Blow-up Of Solutions To Some Parabolic Systems

In this paper, mainly applying the basic theory to parabolic systems, we give a systematic discussion on the global existence and the blowing-up at a finite time to four class of parabolic systems, including the problems with nonlocalsource, non-local boundary conditions, with nonlinear local source, with non-linear boundary flux, and with absorption term.The first chapter is the introduction. Firstly, we introduce the development status and background of the parabolic systems.In the second chapter, we consider the blow-up property of the solution to the parabolic systems with nonlinear local source.whereΩ(?) R^N is a bounded domain with smooth boundary,α,β, p, q are nonnegative constants satisfyingα+ p > 0,β+ q > 0, and u₀(x),v₀(x)∈C₀(Ω) are nonnegative and nontrivial functions. x₀∈Ωis a fixed point.We essentially study: (1) the condition that positive solution blows up in finite time; (2) necessary and sufficient conditions that u and v blow up at the same time; (3) the mode that the solution blows up uniformly in the interior of the domain; (4) asymptotic behavior of the blow up solution on boundary layer and the estimate of the size of the boundary layers. the domain; (4) asymptotic behavior of the blow up solution on boundary layer and the estimate of the size of the boundary layers.Theorem 1. If one of the following conditions hold:(1)α> 1,p>0,q = 0,β=1,λ< 1,α≤1+p(1-λ)/λ;(2)β> 1,q>0,p = 0,α= 1,λ<1,β≤1 + q(1 -λ)/λ;then the solution of problem (1) blows up at finite time fou any nontrivial initial.Theorem 2. Suppose that (u,v) is a classical solution to (1) onΩ×(0,T),and blows up at a finite time. if u and v blow up at the same time, then the following conclusions hold uniformly on any compact subset ofΩ:(i) If p >β-1, q >α-1 andpq > (α-1)(β-1), orp <β-land q <α-1, thenwhereθ= (p + 1 -β)/[pq - (α- 1)(β- 1)],σ= (q + 1-α)/[qp - (α-1)(β- 1)];(ii) If p >β- 1, and q =α- 1 > 0, then(iii) If p =β- 1 > 0, and q >α- 1,then (iv) If p =β-1> Oand q =α- 1 > 0,thenIn the third chaper, we consider the blow-up property to the solution of parabolic systems with non-local source non-local boundary conditions.whereΩ(?)R^N is a bounded domain with smooth boundary, m, n, p, q are nonnegative constants satisfying m + n > 0, p + q > 0;(?) and (?) are a nonnegative and nontrivial functions on(?)Ωx,x₀∈Ωis a fixed point. u₀(x), v₀(x)∈C^2,v(Ω)(0 < v < 1), u₀(x), v₀(x)≥0, and satisfy some compatible conditions.We mainly study: (1) conditions under which the global solution exists; (2) the conditions under which positive solutions blow up in finite time; (3) blow-up set of solution; (4) necessary and sufficient conditions under which u and v blow up at the same time; (5) the mode that the solution blows up uniformly in the interior of the domain.We haveTheorem 3. If∫_Ω(?)(x,y)dy≥1 and∫_Ω(?)(x,y)dy≥1 for any x∈(?)Ωone of the following conditions is valid:(a)m>1; (b) q > 1; (c) np > (1 - m)(1 - q),then the solution of problem (3) blows up at finite time for any positive initials. Theorem 4. Suppose∫_Ω(?)(x, y)dy≤1 and∫_Ω(?)(x, y)dy≤1 for all x∈(?)Ω, (u,v)is a classical solution of (3), if u and v blow up at the same time T, then the following conclusions hold uniformly on any compact subset ofΩ:(i) if n > q -1, p > m -1, or n < q- 1 and p < m -1, thenwhereθ= (p + 1 -β)/[pq - (α- 1)(β- 1)],σ= (q+1-α)/[qp - (α-1)(β- 1)];(ii) if n> q-1 > 0 and q = m - 1 > 0, then(iii) if n = q - 1 > 0 and q > m - 1 > 0,then(iv) if n = q - 1 > 0 and g = m - 1 > 0,thenIn the fouth chapter, we consider the global existence of solutions and the property of blowing-up in finite time to the solution of the parabolic systems with terms of nonlinear reaction, absorption and boundary flux. whereΩ(?) R^N is a bounded domain with smooth boundary, p₁, p₂, q₁, q₂,α,βand a, b are positive constants, (?)/(?)v represents the out normal derivatives at the boundary (?)Ω, u₀(x), v₀(x)∈C¹(Ω) are positive functions satisfying some compatibleconditions. We study the conditions under which positive global solution exists or the positive solution blows up in finite time.Theorem 5. The solution of (6) for large initials blows up in finite time if one of the following conditions hold:(a) p₁q₁ >μ₁γ₁;(b) p₂q₂>μ₂γ₂;(c)α≤1,β≤1,p₁q₂ >μ₁γ₂ or p₂q₁>μ₂γ₁;(d)α<1,β≤1,p₁q₂ >μ₁γ₂ or p₂q₁>μ₂γ₁+p₂(α-1);(e) a≤1,β> 1,p₁q₂ >μ₁γ₂+p₂(β-1) or p₂q₁>μ₂γ₁;(f)α< 1,β>1,p₁q₂ > max{μ₁γ₂, (β+1)/2+q₂(β-1)} or p₁q₂>max{μ₁γ₂,(α+1)/2+p₂(α-1)}.We also haveTheorem 6. supposeα> 1,β> 1.(i) For p₁q₁ =η₁λ₁,p₂q₂<μ₂μ₂,p₁q₂<μ₁λ₂,p₂q₁<μ₂λ₁,ifα^βb^p1 < 1(or a^q2b^α< 1), then the solution of (6) for large initials blows up in finite time;ifα^βb^p1≥2^{β(α+p1)+β+p1}(or a₁^qb^α≥2^{α(β+q1)+α+q1}) then the global solution of (6) exists.(ii) For p₁q₁<η₁λ₁,p₂q₂=μ₂λ₂,p₁q₂<μ₁λ₂,p₂q₁<μ₂λ₁, if a < min{(α-1)λ₀c₃²(2c₂²)^-1, (α+1)c₁²(8c₂²)^-1}, b < min{(β-1)λ₀c₃²(2c₂²)^-1,(β+ 1)c₁²(8c₂²)^-1}, the solution of (6) with large initials blow up in finite time;if a≥2^α+1(λ₀ + 3c₁^-1c₂²)c₁^-1,b≥2^β+1(λ₀ + 3c₁^-1c₂²)c₁^-1 then the globalsolution of (6) exists.(iii) For p₁q₁<η₁λ₁,p₂q₂ <μ₂λ₂,p₁q₂ =μ₁λ₂,p₂q₁ <μ₂λ₁,if a≥max{2^α+1(λ₀ + 3c₂²), 2^α+1c₁^{-2p1/(β+1)}}, b≥2^β+1(λ₀ + 3c₂²), then theglobal solution of (6) exists.(iv) For p₁q₁ <η₁λ₁,p₂q₂ <μ₂λ₂,p₁q₂ <μ₁λ₂,p₂q₁ =μ₂λ₁,if b≥max{2^β+1(λ₀ + 3c₂²),2^β+1c₁^{-2q1/(α+1)}},b≥2^α+1(λ₀ + 3c₂²,), then theglobal solution of (6) exists.In the fifth chapter, we consider a class of parabolic systems with fast diffusion,and with exponential reaction and boundary flux terms.We study the global existence and blow-up property of the solutons, whereΩ(?) R^N is a bounded domain with smooth boundary, m q₁, p₂, q₃, p₄ are real numbers, n is the outnormal to (?)Ω, u₀(x), v₀(x) are continuous functions satisfying thefollowing compatible conditions.(?)u/(?)v=e^p3u0+q3v0,(?)v/(?)v=e^p4u0+q4v0,x∈(?)Ω.We haveTheorem 7. Let 1≤m < 2. If 0 < p₁q₂ < 1,0 <α₁,α₂ < 1,p₁≤p₃,q₁≤q₃,p₂≤p₄,q₂≤q₄; then the global solution of (7) exists.Theorem 8. Let m > 2. If p₁q₂ > 1,0 <α₁,-1 <α₁,α₂ < 0,p₁≥2_p3,q₁≥2_q3,p₂≥2_p4,q₂≥2_q4,then the solution of (7) blows up in finite time.