Several Classes Of Nonlinear Evolution Equations Definite Solution Of The Problem

In this paper, we study the existence, uniqueness of local and global solutions, the asymptotic behavior of global solutions for the initial boundary value problems and Cauchy problem to some nonlinear evolution equations, and give the sufficient conditions of blowup of solutions for some problems above. The main results include the following five parts:In Chapter 2, the existence, uniqueness and regularities of the global generalized solution and global classical solution for the periodic boundary value problem and the Cauchy problem of the general cubic double dispersion equationu_tt-u_xx-au_xxtt+bu_x⁴-du_xxt=f(u)_xx, x∈R, t＞0 (1)are proved and the sufficient conditions of blow-up of the solutions for the Cauchy problems in finite time are given. The initial values of equation (1) are given byu(x,0)=u₀(x), u_t(x,0)=u₁(x), x∈R. (2)The main results are stated as follows:Theorem 1 Suppose that f∈C^k+1(R), f'(s) is bounded below, u₀∈H^k+1(R), u₁∈H^k(R). If k-1≥2(k=1+P₁= 2+p₂, P₁,P₂≥0), then the Cauchy problem (1), (2) has a unique generalized global solution u(x,t)∈V([0,T]×R, k-1≥2)={when k-1≥2, v(x, t) has continuous derivatives v_(x^s)(t^r)(x,t), 0≤s+r≤k-1, r=0, 1, 2 and generalized derivatives v_(x^s)(t^r)(x,t), 0≤s+r≤k+1, r=0, 1, 2, 3, (x, t)∈[0,T]×R}Theorem 2 Suppose that f∈C^k+1(R), f'(s) is bounded below, u₀∈H^k+1(R), u₁∈H^k(R). If k-1≥4(k=1+p₁=2+p₂, P₁, P₂≥0), then the Cauehy problem (1), (2) has a unique classical global solution u(x,t)∈V([0,T]×R, k-1≥4).Theorem 3 Assume that d≥0, f∈C(R), u₀∈H¹(R), u₁∈L²(R), G(s)=∫₀^sf((?))d(?), G(u₀)∈L¹(R) and there areβ＞0 andε>0 such that2f(s)s≤[4+8β+2a^-1ε^-1d²]G(s)+(4β+ε^-1a^-1d²-ε)s², (?)s∈R.Then the generalized solution u(x,t)∈V([0,T]×R, k-1≥3) or the classical solution u(x,t)∈V([0,T]×R, k^-1≥4) of the Cauchy problem (1), (2) blows-up in finite time if one of the following conditions holds: (1) E₁(0)=(?):(2) E₁(0)=0 and (?);(3) E₁(0)＞0 and (?) where E₁(t)=(?)Theorem 4 Assume that d>O, f∈C(R), G(s)=∫₀^sf((?))d(?), u₀∈H¹(R), u₁∈L²(R), G(u₀)∈L¹(R) and there are constantsβ>0 andε>0 such thatsf(s)≤(2+4β+a^-1ε(-1_d²)G(s), (?)_s∈R.Then the solution u(s,t)(see Theorem 3) of the problem (1), (2) blows-up in finite time if the following conditions hold:(1) E₁(0)≤0;In Chapter 3. we first prove the existence and uniqueness of the local classical solution to the periodic boundary value problem for a system of generalized IMBq equations arised from DNAu_tt=a₁u_xx-a₂(u²)_xx+(a₁/2)(v²)_xx+a₃u_xxtt, x∈R, t＞0, (3)v_tt=a₁(uv)_xx+a₃v_xxtt, x∈R. t＞0. (4)Next the existence and uniqueness of the local classical solution for the Cauchy problem to this system are proved by the sequence of the periodic boundary value problems. The system (3), (4) owns the initial conditions:u(x,0)=u₀(x), u_t(x,0)=u₁(x), x∈R, (5)v(x,0)=v₀(x), v_t(x,0)=v₁(x), x∈R. (6)The main results are as follows:Theorem 5 If u₀, u₁, v₀, v₁∈H⁵(R), then there exists a unique classical solution u(x,t), v(x,t) of the Cauchy problem (3)-(6) in [0,t₂]×R; where 0＜t₂＜t_?, t_?=2/(K(?)^1/2) and‖·‖=‖·‖L₂(R),(?)=‖u₁‖²+a₃‖u_1x‖²+‖u_1x³â€–²+a₃‖u_1x⁴‖²+‖u₀‖²+a₁‖u_0x‖²+a₁‖u_0x⁴‖²+‖v₁‖²+a₃‖v(1x)‖²+‖v(1x³)‖²+a₃‖v_1x⁴‖²+‖v₀‖²+‖v_0x⁴‖²+1In Chapter 4, we prove the Cauchy problem for the nonlinear pseudo-paraboic equationv_t-av_xxt-βv_xx+(?)v_x+f(v)_x=(?)(v_x)+g(v)-αg(v)_xx, x∈R, t＞0,v(x,0)=v₀(x), x∈Radmits a unique global generalized solution in C¹([0,∞); H^s(R)), a unique global classical solution and asymptotic behavior of the solution. We also prove the Cauchy problemv_t-αv_xxt-βv_xx=g(v)-αg(v)_xx, x∈R, t＞0,v(x,0)=v₀(x), x∈Rhas a unique global generalized solution in C¹([0,∞); W^m,P(R)∩L^∞(R)), a unique global classical solution and asymptotic behavior of the solution. By the scaling transformationv(x,t)=u((1/(?))x,t),The above mentioned two Cauchy problem become Cauchy problemu_t-U_xxt-(β/α)u_xx+(Î¥/(?))u_x+(1/(?))f(u)_x=(1/(?))(?)((1/(?))u_x)_x+g(u)-g(u)_xx, x∈R, t＞0, (7)u(x,0)=u₀(x), x∈R (8)and Cauchy problemu_t-(β/α)u_xx-u_xxt=g(u)-g(u)_xx, x∈R, t＞0, (9) u(x,0)=u₀(x), x∈R, (10)respectively. The main results are as follows:Theorem 6 Assume that the following conditions hold:(1) s≥2, u₀∈H^s;(2) f∈C^[s](R);(3) g∈C^[s]+1(R), g(0)=0 and there is a constant C₀, such that for any s∈R, (d/(ds))g(s)=g'(s)≤C₀;(4) (?)∈C^[s](R) and for any s∈R, (?)(s)s≥0 or there is a constant C₁, such that (?)'(s)≥C₁.Then the Cauchy problem (7), (8) admits a unique global generalized solution u∈C₁([0,∞); H^s.Theorem 7 Assume that u(x,t) is a global generalized solution or a global classical solution for the problem (7), (8). If u₀∈H¹, f∈C¹(R), g∈C²(R), g(0)=0 and (?)s∈R, g'(s)≤C₀＜0; (?)(s)s≥0 or (?)∈C¹(R), there is a constant C₁≥βsuch that (?)'(s)≥C₁, (?)s∈R. Then u(x,t) satisfies‖u(.,t)‖²+‖u_x(.,t)‖²â‰¤(‖u₀‖²+‖U_0x‖²)e(2c₀t)Theorem 8 Suppose that(1) u₀∈W^m,p∩L^∞(m≥2 is an integer);(2) g∈C^m+1(R), g(0)=0 and (?)s∈R, g(s)s≤0 or there is a constant C₀ such that g'(s)≤C₀, (?)s∈R.Then the Cauchy problem (9), (10) admits a unique generalized global solution u∈C²([0,∞); W^m,p∩L^∞).Theorem 9 Suppose that the conditions of Theorem 8 hold, g∈C^k+m+1(R), where k≥0 is arbitrary integer. Then the generalized solution u(x,t) for the problem (9), (10) belongs to C^k+2+l([0,T]; W^m-l,p∩L∞)((?)T＞0), 0＜l＜m. If k=0, l=0, m＞2+(1/p), then the problem (9), (10) has a unique global classical solution u∈C²([0,∞); W^m,p∩L^∞), i.e., u∈C²([0,∞); C²(R)∩L∞).Theorem 10 Suppose that u(x,t) is a global generalized solution or a global classical solution for the problem (9), (10). If u₀∈H¹, g∈C²(R), g(0)=0 and (?)s∈R, g'(s)≤C₀≤0, then u(x,t) satisfies‖u(.,t)‖²+‖u_x(.,t)‖²â‰¤(‖u₀‖²+‖u_0x‖²)e^2C₀t, t≥0 In Chapter 5, we consider the existence and the uniqueness of the solution for the Cauchy problem of a class of Boussinesq equationu_tt-u_xxtt+U_xxxxtt=-αu_xxxx+u_xx+f(u)_xx, x∈R, t＞0, (11)u(x,0)=(?)(x), u_t(x,0)=Ψ(x), x∈R. (12)We prove that the global existence and finite time blow up of the solution for the problem by aid of the potential well method.The main results are stated as follows:Theorem 11 Suppose that 1≤s＜p, (?)∈H^s,Ψ∈H^s∩H^-1, E(0)≤d. When‖(?)‖²+α‖(?)_x‖²â‰¤(2(p+1)d)/(p-1), the Cauchy problem (11), (12) admits a unique global solution u∈C¹([0,∞); H^s); When‖(?)‖²+α‖(?)_x‖²ï¼ž(2(p+1)d)/(p-1),and ((-(?)_x²)^-1/2(?),(-(?)_x²)^-1/2Ψ+((?),Ψ)+((?)_x,Ψ_x)≥0 as E(0)=d, the solution of the problem (11), (12) blows up in finite time.In Chapter 6, we dicuss the following initial boundary value problem of BBM-Burgers-Ginzburg-Landau equationu_t-α△u_t+β△u+γ△²u+(?)f_j(u)_xj=δ△g(u)+G(u), (x,t)∈Q_T, (13)u=O,â–³u=O, (x,t)∈δΩ×[O,T], (14)u(x,0)=u₀(x), x∈(?); (15)the boundary value problem of equation (13)u=O, (δu/δv)=0, (x,t)∈δΩ×[O,T], (16)u(x,O)=u₀(x), x∈(?) (17)and the boundary value problem of equation (13)(δu/δv), (δ△u/δv)=0, (x,t)∈[0,T], (18)u(x,0)=u₀(x), x∈(?). (19)We prove the existence and the uniqueness of the global generalized solution of the above problems, and give the sufficient conditions of blowup of the solution for the problem (13), (18), (19) in finite time; we also show the asymptotic behavior of the solutions for the problem (13)-(15), the problem (13), (16), (17) and the problem (13), (18), (19).The main results are as follows: Theorem 12 Assuming that the following conditions hold:(1) u₀∈H⁴(Ω);(2) f_i∈C³(R)(i=1,2,3) and there exists a constant K＞0, such that (?)s∈R,|f'_j(s)|≤K, i=1,2,3;(3) g∈C⁴(R), g'(s)≥A, and |g'(s)|≤K₁|s|^ξ₁+1, whereξ₁＞0, K₁＞0 and A are constants, 0＜ξ₁＜3;(4) G∈C²(R), G'(s)≤B, and |G'(s)|≤K₂|s|^ξ₂, whereξ₂＞0, K₂＞0 and B are constants, 0＜ξ₂＜8.Then the boundary value problem (13)-(15), or the boundary value problem (13), (16), (17) or the boundary value problem (13), (18), (19) has a unique generalized solution u∈C([0,∞); H⁴(Ω)) and u_t∈L²([0,∞); H²(Ω)).Theorem 13 Suppose that u(x,t) is a generalized solution of problem (13), (18), (19). And the following conditions hold:(1)g∈C²(R), f_j(j=1,2,3)=0;(2)∫_Ωu₀(x)dx=ζ＞0;(3)G∈C²[0,∞) is a positive convex function on [0,∞);(4) G(s) grows fast enough as s→∞so that the integralconverges, then there is a finite time t₀≤T₀, such that(?)(?)|u(x,t)|=∞where T₀ is given by (20).Theorem 14 Assume that(1) u₀∈H¹(Ω);(2) f_j∈C¹(R), (?)_s∈R, |f_j(s)|≤K, j=1,2,3, where K is a positive constant;(3) g∈C²(R), there exists a constant A, such that (?)_s∈R, g'(s)≥A;(4) G∈C¹(R), G(0)=0 and there exists a constant B＞0, such that for all s∈R, G'(s)≤-B;(5) There exists a constant 0＜ε₀＜2B, such that 2δA+2β-K²/ε₀＞0.Then the global generalized solution of the initial boundary value problem (13)-(15), or initial the boundary value problem (13), (16), (17) or the initial boundary value problem (13), (18), (19) has the asymptotic behavior‖u(.,t)‖²+α‖▽u(.,t)‖²â‰¤(‖u₀‖²+α‖▽u₀‖²)e^-λt,whereλ=min{2B-ε₀,(2δA+2β-(K²/ε₀))/α}.