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Blow-up estimates of positive solutions of a reaction-diffusion system

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  • This paper is concerned with positive solutions of the reaction-diffusion system

    $u_t - \Delta u = u^(m_1)v^(n_1)$ ,
    $v_t - \Delta v = u^(m_2)v^(n_2)$ ,



    which blow up at $t = T$. We obtain the following estimates on the blow-up rates:

    $c(T - t)^(-(n_1-n_2+1)/\gamma) <= max_(x\in\Omega) u(x, t) <= C(T - t)^(-(n_1-n_2+1)/\gamma)$,
    $c(T - t)^(-(m_2-m_1+1)/\gamma) <= max_(x\in\Omega) v(x, t) <= C(T - t)^(-(m_2-m_1+1)/\gamma)$,



    for some positive constants $c,C$ and $\gamma = m_2n_1 - (1 - m_1)(1 - n_2)$.

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