}{g_1(u)} = +\infty\quad {\rm and}\quad \int^{+\infty}_{c_2}\frac{dv}{f_2(v)} = +\infty, $

and the second watershed is

$ \int_{\tilde{c}_1}^{+\infty}\frac{dU}{\tilde{f}(\tilde{F}^{-1}(K\tilde{G}(U)))} = +\infty\quad{\rm and}\quad \int_{\tilde{c}_2}^{+\infty}\frac{dV}{\tilde{g}(\tilde{G}^{-1}(\frac{1}{\epsilon}\tilde{F}(V)))} = +\infty. $

Here $ \tilde{f}, \tilde{g}, \tilde{F} $ and $ \tilde{G} $ will be defined in Section 2.2. (2) If there exist nonnegative smooth functions $ h(u) $, $ l(v) $ and $ H(s) $ such that

$ f(u,v)h'(u)l(v)+g(u,v)h(u)l'(v) = H[h(u)l(v)]\geq 0, $

then the watershed for blowup in finite time and global existence of the solution is

$ \int^{+\infty}_0\frac{ds}{H(s)} = +\infty. $