A Singular Serturbation Problem For An Envelope Equation In Plasma Physics

This paper includes four chapter: the introduction is in the first chapter;the existence and the uniqueness of the local solution and the global solution for the Cauchy problem of the nonlinear plasma ware equation are proved in the second chapter,and the blow-up of the solution for the Cauchy problem of the nonlinear plasma ware equation is discussed by the concavity methed under some conditions in this chapter. the convergence of (NLW_ω)is discussed in the third chapter. Case of the unbounded initial data is studied in the fourth,these are new resultsIn the second chapter,we study the existence and uniqueness of the local solution in the following Cauchy problem for the nonlinear wave equation where E_ω(x, t = 0) and (?)_tE_ω(x, t = 0) are given intial value functions, the first equation is writed (NLW_ω). For this purpose,we first conside the following linearAfter the existence and uniqueness of the local solution to the problem (0.2) are proved, using the contraction mapping principle we can prove the existence and uniqueness of the local solution to the nonlinear problem. we apply Levine's concavity methods [21]to prove the solution of problem (0.1) blows-up in finite time,The main results are the following:Theorem 2.1 Assume that E₀∈H^s(Rⁿ), E₁∈H^s-1(Rⁿ), then (?)T₀ > 0, such that there exists a unique solution E_ω∈C([0, T₀]; H^s), (?)_tE_ω∈C([0,T₀]; H^s-1), (?)_t²E_ω∈C([0,T₀];L²) satisfying (NLW_ω) with E_ω(0) = E₀å’Œ(?)_tE_ω(0) = E₁, ||E_ω||_L∞(0,T₀;H^s) is moreover bounded, and this bound as well as T₀ only depend on ||E₀||_H^s and ||E₁||_H^s-1/ω.Theorem 2.2 Suppose that the nonlinearity satisfies f(|u|)·u≥(1+2α)F(u), then (1⁰) if∈(0)≤0,β= -∈(0), then there exists t→T < T₀^*, such as ||E_ω||₂→∞. where Theorem 2.3 If f satisfies |f(|E|²)|≤K|E|^2σ,σn < 2,E₀∈H^s(Rⁿ),E₁∈H^s-1 (Rⁿ) ,thenωis sufficiently large , the solution to (NLW_ω) with initial data E_ω(0) = E₀ and (?)_tE_ω(0) = E₁ exists globally and E_ω∈L^∞(0,∞;H^s).In chapter three,we study the relation of the equation and its limit equation this eqution is writed NLS ,the main results are following:Theorem 3.1 Let E₀^ω→E₀, (ω→∞), in H^s(Rⁿ),E₁^ω, E'₁^ωin H^s-1(Rⁿ) such that E₁^ω, E'₁^ωare bounded in L²(Rⁿ) and 1/ωD^s-1→0(ω→∞) , 1/ωD^s-1E'₁^ω→0(ω→∞) in L²(Rⁿ)λ∈R, then exists a time T₁ ,ωis sufficiently large , depending only on ||E₀||_H^s, such that solution E_ωto NLW_ωwith the initial data E_ω(0) = E₀^ω,(?)_tE_ω(0) = E₁^ω+ e^iω2λE'₁^ωexists on [0, T₀] , ifωis sufficiently large,and the solution E to (NLS) with the initial datum E(0) = E₀ exists on the same interval. Furthermore E_ωconverges to E asω→∞in L^∞(0,T₁;H²).Theorem3.2 Let E₀^ω→E₀(ω→∞) in H^s(Rⁿ), E₁^ω, E'₁^ωin H^s-1(Rⁿ) such that E₁^ω→E₁(ω→∞), E'₁^ω→E'₁(ω→∞) in L²(Rⁿ) and 1/ωD^s-1E₁^ω→0(ω→∞), 1/ωD^s-1E'₁^ω→0(ω→∞) in L²(Rⁿ). Let E_ωbe the solution to Let g be the solution toThen (?)_tE_ω- (?)_tE - e^2iω2tS(-t)[E₁ + e^iω2λE'₁ - i/2ΔE₀ + 1/2if(|E₀|²)E₀] - e^2iω2tg(x,t)→0(ω→∞),in L^∞(0,T₂,L²) for T₂ depending only on ||E₀||_H^s.Theorem3.3 Under the same asumption as for Thorem 3.2,let T_ωbe the existence time of to E_ωand T(E₀) the existence time of the solution E to (NLS)with the datum E(0) = E₀. thenMoreover (?)T < T(E₀), E_ω→E in L^∞(0,T;H²), (?)_tE_ω- (?)_tE - e^2iω2tS(-t)(E₁ - i/2ΔE₀ + 1/2i|E₀|^2(r-1)E₀) - e^2iω2tg(x,t)→0,ω→+∞in L^∞(0,T;L²). Where g is definded as in Theorem3.2, and 1/ω(?)_tE_ω→0(ω→∞) in L^∞(0,T;H^s-1).In chapter four , Case of unbounded initial data .The main results are following :Theorem4 Let E₀^ω→E₀, (ω→∞) in H^s and E₁^ω→E₁,(ω→∞) in H^s-1, E_ω, is the solution toLet (T|⁾_ω= T_ω(E₀^ω,E₁^ω) is the existence time of E_ω. T(E₀) is the existence time of E. E is the solution to Then Moreover (?)T < T(E₀), E_ω→E(ω→∞) in L^∞(0, T(E₀),H^s), and 1/ω(?)_tE_ω- e^iω2λ.e^2iω2t[S(-t)E₁ +φ(x,t)]→0(ω→∞) in L^∞(0,T;H^s-1) Whereφis the solution to...