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Global existence and blow-up for a nonlinear reaction-diffusion system. (English) Zbl 0884.35068
The author considers the system \[ u_t-\Delta u=u^{m_1} v^{n_1}, \quad v_t-\Delta v=u^{m_2} v^{n_2} \] for \(x\in\Omega\), \(t>0\), where \(\Omega\subset \mathbb{R}^n\) is a bounded domain with a smooth boundary, \(u\) and \(v\) have given nonnegative values initially and are zero when \(x\in \partial\Omega\), \(t>0\). Solutions here are classical solutions. The constants \(m_1\), \(n_1\), \(m_2\), \(n_2\) satisfy \(m_1\), \(n_2\geq 0\), and \(n_1,m_2>0\). The point of the paper is the proof of:
i) If \(m_1\leq 1\), \(n_2\leq\) and \(m_2n_1 \leq(1-m) (1-n_2)\), then all nonnegative solutions are global.
ii) If \(m_1>1\) or \(n_2>1\) or \(m_2n_1> (1-m_1) (1-n_2)\), then there are both global solutions and solutions which blow up in finite time, depending on the magnitude of the initial values.

MSC:
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35K57 Reaction-diffusion equations
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