Blow-up Estimates of the Positive Solution of a Parabolic System

This paper establishes the blow-up estimates for the systems ut − Δu = 0, vt − Δv = 0 in BR × (0, T), BR ⊂ Rn, with the nonlinear boundary conditions ∂u∂η=um1vn1 and ∂v∂η=um2vn2 on SR × (0, T) where 0 ≤ m1 < 1 + m2 and 0 ≤ n2 < 1 + n1. We prove that c(T − t)− α/2 ≤ max u(x, t) ≤ C(T − t)− α/2 and c(T − t)− β/2 ≤ max v(x, t) ≤ C(T − t)− β/2 under some monotonicity assumptions on the initial values, where α = (n1 − n2 + 1)/γ, β = (m2 − m1 + 1)/γ, and γ = n1m2 − (1 − m1)(1 − n2).

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