Blow-up and global existence of solutions for a nonlocal degenerate parabolic system
This paper deals with the blow-up and global existence of nonnegative solutions of the degenerate parabolic system with nonlocal sources \[ \begin{cases} u_t=\Delta u^m+ v^{p_1}||u||^{p_2}_\alpha, & (x,t)\in \Omega\times (0,T), \\ v_t=\Delta v^n+u^{q_1}||v||^{q_2}_\beta,& (x,t)\in \Omega\times (0,T), \\ u(x,t)=v(x,t)=0,\quad &(x,t) \in \partial\Omega\times (0,T), \\ u(x,0)=u_0(x),\;v(x,0)=v_0(x), & x\in \Omega,\end{cases}\tag{1} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with smooth boundary \(\partial \Omega\), \(m,n>1\), \(\alpha, \beta \geq 1\), \(p_1\), \(q_1\), \(p_2\), \(q_2>0\), and \(u_0\), \(v_0\) are nonnegative bounded functions. The main results of the paper are given by the following theorems. Theorem 1. If one of the following conditions holds: (1) \(m > p_2, n > q_2\) and \(p_1q_1 < (m-p_2)(n-q_2)\), (2) \(m > p_2, n > q_2 , p_1q_1 = (m-p_2)(n-q_2)\) and the domain is sufficiently small, (3) \(m > p_2, n > q_2 , p_1q_1 > (m-p_2)(n-q_2)\) and the initial data \(u_0(x), v_0(x)\) are sufficiently small, (4) \(m \leq p_2\) or \(n \leq q_2\) and the initial data \(u_0(x), v_0(x)\) are sufficiently small, then the nonnegative solution of the system exists globally. Theorem 2. If one of the following conditions holds: (1) \(m > p_2, n > q_2, p_1q_1 > (m-p_2)(n-q_2)\) and the initial data \(u_0(x), v_0(x)\) are sufficiently large, (2) \(m >p_2, n > q_2, p_1q_1 = (m-p_2)(n-q_2)\) and the domain contains a sufficiently large ball, \(u_0(x), v_0(x)\) are positive and continuous in \(\Omega\), (3) \(m \leq p_2\) or \(n \leq q_2\) and initial data \(u_0(x), v_0(x)\) are sufficiently large, then the nonnegative solution of system blows up in a finite time.
scientific article; zbMATH DE number 6145522