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Precisely for $\\chi<1$ and any sufficiently regular initial data $u(x,0)\\geq 0$ and $v(x,0)>0$ on $\\bar{\\Omega}$, we show the existence of global classical solutions. 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Precisely for $\\chi<1$ and any sufficiently regular initial data $u(x,0)\\geq 0$ and $v(x,0)>0$ on $\\bar{\\Omega}$, we show the existence of global classical solutions. 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