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Global existence and finite time blow up for a reaction-diffusion system. (English) Zbl 0984.35088
The reaction-diffusion system considered here is the following: \[ u_t= \Delta u+u^\alpha v^p,\;v_t=\Delta v+u^qv^\beta \quad\text{for}\quad x \in\Omega,\;t>0, \] with initial values \(u(x,0)=u_0 (x)\geq 0\), \(v(x,0)=v_0 (x)\geq 0\) and zero Dirichlet boundary conditions. The domain \(\Omega\subset\mathbb{R}\) is bounded and has a smooth boundary, \(\alpha,\beta, p,q\geq 0\), \(\alpha+p>0\) and \(\beta +q>0\). Depending on the coefficients either (1): all solutions exist globally, (2): all solutions with nonzero initial data blow up in finite time, or (3): solutions exist globally for small initial data and blow up for large initial data. Let \(\lambda_1\) denote the first eigenvalue for \(-\Delta\varphi =\lambda \varphi\) in \(\Omega\) with \(\varphi=0\) on \(\partial\Omega\). Assuming \(\alpha \geq\beta\) his theorem states that, if \(\alpha\leq 1\) and \(pq\leq(1-\alpha) (1-\beta)\), then (1) holds; if \(1+p(\lambda_1^{-1} -1)>\alpha >\beta=1\) and \(p>q =0\) then (2) holds. In case that \(1+p(\lambda^{-1}_1 -1)=\alpha >\beta=1\) (and \(p> q=0)\) the author needs \(\lambda_1<{2\over 3}\) such that (2) holds; if \(\lambda_1 \in[{2\over 3},1)\) the problem is still open. By symmetry a similar condition appears when interchanging the roles of \(u,\alpha,p\) and \(v,\beta,q\). In the remaining cases, (3) holds. The basic instruments in the proofs are comparison principles and testing with \(\varphi_1\). Closely related results appeared in [H. W. Chen, J. Math. Anal. Appl. 212, 481-492 (1997; Zbl 0884.35068)].
Reviewer: G.H.Sweers (Delft)

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35K57 Reaction-diffusion equations
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