Blow-up Estimates Of The Nonnegative Solutions To A Reaction-diffusion System

Diffusion phenomena appear wildly in nature. In applied science, many problems related diffusion can be modelled by reaction-diffusion equations (systems), which are parabolic equations (systems) in the main. In view of mathematical theory and its applications, it is very important to study the reaction-diffusion equations, which results in new mathematical theory and methods. In the past years, the study in this direction attracts a large number of mathematicians both in China and aboard, remarkable progress has been achieved, many new ideas and methods have been developed, which enriched the theory of partial differential equations.In this paper, we study the blow-up properties of the solution to reaction-diffusion system with nonlinear localized reaction:where p_i, q_i(i= 1,2) are constants, p₂ > P₁ — 1 > 0, q₁ >q₂ — 1 > 0, and x_o : R⁺ → Ω is Holder continuous. When Ω = Rⁿ, the diffusion system is subject to the initial conditionsu(x,0) = u_o(x), v(x,0) = v_o(x), x ∈Rⁿ (2)where u₀(x) and v₀(x) are nonnegative continuous bounded functions in Rⁿ . When Ω(?) Rⁿ is a bounded domain with smooth boundary. In this case, the diffusion system is subject to the following boundary and initial conditionwhere u_o(x) and v_o(x) are continuous nonnegative functions in Ω vanishing on (?)Ω.