The Cauchy Problem For Generalized Bbm-burgers-ginzburg-landau Equation

The paper consists of three chapters. The first chapter is the introduction. In thesecond chapter, we will study the existence and uniqueness of the local solution to theCauchy problem for generalized BBM-Burgers-Ginzburg-Landau Equation; In the thirdchapter, we will study the existence and uniqueness of the global solution to the Cauchyproblem for the above mentioned equation by the extension theorem of solution, and wewill also study the decay property of the solution.We will study the following Cauchy problem for generalized BBM-Burgers-Ginzburg-LandauEquationwhere v(x,t) denotes the unknown function, G(s),g(s),f_j(s),(j = 1,2,…,n) are givennonlinear function,α＞0,β＞0,γ＞0,δ≠0 are constants, v₀(x) is given initial valuefunction.For simplicity, by the scaling transformequation (1) can be written asWithout loss of generality, we will study the following Cauchy problem The main results axe the following:Theorem 1 Suppose that s＞n/2,φ(x)∈H^s(Rⁿ),h(ξ),f_j(ξ), (j=1,2,…,n),G(ξ)∈C^k(R),k = [s] + 1, h(0) = f_j(0) = G(0) = 0, (j=1,2,…,n),then the problem (3), (4) hasa unique local solutionwhere [0, T₀) is the maximal time interal of existence of u(x, t). Moreover, ifthen T₀ =∞.Theorem 2 Suppose that s＞n/2,φ(x)∈H^s(Rⁿ),h(ξ),f_j(ξ), (j=1,2,…,n),G(ξ)∈C^k(R),k= [s] + 1,h(0) = f_j(0) = G(0) = 0, (j = 1,2,…,n), [0,T₀) is the maximal timeinterval of the solution u(x,t) to the Cauchy problem (3),(4). Then T₀＜∞if only ifTheorem 3(n=3) Suppose that s＞9/2,φ(x)∈H^s(R³),h(ξ),f_j(ξ),(j=1,2,3),G(ξ)∈C^k(R),k=[s]+1,h(0)=f_j(0)=G(0)=0,|f_j(ξ)|≤C|ξ|³,(j=1,2,3),ξG(ξ)≤-C|ξ|^2+λ,(λ＞0,ξ∈R),h'(ξ)≥0,|h"(ξ)|≤C(1+|ξ|^μ),μ＞0,10μ＜λ+2,whereC＞0 is constant,then the problem (3), (4) has a unique global solutionRemark 1 Under the conditions of Theorem 3, if s＞11/2, then the Cauchy problem(3), (4) admits a unique global classical solutionTheorem 4(n=2) Suppose that s＞4, s＞4,φ(x)∈H^s(R²),h(ξ),f_j(ξ),(j=1,2),G(ξ)∈C^k(R),k=[s] + 1,h(0) = f_j(0) = G(0) = 0,(j = 1,2),ξG(ξ)≤0,h'(ξ)≥0,|h'(ξ)|≤ C(1+|ξ|²),|G(ξ)|≤C|ξ|³,|f_j(ξ)|≤C|ξ|³,(j=1,2),where C＞0 is constant,then theproblem (3), (4) has a unique global solutionRemark 2 Under the conditions of Theorem 4, if s＞5, then the Cauchy problem(3), (4) admits a unique global classical solutionTheorem 5(n=1) Suppose that s＞7/2,φ(x)∈H^s(R),h(ξ),f(ξ), G(ξ)∈C^k(R),k=[s]+1,h(0)=f(0)=G(0)=0,ξG(ξ)≤0,h'(ξ)≥0,then the problem (3), (4) has a uniqueglobal solutionRemark 3 Under the conditions of Theorem 5, if s＞9/2, then the Cauchy problem(3), (4) admits a unique global classical solutionTheorem 6 Suppose that n = 3,s＞9/2,φ(x)∈H^s(Rⁿ),h(ξ),f_j(ξ),(j=1,2,…,n),G(ξ)∈C^k(R),k=[s]+1,h(0)=f_j(0)=G(0)=0,|f_j(ξ)|≤C|ξ|³,(j=1,2,…,n),ξG(ξ)≤-C|ξ|^2+λ,(λ＞0,ξ∈R),G'(ξ)≤-B＜0,h'(ξ)≥A＞0,|h"(ξ)|≤G(1+|ξ|^μ),μ＞0,10μ＜λ+2, where C＞0 is constant, then the global solution of the problem (3), (4)has the decay property:whereλ=min{A+1,bB} is constant.Remark 4 Under the conditions of Theorem 4 (n = 2) or Theorem 5 (n = 1),ifthen the global solution of the problem (3), (4) also has the decay property (7).