Construction of asymptotic solutions of certain boundary-value problems for the nonautonomous wave equation

Applying the periodic Ateb-functions we construct single-frequency asymptotic approximations of solutions of problems for the nonlinear nonautonomous wave equation.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 3, 1997, pp. 93–97.     

On asymptotic approximation of a solution of a boundary-value problem for a nonlinear nonautonomous equation

On the basis of periodic Ateb functions, in the resonance and nonresonance cases, we construct the asymptotic approximation of one-frequency solutions of a boundary-value problem for a nonlinear nonautonomous equation.     

On asymptotic approximations for slow wave processes in nonlinear dispersive media

We consider the application of asymptotic methods of nonlinear mechanics (the Krylov-Bogolyubov-Mitropol'skii method) and the method of separation of motions in nonlinear systems for the construction of an approximate solution of a nonlinear equation that describes a nonstationary wave process. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 3, pp. 357–371, March, 1998.     

On the construction of an asymptotic solution of a perturbed Bretherton equation

We consider the application of the asymptotic method of nonlinear mechanics to the construction of the first and second approximations of a solution of the Bremerton equation. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 1, pp. 58–71, January, 1998.     

On the existence and uniqueness of solutions to an asymptotic equation of a variational wave equation

We prove the global existence and uniqueness of admissible weak solutions to an asymptotic equation of a nonlinear hyperbolic variational wave equation with nonnegative L ²(ℝ) initial data. The work of Ping Zhang is supported by the Chinese postdoctor’s foundation, and that of Yuxi Zheng is supported in part by NSF DMS-9703711 and the Alfred P. Sloan Research Fellows award.     

On construction of asymptotic solution of the perturbed Klein-Gordon equation

We consider an application of the asymptotic method of nonlinear mechanics to the construction of an approximate solution of the Klein-Gordon equation.Academician.Translated from Ukrainskii Matematicheskii Zhumal, Vol. 47, No. 9, pp. 1209–1216, September, 1995.This research is partially supported by the International Soros Foundation for Support of Education Program in Natural Sciences.     

Construction of one-frequency solutions of boundary-value problems for a nonautonomous wave equation

By using special periodic Ateb-functions, we construct asymptotic representations of one-frequency solutions of boundary-value problems for a nonautonomous wave equation.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 9, pp. 1275–1279, September, 1994.     

Oscillatory asymptotic expansion for semilinear wave equation

We study oscillatory properties of the solution to semilinear wave equation, assuming oscillatory terms in initial data have sufficiently small amplitude. The main result gives an a priori estimate of the remainder in the approximation by means of the method of geometric optics. The method of establishing this estimate is based on a combination between energy type estimates for transport equation and Sobolev embedding. Copyright © 2001 John Wiley & Sons, Ltd.     

Some remarks on the asymptotic behavior of a nonautonomous,nonlinear functional differential equation

The asymptotic behavior of the abstract nonautonomous, nonlinear functional differential equation $\text{x}\text{?}\text{(t) = f(t, x(t)) + g(t, x}{}_{\mathrm{t}}\text{), x}{}_{\mathrm{s}}\text{= ?}$ is considered. Estimates on the growth of solutions are given and these estimates are shown to be the best possible.     

On asymptotic approximations of the residual currents

We use a -module approach to discuss positive examples for the existence of the unrestricted limit of the integrals involved in the approximation to the Coleff-Herrera residual currents in the complete intersection case. Our results also provide asymptotic developments for these integrals.

     

Long-time asymptotic expansion for the damped semilinear wave equation

The first initial-boundary value problem is considered for the damped semilinear wave equation with the quadratic nonlinearity. For small initial data and homogeneous boundary conditions its solution is constructed in the form of a series in the eigenfunctions of the Laplace operator. The long-time asymptotic expansion is obtained which shows the nonlinear effects of amplitude and frequency multiplication. The same results hold for the admissible initial data that are not small.     

On a nonautonomous difference equation with bounded coefficient

In this paper we investigate the boundedness, the persistence and the attractivity of the positive solutions of the nonautonomous difference equation

     

On a semilinear wave equation: The Cauchy problem and the asymptotic behavior of solutions

The semilinear wave equation

$\text{□u + m}{}^2\text{u + ¦u¦}{}^{\mathrm{p\; ?\; 2}}\text{u(V1 ¦u¦}{}^{\mathrm{p}}\text{) = 0}$

in $\text{Ω= R}{}^3\text{, ?∞ < t < ∞}$, is studied where □ denotes the d'Alembertian operator and 1 means spatial convolution. Under mild assumptions on the real-valued function V and 2 ? p ? 3 the well-posedness of the Cauchy problem is proved. Furthermore, some properties of the solutions of the equation are analyzed such as the asymptotic behavior of local energy as $\text{¦t¦ → + ∞}$ in the case of zero mass. Our results extend that of Perla Menzala and Strauss, where case p = 2 was studied.     

Global existence and asymptotic behavior of solutions for a semi-linear wave equation

In this paper, we investigate the initial value problem for a semi-linear wave equation in n-dimensional space. Based on the decay estimate of solutions to the corresponding linear equation, we define a set of time-weighted Sobolev spaces. Under small condition on the initial value, we prove the global existence and asymptotic behavior of the solution in the corresponding Sobolev spaces by the contraction mapping principle.     

Higher order asymptotic expansions to the solutions for a nonlinear damped wave equation

We study the Cauchy problem for a nonlinear damped wave equation. Under suitable assumptions for the nonlinearity and the initial data, we obtain the global solution which satisfies weighted L ¹ and \({L^\infty}\) estimates. Furthermore, we establish the higher order asymptotic expansion of the solution. This means that we construct the nonlinear approximation of the global solution with respect to the weight of the data. Our proof is based on the approximation formula of the linear solution, which is given by Takeda (Asymptot Anal 94:1–31, 2015), and the nonlinear approximation theory for a nonlinear parabolic equation developed by Ishige et al. (J Evol Equ 14:749–777, 2014).     

On the asymptotic behavior of the solutions of semilinear nonautonomous equations

We consider nonautonomous semilinear evolution equations of the form $$\frac{dx}{dt}= A(t)x+f(t,x) . $$ Here A(t) is a (possibly unbounded) linear operator acting on a real or complex Banach space $\mathbb{X}$ and $f: \mathbb{R}\times\mathbb {X}\to\mathbb{X}$ is a (possibly nonlinear) continuous function. We assume that the linear equation (1) is well-posed (i.e. there exists a continuous linear evolution family {U(t,s)}_(t,s)∈Δ such that for every s∈?₊ and x∈D(A(s)), the function x(t)=U(t,s)x is the uniquely determined solution of Eq. (1) satisfying x(s)=x). Then we can consider the mild solution of the semilinear equation (2) (defined on some interval [s,s+δ),δ>0) as being the solution of the integral equation $$x(t) = U(t, s)x + \int_s^t U(t, \tau)f\bigl(\tau, x(\tau)\bigr) d\tau,\quad t\geq s . $$ Furthermore, if we assume also that the nonlinear function f(t,x) is jointly continuous with respect to t and x and Lipschitz continuous with respect to x (uniformly in t∈?₊, and f(t,0)=0 for all t∈?₊) we can generate a (nonlinear) evolution family {X(t,s)}_(t,s)∈Δ , in the sense that the map $t\mapsto X(t,s)x:[s,\infty)\to\mathbb{X}$ is the unique solution of Eq. (4), for every $x\in\mathbb{X}$ and s∈?₊. Considering the Green’s operator $(\mathbb{G}{f})(t)=\int_{0}^{t} X(t,s)f(s)ds$ we prove that if the following conditions hold

• the map $\mathbb{G}{f}$ lies in $L^{q}(\mathbb{R}_{+},\mathbb{X})$ for all $f\in L^{p}(\mathbb{R}_{+},\mathbb{X})$ , and
• $\mathbb{G}:L^{p}(\mathbb{R}_{+},\mathbb{X})\to L^{q}(\mathbb {R}_{+},\mathbb{X})$ is Lipschitz continuous, i.e. there exists K>0 such that $$\|\mathbb{G} {f}-\mathbb{G} {g}\|_{q} \leq K\|f-g\|_{p} , \quad\mbox{for all}\ f,g\in L^p(\mathbb{R}_+,\mathbb{X}) , $$

then the above mild solution will have an exponential decay.     

Pullback attractors for a nonautonomous damped wave equation with infinite delays in weighted space

In this paper, we investigate the existence of solutions for a damped wave equation with infinite delays in the weighted space and , respectively. In addition, we study the existence of pullback attractor for the process associated to the problem by a direct and simple compactness method.