A Class Of Nonlinear Parabolic Equations Blow And Overall Solution

In this paper,we study the following problem (P) for reaction diffusion model,whereΩis a bounded domain in R³ with smooth boundry (?)Ω, 0 < T < +∞. Under suitable assumption on F(u),a(u),g(x,t),f(u) and initial data u₀(x), by constructing an auxiliary funtion and using maximum principles and using differetial inequality technique, we proved the existence theorems of blow-up solutions, upper bound of blow-up time, upper estimates of blow-up rate, lower bound of blow-up time, and exeistence theorems of global positive solutions, upper estimates of global positive solutions.The content of this paper is organized as follows.In section 1, we introduce the present conditions of some relative problems.In section 2, we give some preliminary results, including lemma and differetial inequalitywhich will be used in our main result.In section 3, we proved the estitence of blow-up solutions , upper bound of blow-up time of problem (P), gived the main result.Theorem 1. Let u(x, t) be a solution of (P). Suppose that(1) f(0) = 0; if s∈R⁺,(2) For (x,t)∈(?)×(0,T),(3) The constant(4) The integration whereThen u(x, t) must blow up in finite time T andandwherewhereΦ^-1 is inverse function ofΦ.In section 4, we discussed the lower bound of blow-up time of (P) and gived the main result.Theorem 2. Let u(x,t) be a solution of (P). Suppose that(1) f(0) = 0, f(s) > 0 if s > 0 .(2)∫_s^+∞(?)dηis bounded for s≥s₀ > 0.(3) g(x,t) is bounded, namely |g(x,t)|≤M.(4) There exist postive constants n≥2 andξsuch that(5) a and f are related bywhere K is positive constant ,γ∈(0,1) . Thenwhere K₂ = (?)K₁θ^-1,K₁ = nMC^?λ₁^?|Ω|^?,θ= (?), C = 4^?·3^?·π^?, |Ω| denotes the volume ofΩ,λ₁ is the first eigenvalue in the fixed membrane problem △ω+λω= 0,ω>0 inΩ;ω= 0 on (?)Ω.In section 5, we proved the estitence of global solotions , upper estimates of global positive solutions of problem (P) and gived the main result.Theorem 3. Let u(x,t) be a solution of (P). Suppose that(1) f(0)= 0; for s∈R⁺,(2) For (x,t)∈(?)×(0,T),(3) The constant(4) The integrationwhereThen u(x, t) must be a global solution andwherewhereψ^-1 is the inverse function ofψ.The last section, we discussed the case of nonlinear boundary conditions and gived the main result. Theorem 4. Let u be a solution of (P'). Suppose that(1) For s∈R⁺,(2) For (x,t)∈(?)×(0,T),(3) The constant(4) The integrationwhere(5) (?) is boundedThen u(x, t) must blow-up in finite time T andandwherewhereΦ^-1 is inverse function ofΦ.