On the question of global existence for the two-component reaction-diffusion systems with mixed boundary conditions (Q1968660)

This paper considers the global existence versus finite time blowup of solutions of the two-component system \[ \left.\begin{aligned} u_t- d_1\Delta u & = f(u,v)\\ v_t- d_2\Delta v & = g(u,v)\end{aligned}\right\}\quad\text{in }\Omega\times \{t> 0\}, \] \[ \begin{aligned}\alpha_1u &+ (1-a_1){\partial u\over\partial n}= \beta_1,\\ \alpha_2v &+ (1- a_2){\partial v\over\partial n}= \beta_2\quad\text{on }\partial\Omega\times \{t>0\},\end{aligned} \] \[ u(x,0)= u_0(x),\quad v(x,0)= v_0(x)\quad\text{in }\Omega, \] where \(0\leq \alpha_i\leq 1\) and \(\beta_i\geq 0\) for \(i= 1,2\), and \(\Omega\) is a bounded domain in \(\mathbb{R}^n\) with smooth boundary \(\partial\Omega\); \(u_0, v_0\in L^\infty(\Omega; \mathbb{R}^1_+)\); \(d_1, d_2>0\); \(f\), \(g\) are locally Lipschitz, and \(f(0,v)\geq 0\), \(g(u,0)\geq 0\) for all \(u, v\geq 0\). The existence of a unique nonnegative solution of this system for \(0\leq t< T^*\leq\infty\) is well known -- the question is under what conditions does \(T^*=\infty\)? Given additional technical assumptions about the growth of \(f\) and \(g\), some advances have been made in the case that \(u\) and \(v\) satisfy the same boundary conditions, either Dirichlet, Neumann or mixed. This paper extends the result in case \(u\) and \(v\) satisfy different boundary conditions. The proof involves establishing a priori estimates for the solutions in \(V^{1,0}_2(\Omega\times [0,T])\) and bootstrapping these estimates into \(L^p(\Omega\times [0,T])\) for \(p\) large enough to use classical parabolic regularity to guarantee pointwise bounds.