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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
##### Keywords:
blow-up; classical solutions
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##### References:
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