Global existence and blowup for a coupled parabolic system with time-weighted sources on a general domain

Abstract

We consider the parabolic problem \(\mathbf{u}_{t}-\Delta \mathbf{u} = F(t, \mathbf{u})\) in \(\Omega \times (0,T)\) with homogeneous Dirichlet boundary conditions. The nonlinear term is given by $$\begin{aligned} F(t, \mathbf{u})=(f_1(t) u_2^{p_1}, \ldots , f_m(t) u_1^{p_m}), \end{aligned}$$ where \(\mathbf{u}=(u_1, \ldots , u_m) \), \( p_i \ge 1\), and \( f_i \in C[0,\infty ),\) for \(i=1,\ldots ,m\). The set of initial data is \(\left\{ \mathbf{u}_0=(u_{0,1},\ldots ,u_{0,m}) \in \right. \left. C_0(\Omega )^m; u_{i,0}\ge 0\right\} \), where \(\Omega \) is an arbitrary domain (either bounded or unbounded) with smooth boundary. We determine conditions that guarantee either the global existence or the blowup in finite time of nonnegative solutions. These conditions are given in terms of the asymptotic behavior of \(\Vert \mathbf{S }(t) { \mathbf u}_0\Vert _{L^\infty (\Omega )}\), where \((\mathbf{S }(t){ \mathbf u}_0)_{t\ge 0}\) is the heat semigroup on \(C_0(\Omega )^m\).