International Seminar. Day on Diffraction. Proceedings (IEEE Cat. No.99EX367), 1999
We study the following initial-boundary value problem for the nonlocal Whitham equation u/sub t/+... more We study the following initial-boundary value problem for the nonlocal Whitham equation u/sub t/+N(u)+Ku=0, (x,t) /spl isin/R/sup +//spl times/R/sup +/, u(x,0)=u~(x), x/spl isin/R/sup +/, where the nonlinearity is N(u)=u/sub x/u and K is the pseudodifferential operator on the half-line of order /spl alpha/ satisfying 1</spl alpha/<2 and some dissipative conditions. We prove that if the initial data are such that x/sup /spl delta//u~/spl isin/L/sup 1/, with /spl delta//spl isin/(0, 1/2 ) and the norm /spl par/u~/spl par/X/sup +//spl par/x/sup /spl delta//u~/spl par/(L/sup 1/) is sufficiently small, where X={/spl psi//spl isin/(L/sup 1/), /spl psi/'/spl isin/L/sup 1/;/spl par//spl psi//spl par/x=/spl par//spl psi//spl par/(L/sup 1/)+/spl par//spl psi//sub x//spl par/(L/sup 1/)</spl infin/}, then there exists a unique solution u/spl isin/C ([0, +/spl infin/); L/sup 2/)/spl cap/C(R/sup +/, H/sup 1/) of the initial-value problem (1), where H/sup k/ is the Sobolev space with norm /spl par//spl phi//spl par/(H/sup k/)=/spl par/(1-/spl part//sub 2//sup x/)k/2/spl phi//spl par/(L/sup 2/). We also study large time asymptotics of the solutions.
This paper explores an asymptotic approach to the solution of a non-linear transmission line mode... more This paper explores an asymptotic approach to the solution of a non-linear transmission line model. The model is based on a set of non-linear partial differential equations without analytical solution. The perturbations method is used to reduce the system of non-linear equations to a single non-linear partial differential equation, the modified Korteweg–de Vries equation (KdV). By using the Laplace transform,
Journal of Mathematical Analysis and Applications, 2012
Abstract We consider the initial-boundary value problem on a half-line for the evolution equation... more Abstract We consider the initial-boundary value problem on a half-line for the evolution equation ( ∂ t + R 1 2 ∂ x 2 + K ) u ( x , t ) = − u x u , where R α ϕ = 1 2 Γ ( α ) sin ( π 2 α ) ∫ 0 + ∞ ϕ ( y ) | x − y | 1 − α d y is the modified Riesz potential and K u = 1 2 π i θ ( x ) ∫ − i ∞ i ∞ e p x p tanh | p | | p | ( u ˆ ( p , t ) − u ( 0 , t ) p ) d p , where θ ( x ) is the Heaviside step function. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial-boundary value problem and the asymptotic behavior of solutions for large time.
We study the initial-boundary value problem for the fractional Landau–Ginzburg equations on a seg... more We study the initial-boundary value problem for the fractional Landau–Ginzburg equations on a segment. The aim of this paper is to prove the global existence of solutions to the inital-boundary value problem and to find the main term of the asymptotic representation of solutions.
The large-time asymptotic behaviour is studied for a system of non-linear evolution dissipative e... more The large-time asymptotic behaviour is studied for a system of non-linear evolution dissipative equations ; u{sub t}+N(u,u)+Lu=0, x element of R{sup n}, t>0; u(0,x)=u-tilde(x), x element of R{sup n}; where L is a linear pseudodifferential operator Lu=F-bar{sub {xi}}{sub {yields}}{sub x}(L({xi}u-bar({xi})) and the non-linearity N is a quadratic pseudodifferential operator N(u,u{sub =}F-bar{sub {xi}}{sub {yields}}{sub x}{integral}{sub R{sup n}}A{sup kl}(t,{xi},y)u-bar{sub k}(t,{xi}-y)u-bar{sub l}(t,y)dy, where u-bar{identical_to}F{sub x{yields}}{sub {xi}}u is the Fourier transform. Under the assumptions that the initial data u-tilde element of H{sup {beta}}{sup ,0} intersection H{sup 0,{beta}}, {beta}>n/2 are sufficiently small, where H{sup n,m}={l_brace}{phi} element of L{sup 2}: || {sup m} {phi}(x)||{sub L{sup 2}} ={radical}(1+x{sup 2}), is a Sobolev weighted space, and that the total mass vector M={integral}u-tilde(x)x{ne}0 is non-zero it is proved that the leading term in the large-time asymptotic expansion of solutions in the critical case is a self-similar solution defined uniquely by the total mass vector M of the initial data.
The Cauchy problem is studied for the nonlinear equations with fractional power of the negative L... more The Cauchy problem is studied for the nonlinear equations with fractional power of the negative Laplacian { ut+(−Δ)α/2u+u1+σ=0, x∈Rn, t>0,u(0,x)=u0(x), x∈Rn, where α ∈ (0,2), with critical σ = α/n and sub‐critical σ ∈ (0,α/n) powers of the nonlinearity. Let u0∈ L1,a∩ L∞ ∩ C, u0(x) ⩾ 0 in Rn, θ = ∫Rnu0(x)dx>0. . The case of not small initial data is of interest. It is proved that the Cauchy problem has a unique global solution u ∈ C([0,∞); L∞ ∩ L1,a∩ C) and the large time asymptotics are obtained.
International Seminar. Day on Diffraction. Proceedings (IEEE Cat. No.99EX367), 1999
We study the following initial-boundary value problem for the nonlocal Whitham equation u/sub t/+... more We study the following initial-boundary value problem for the nonlocal Whitham equation u/sub t/+N(u)+Ku=0, (x,t) /spl isin/R/sup +//spl times/R/sup +/, u(x,0)=u~(x), x/spl isin/R/sup +/, where the nonlinearity is N(u)=u/sub x/u and K is the pseudodifferential operator on the half-line of order /spl alpha/ satisfying 1</spl alpha/<2 and some dissipative conditions. We prove that if the initial data are such that x/sup /spl delta//u~/spl isin/L/sup 1/, with /spl delta//spl isin/(0, 1/2 ) and the norm /spl par/u~/spl par/X/sup +//spl par/x/sup /spl delta//u~/spl par/(L/sup 1/) is sufficiently small, where X={/spl psi//spl isin/(L/sup 1/), /spl psi/'/spl isin/L/sup 1/;/spl par//spl psi//spl par/x=/spl par//spl psi//spl par/(L/sup 1/)+/spl par//spl psi//sub x//spl par/(L/sup 1/)</spl infin/}, then there exists a unique solution u/spl isin/C ([0, +/spl infin/); L/sup 2/)/spl cap/C(R/sup +/, H/sup 1/) of the initial-value problem (1), where H/sup k/ is the Sobolev space with norm /spl par//spl phi//spl par/(H/sup k/)=/spl par/(1-/spl part//sub 2//sup x/)k/2/spl phi//spl par/(L/sup 2/). We also study large time asymptotics of the solutions.
This paper explores an asymptotic approach to the solution of a non-linear transmission line mode... more This paper explores an asymptotic approach to the solution of a non-linear transmission line model. The model is based on a set of non-linear partial differential equations without analytical solution. The perturbations method is used to reduce the system of non-linear equations to a single non-linear partial differential equation, the modified Korteweg–de Vries equation (KdV). By using the Laplace transform,
Journal of Mathematical Analysis and Applications, 2012
Abstract We consider the initial-boundary value problem on a half-line for the evolution equation... more Abstract We consider the initial-boundary value problem on a half-line for the evolution equation ( ∂ t + R 1 2 ∂ x 2 + K ) u ( x , t ) = − u x u , where R α ϕ = 1 2 Γ ( α ) sin ( π 2 α ) ∫ 0 + ∞ ϕ ( y ) | x − y | 1 − α d y is the modified Riesz potential and K u = 1 2 π i θ ( x ) ∫ − i ∞ i ∞ e p x p tanh | p | | p | ( u ˆ ( p , t ) − u ( 0 , t ) p ) d p , where θ ( x ) is the Heaviside step function. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial-boundary value problem and the asymptotic behavior of solutions for large time.
We study the initial-boundary value problem for the fractional Landau–Ginzburg equations on a seg... more We study the initial-boundary value problem for the fractional Landau–Ginzburg equations on a segment. The aim of this paper is to prove the global existence of solutions to the inital-boundary value problem and to find the main term of the asymptotic representation of solutions.
The large-time asymptotic behaviour is studied for a system of non-linear evolution dissipative e... more The large-time asymptotic behaviour is studied for a system of non-linear evolution dissipative equations ; u{sub t}+N(u,u)+Lu=0, x element of R{sup n}, t>0; u(0,x)=u-tilde(x), x element of R{sup n}; where L is a linear pseudodifferential operator Lu=F-bar{sub {xi}}{sub {yields}}{sub x}(L({xi}u-bar({xi})) and the non-linearity N is a quadratic pseudodifferential operator N(u,u{sub =}F-bar{sub {xi}}{sub {yields}}{sub x}{integral}{sub R{sup n}}A{sup kl}(t,{xi},y)u-bar{sub k}(t,{xi}-y)u-bar{sub l}(t,y)dy, where u-bar{identical_to}F{sub x{yields}}{sub {xi}}u is the Fourier transform. Under the assumptions that the initial data u-tilde element of H{sup {beta}}{sup ,0} intersection H{sup 0,{beta}}, {beta}>n/2 are sufficiently small, where H{sup n,m}={l_brace}{phi} element of L{sup 2}: || {sup m} {phi}(x)||{sub L{sup 2}} ={radical}(1+x{sup 2}), is a Sobolev weighted space, and that the total mass vector M={integral}u-tilde(x)x{ne}0 is non-zero it is proved that the leading term in the large-time asymptotic expansion of solutions in the critical case is a self-similar solution defined uniquely by the total mass vector M of the initial data.
The Cauchy problem is studied for the nonlinear equations with fractional power of the negative L... more The Cauchy problem is studied for the nonlinear equations with fractional power of the negative Laplacian { ut+(−Δ)α/2u+u1+σ=0, x∈Rn, t>0,u(0,x)=u0(x), x∈Rn, where α ∈ (0,2), with critical σ = α/n and sub‐critical σ ∈ (0,α/n) powers of the nonlinearity. Let u0∈ L1,a∩ L∞ ∩ C, u0(x) ⩾ 0 in Rn, θ = ∫Rnu0(x)dx>0. . The case of not small initial data is of interest. It is proved that the Cauchy problem has a unique global solution u ∈ C([0,∞); L∞ ∩ L1,a∩ C) and the large time asymptotics are obtained.
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Papers by Elena Kaikina