The Blow-up Properties To Two Parabolic Systems With Nonlinear Nonlocal Sources

In this paper, we consider the blow-up properties of the solutions to u_t=â–³u^m + u^p1∫ΩV^q1(x, t)dx,V_t=â–³Vⁿ + V^p2∫Ωu^q2(x, t)dx, with null Dirichlet bound-ary conditions, where m, n≥1.In Chaper two, we consider the case m=n=1. Applying the methods in [31], we prove that: (i) when P1, P2≤1 this system possesses uniform blow-up profiles. In other words, the nonlocal terms∫ΩV^q1(x,t)dx,∫Ωu^q2(x,t)dx play a leading role in the blow-up profiles for this case. (ii) when p1, P2＞1, this system presents single point blow-up patterns, which shows that the local terms u^p1, V^q1 dominates the nonlocal terms∫ΩV^q1(x, t)dx,∫ΩV^q2(x, t)dx in the blow-up profiles.In Chaper three, we consider the case m, n＞1. Applying the methods in [31], we find that: when P1, P2≤1 and q1q2＞(m-p1)(n-P2), the solution of the system uniformly blows up on any compact subset ofΩ. Furthermore, we find that the precise blow-up rates is independent of m, n.