Asymptotics of a solution of a nonlinear system of diffusion of a magnetic field into a substance

We study the asymptotic behavior at large time of a solution to a system of nonlinear integro-differential equations which arises in mathematical modeling of diffusion of a magnetic field into a substance. We establish the corresponding stabilization rate.     

Non-linear parabolic problem associated with the penetration of a magnetic field into a substance

We study the following initial and boundary value problem: In section 1, with u₀ in L²(Ω), f continuous such that f(u) + ? non-decreasing for ? positive, we prove the existence of a unique solution on (0,T), for each T > 0. In section 2 it is proved that the unique soluition u belongs to L²(0, T; H ∩ H²) ∩ L^∞(0, T; H) if we assume u₀ in H and f in C¹(?,?). Numerical results are given for these two cases.     

Large time behavior of solutions to a nonlinear integro-differential system

Asymptotic behavior of solutions as t→∞ to the nonlinear integro-differential system associated with the penetration of a magnetic field into a substance is studied. Initial-boundary value problems with two kinds of boundary data are considered. The first with homogeneous conditions on whole boundary and the second with non-homogeneous boundary data on one side of lateral boundary. The rates of convergence are given too.     

Large time behaviour of solutions for a nonlinear thermoelastic system

We consider a nonlinear thermoelastic system in $\text{R}{}^{\mathrm{n}}$. We prove the well-posedness and give the decay rate of the energy as t→+∞. Our main tools are the semi-group theory and the Fourier splitting method. We extend some results of Kim (SIAM J. Math. Anal. 23 (1992) 889), Liu and Renardy (Appl. Math. Lett. 8 (1995) 1), and improve the work in Lagnese (SIAM Stud. Appl. Math. 10 (1989)) by proving that it is not necessary to add damping term to get the exponential decay of the solution.     

Asymptotics of solutions to the periodic problem for the nonlinear damped wave equation

We study large time asymptotic behavior of solutions to the periodic problem for the nonlinear damped wave equation

On the behavior of solutions to the Cauchy problem for nonlinear degenerate parabolic systems with large time values

Translated from Matematicheskie Zametki, Vol. 57, No. 1, pp. 40–47, January, 1995.     

Global well-posedness of strong solutions to a two-phase model with magnetic field for large oscillations in three dimensions

In this paper, we consider Cauchy problem for a two-phase model with magnetic field in three dimensions. Under some smallness assumptions on the initial data but possibly large oscillations, we obtain the global well-posedness of strong solution as well as its large-time behavior. Compared with Wen and the first author's work [28] where global well-posedness and large time behavior of strong solutions were obtained subject to smallness of initial data in $H^2$ norm, we only need the smallness of initial energy which allows large oscillations of the initial data.     

Large time behavior of solutions to a nonlinear hyperbolic relaxation system with slowly decaying data

We consider the large time asymptotic behavior of the global solutions to the initial value problem for the nonlinear damped wave equation with slowly decaying initial data. When the initial data decay fast enough, it is known that the solution to this problem converges to the self-similar solution to the Burgers equation called a nonlinear diffusion wave, and its optimal asymptotic rate is obtained. In this paper, we focus on the case that the initial data decay more slowly than previous works and derive the corresponding asymptotic profile. Moreover, we investigate how the change of the decay rate of the initial values affect its asymptotic rate.     

Asymptotics of the solutions of the one-dimensional nonlinear system of equations of shallow water with degenerate velocity

A system of one-dimensional nonlinear equations of shallow water with degenerate velocity is considered. The change of variables taking the given system to a nonlinear system with small nonlinearity is proposed. Formal asymptotic solutions near the point of degeneracy are obtained.     

Virial type blow-up solutions for the Zakharov system with magnetic field in a cold plasma

In this paper, we study blow-up solutions of virial type to the Zakharov system with magnetic field in a cold plasma in RN (N=2,3). After obtaining some a priori estimates on those terms generated by the magnetic field, we obtain a virial type blow-up result to the system under consideration. The result suggests that the magnetic field in a cold plasma doesn?t affect the virial type blow-up character of the Zakharov system.     

Multiple solutions to a magnetic nonlinear Choquard equation

We consider the stationary nonlinear magnetic Choquard equation

$(- {\rm i}\nabla+ A(x))^{2}u + V (x)u = \left(\frac{1}{|x|^{\alpha}}\ast |u|^{p}\right) |u|^{p-2}u,\quad x\in\mathbb{R}^{N}$

where A is a real-valued vector potential, V is a real-valued scalar potential, N ≥ 3, \({\alpha \in (0, N)}\) and 2 ? (α/N) < p < (2N ? α)/(N?2). We assume that both A and V are compatible with the action of some group G of linear isometries of \({\mathbb{R}^{N}}\) . We establish the existence of multiple complex valued solutions to this equation which satisfy the symmetry condition

$u(gx) = \tau(g)u(x)\quad{\rm for\, all }\ g \in G,\;x \in \mathbb{R}^{N},$

where \({\tau : G \rightarrow \mathbb{S}^{1}}\) is a given group homomorphism into the unit complex numbers.

     

一类带记忆边界条件波动系统解的长时间性态

研究一类带记忆边界条件波动系统解的长时间性态.在初边值满足一定条件时,利用Faedo-Galerkin近似方法得到了系统解的存在唯一性.使用扰动能量方法,证明了系统解的一致衰减性.     

Large time behavior of solutions for a system of nonlinear damped wave equations

We consider the Cauchy problem of the semilinear damped wave system:

     

Existence of solutions for a nonlinear system with a parameter

In this paper, the existence of solutions for a system of nonlinear equations is considered. n² nonzero real solutions are obtained by using the critical point theory. Additionally, the Dirichlet boundary value problems of even order difference equations and partial difference equations are investigated.     

Asymptotics of solutions of a linear singularity perturbed system with degeneracy

Asymptotic expansions with respect to a small parameter e are constructed for a fundamental system of solutions of a linear singularity perturbed system of equations of the form

Asymptotics of the fundamental system of solutions of a linear functional-differential equation with respect to a parameter

We study a functional-differential equation, whereF is a linear operator acting from the Hölder spaceH into the Sobolev space W _p ^s [0, 1] and is a complex parameter. For large absolute values of , we construct a one-to-one correspondence between the solutionsx(;t) andy(;t) of the equations andy ⁽ⁿ⁾+y ⁿ=0. We also establish conditions that should be imposed on the operatorF in order that specially selected fundamental systems of solutions of these equationsx _j (;t) andy _j (;t), j=1,...,n, satisfy the estimate with constantsc, >0 for the functional space=W _q ^l [0, 1] or.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 6, pp. 811–836, June, 1995.This work was financially supported by the International Science Foundation and the Foundation for Fundamental Research of the Ukrainian State Committee on Science and Technology.