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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)

##### MSC:
 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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