Blow-up and global existence for a nonlocal degenerate parabolic system. (English) Zbl 1030.35096
This paper studies the global existence and nonexistence of nonnegative solutions of the nonlocal degenerate parabolic system
\[
u_t= \Delta(u^m)+ a\|v\|^p_\alpha,\quad v_t= \Delta(v^n)+ b\|u\|^q_\beta,\quad (x,t)\in \Omega\times (0,T)
\]
with homogeneous Dirichlet boundary condition and for nonnegative bounded initial data, where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with smooth boundary \(\partial\Omega\), \(m,n> 1\), \(\alpha,\beta\geq 1\), \(p,q,a,b> 0\) and \(\|\cdot\|_\alpha\) denotes \(L^\alpha(\Omega)\) norm. It is proved that if \(pq< mn\), every nonnegative solution is global, and whereas if \(pq> mn\), there exist both global and blow-up nonnegative solutions depending on the initial data. When \(pq= mn\), there exist both global and blow-up nonnegative solutions depending on the domains. The results are proved by comparison arguments.
Reviewer: Shigeru Sakaguchi (Ehime)
MSC:
| 35K50 | Systems of parabolic equations, boundary value problems (MSC2000) |
| 35K55 | Nonlinear parabolic equations |
| 35B33 | Critical exponents in context of PDEs |
| 35K65 | Degenerate parabolic equations |
Keywords:
nonnegative solutions; homogeneous Dirichlet boundary condition; comparison arguments; Nonlocal sourceReferences:
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