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We consider the nonlocal coupled parabolic systemary. We assume that 0 γ 1 , γ 2 < 1, p, q 1 and the initial data u(0), v(0) ∈ C 0 (Ω). We obtain the Fujita critical exponent for the system above which is not given by the scaling argument.

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We study the existence, uniqueness and regularity of positive solutions of the parabolic equation u t − u = a(x)u q + b(x)u p in a bounded domain and with Dirichlet's condition on the boundary. We consider here a ∈ L α (Ω), b ∈ L β (Ω) and 0 < q 1 < p. The initial data u(0) = u 0 is considered in the space L r (Ω), r 1. In the main result (0 < q < 1), we assume a, b 0 a.e. in Ω and we assume that u 0 γ d Ω for some γ > 0. We find a unique solution in the space C([0, T ], L r (Ω)) ∩ L ∞ loc ((0, T ), L ∞ (Ω)).