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Nonexistence of positive solutions to a semilinear elliptic system and blow-up estimates for a reaction-diffusion system. (English) Zbl 0935.35042
Author’s abstract: We get simple conditions under which the elliptic system $$-\Delta u = u^{p_1} v^{q_1}$$, $$-\Delta v = u^{p_2}v^{q_2}$$ in $$\mathbb{R}^n$$, $$n\geq 3$$ with $$p_i + q_i > 1$$, $$i = 1,2$$ has no positive radially symmetric solutions. Then by using this nonexistence result, we establish blow-up estimates for a reaction-diffusion system of Fujita type $$u_t = \Delta u + u^{p_1}v^{q_1}$$, $$v_t = \Delta v + u^{p_2}v^{q_2}$$ with the homogeneous Dirichlet boundary condition.

##### MSC:
 35J55 Systems of elliptic equations, boundary value problems (MSC2000) 35K50 Systems of parabolic equations, boundary value problems (MSC2000) 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35K57 Reaction-diffusion equations
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