Pavel I. Naumkin

Damped wave equation with a critical nonlinearity

We study large time asymptotics of small solutions to the Cauchy problem for nonlinear damped wave equations with a critical nonlinearity { ∂ 2 t u + ∂ t u - Δu + λu 1+2 n = 0, x ∈ R n , t > 0, u(0, x) = eu 0 (x), ∂ t u(0, x) = eu 1 (x), x ∈ R n , where e > 0, and space dimensions n = 1, 2,3. Assume that the initial data uo ∈ H δ,0 n H 0,δ , u 1 e H δ-1,0 n H -1,δ , where δ > n 2, weighted Sobolev spaces are H l,m = {Φ e L2; m l Φ(x)∥ L 2 = √1 + x 2 . Also we suppose that λθ 2/n > 0, ∫u 0 (x) dx > 0, where Then we prove that there exists a positive e 0 such that the Cauchy problem above has a unique global solution u ∈ C ([0, oo); H δ,0 ) satisfying the time decay property ∥u(t)-eθG(t,x)e -φ(t) ∥ Lp ≤ Ce 1+2 n g -1-n 2 (t) -n 2(1-1/p) for all t > 0, 1 ≤ p ≤ ∞, where e ∈ (0, e 0 ].

Modified Wave Operator for a System of Nonlinear Schrödinger Equations in 2d

We consider a system of nonlinear Schrödinger equations with quadratic nonlinearities in two space dimensions. We prove the existence of modified wave operators or wave operators under some mass conditions.

Global existence and time decay of small solutions to the landau-ginzburg type equations

We study the Cauchy problem for the nonlinear dissipative equations (0.1) uo∂u-αδu + Β|u|2/nu = 0,x ∃ Rn,t } 0,u(0,x) = u0(x),x ∃ Rn, where α,Β ∃ C, ℜα 0. We are interested in the dissipative case ℜα 0, and ℜδ(α,Β)≥ 0, θ = ¦∫ u0(x)dx| ⊋ 0, where δ(α, Β) = ##|α|n-1nn/2 / ((n + 1)|α|2 + α2n/2. Furthermore, we assume that the initial data u0 ∃ Lp are such that (1 + ¦x¦)αu0 ∃ L1, with sufficiently small norm ∃ = (1 + ¦x¦)α u0 1 + u0 p, wherep 1, α ∃ (0,1). Then there exists a unique solution of the Cauchy problem (0.1)u(t, x) ∃ C ((0, ∞); L∞) ∩ C ([0, ∞); L1 ∩ Lp) satisfying the time decay estimates for allt0 u(t)||∞ Cɛt-n/2(1 + η log 〈t〉)-n/2, if hg = θ2/n 2π ℜδ(α, Β) 0; u(t)||∞ Cɛt-n/2(1 + Μ log 〈t〉)-n/4, if η = 0 and Μ = θ4/n 4π)2 (ℑδ(α, Β))2 ℜ((1 + 1/n) υ1-1 υ2) 0; and u(t)||∞ Cɛt-n/2(1 + κ log 〈t〉)-n/6, if η = 0, Μ = 0, κ 0, where υl,l = 1,2 are defined in (1.2), κ is a positive constant defined in (2.31).

On the asymptotics for cubic nonlinear Schrödinger equations

We consider the Cauchy problem for the cubic nonlinear Schrödinger equation where We prove the global existence of small solutions , if the initial data u 1 belong to some analytic function space and are sufficiently small. For the coefficients λ j we assume that there exists θ 0 > 0 such that for all and also we suppose that the initial data are such that where ϵ is a small positive constant depending on the size of initial function in a suitable norm. We also find the large time asymptotic formulas for solutions. In the short range region the solution has an additional logarithmic time decay comparing with the corresponding linear case.

ASYMPTOTICS FOR FRACTIONAL NONLINEAR HEAT EQUATIONS

The Cauchy problem is studied for the nonlinear equations with fractional power of the negative Laplacian [ left{ begin{array}{@{}r@{,}c@{,}l@{qquad}l} u_{t}+( -Delta ) ^{{alpha}/{2}}u+u^{1+sigma } &=&0, & xin {mathbf{R}}^{n},text{ }t>0, [4pt] u( 0,x) &=& u_{0} ( x), x mathbf{L}^{infty }cap mathbf{L}^{1,a}cap mathbf{C})

Quadratic nonlinear Klein-Gordon equation in one dimension

and the large time asymptotics are obtained.

Final state problem for the cubic nonlinear Klein–Gordon equation

We study the initial value problem for the quadratic nonlinear Klein-Gordon equation vtt + v − vxx = λv2, t ∈ R, x ∈ R, with initial conditions v(0, x) = v0(x), vt(0, x) = v1(x), x ∈ R, where v0 and v1 are real-valued functions, λ ∈ R. Using the method of normal forms of Shatah [“Normal forms and quadratic nonlinear Klein-Gordon equations,” Commun. Pure Appl. Math. 38, 685–696 (1985)], we obtain a sharp asymptotic behavior of small solutions without the condition of a compact support on the initial data, which was assumed in the previous work of J.-M. Delort [“Existence globale et comportement asymptotique pour lequation de Klein-Gordon quasi-lineaire a donnees petites en dimension 1,” Ann. Sci. Ec. Normale Super. 34(4), 1–61 (2001)].

Final state problem for Korteweg–de Vries type equations

We study the final state problem for the nonlinear Klein–Gordon equation, utt+u−uxx=μu3,u2002t∊R,x∊R, where μ∊R. We prove the existence of solutions in the neighborhood of the approximate solutions 2u2009Reu2009U(t)w+(t), where U(t) is the free evolution group defined by U(t)=F−1e−it⟨ξ⟩F, ⟨x⟩=1+x2, F and F−1 are the direct and inverse Fourier transformations, respectively, and w+(t,x)=F−1(u+(ξ)e(3/2)iμ⟨ξ⟩2|u+(ξ)|2logu2009t), with a given final data u+ is a real-valued function and ‖⟨ξ⟩3⟨i∂ξ⟩u+(ξ)‖L∞ is small.

A quadratic nonlinear Schrödinger equation in one space dimension

We study the final state problem for the Korteweg–de Vries type equations: ut−1∕ρ∣∂x∣ρ−1ux=λu2ux, (t,x)∊R+×R,∥u(t)−FS(t)∥L2→0 as t→∞, where λ∊R, the function FS(t) we call a final state, defined by the final data u+. We show that there does not exist a nontrivial solution of this equation in the case of FS(t)=U(t)u+, where U(t) is the free evolution group of this equation. We construct the modified wave operator for the Korteweg–de Vries type equations under the conditions that the final data u+ arc real-valued functions and the Fourier transform u+(ξ) vanishes at the origin.

Nonexistence of the usual scattering states for the generalized Ostrovsky-Hunter equation

Abstract In this paper, we study the Cauchy problem for the quadratic derivative nonlinear Schrodinger equation (∗) iu t +u xx =( u x ) 2 , (t,x)∈ R 2 , u(0,x)=u 0 , x∈ R . We suppose that the initial data are small in the weighted Sobolev space H 3,1 ( R ) then we prove that there exist global in time small solutions to the Cauchy problem (∗) . We study the large time behavior of solutions and construct the modified asymptotics for large time values.