Time periodic solutions to a nonlinear wave equation with <Emphasis Type="Italic">x</Emphasis>-dependent coefficients

In this paper, we study the problem of time periodic solutions to the nonlinear wave equation with x-dependent coefficients on under the boundary conditions a ₁ y(0, t)+b ₁ y _x (0, t) = 0, ( for i = 1, 2) and the periodic conditions y(x, t + T) = y(x, t), y _t (x, t + T) = y _t (x, t). Such a model arises from the forced vibrations of a bounded nonhomogeneous string and the propagation of seismic waves in nonisotropic media. For , we establish the existence of time periodic solutions in the weak sense by utilizing some important properties of the wave operator with x-dependent coefficients. This work was supported by the 985 Project of Jilin University, the Specialized Research Fund for the Doctoral Program of Higher Education, and the Science Research Foundation for Excellent Young Teachers of College of Mathematics at Jilin University.     

On the Cauchy problem for D t 2–D x a(t,x) n D x

Abstract The well posedness of the Cauchy problem for the operator P=D_t²–D_xa(t,x)ⁿD_x, with data on t=0 is studied assuming a ∈ C^N( (R)), s₀>1 and sufficiently close to 1, a(t,x)≥ 0. Well posedness is proved in Gevrey classes γ^(s), for , n≥ n₀. Keywords: Partial differential equations, Cauchy problem, Well posedness     

A central limit theorem for solutions of the porous medium equation

We study the large–time behavior of the second moment (energy) for the flow of a gas in a N-dimensional porous medium with initial density v₀(x) 0. The density v(x, t) satisfies the nonlinear degenerate parabolic equation v_t = v^m where m > 1 is a physical constant. Assuming that for some > 0, we prove that E(t) behaves asymptotically, as t , like the energy E_B(t) of the Barenblatt-Pattle solution B(|x|, t). This is shown by proving that E(t)/E_B(t) converges to 1 at the (optimal) rate t^{–2/(N(m-1)+2)}. A simple corollary of this result is a central limit theorem for the scaled solution E(t)^N/2v(E(t)^1/2x, t).     

Analysis of degenerate elliptic operators of Grušin type

We analyse degenerate, second-order, elliptic operators H in divergence form on L ₂(R ⁿ  × R ^m ). We assume the coefficients are real symmetric and a ₁ H _δ  ≥ H ≥ a ₂ H _δ for some a ₁, a ₂ > 0 where

On viscosity solutions of irregular Hamilton-Jacobi equations

It is proved that the initial-value problem for admits a unique continuous viscosity solution under certain conditions which do not exclude that H(x, p) is discontinuous in x. Particular attention is devoted to the linear transport equation , where a may be discontinuous. Received: 21 October 2002     

Multiplicity of standing wave solutions of nonlinear Schrödinger equations with electromagnetic fields

In this paper, we are concerned with the multiplicity of standing wave solutions of nonlinear Schr?dinger equations with electromagnetic fields

Hopf-Lax type formula for non monotonic autonomous Hamilton Jacobi equations

We give an explicit formula for a solution of Hamilton-Jacobi equation u_t+H(u,Du)=0 in with initial condition u(x,0)=u₀(x), where p H(u,p) is convex, positively homogeneous of degree one and the Hamiltonian H need not satisfy the monotonicity condition in u.     

Closed form solution of a periodically forced logistic model

We give the exact closed form solution of the following ordinary differential equation:

Some Explicit Distributions Related to the First Exit Time from a Bounded Interval for Certain Functionals of Brownian Motion

Let (B _t ) _{t≥ 0} be standard Brownian motion starting at y and set X _t = for , with V(y) = y ^γ if y≥ 0, V(y) = −K(−y)^γ if y≤ 0, where γ and K are some given positive constants. Set . In this paper, we provide some formulas for the probability distribution of the random variable as well as for the probability (or b)}. The formulas corresponding to the particular cases x = a or b are explicitly expressed by means of hypergeometric functions.      

Well-posedness for the system of the Hamilton–Jacobi and the continuity equations

Let H (t, x, p) be a Hamiltonian function that is convex in p. Let the associated Lagrangian satisfy the nonstandard minorization condition where m > 0, ω > 0, and C ≥ 0 are constants. Under some additional conditions, we prove that the associated value function is the unique viscosity solution of S _t + H(t, x, ∇S) = 0 in , without any conditions at infinity on the solution. Here ωT < π/2. To the Hamilton–Jacobi equation corresponding to the classical action integrand in mechanics, we adjoin the continuity equation and establish the existence and uniqueness of a viscosity–measure solution (S, ρ) of

Hamilton-Jacobi equations having only action functions as solutions

Let L(x, v) be a Lagrangian which is convex and superlinear in the velocity variable v, and let H(x, p) be the associated Hamiltonian. Conditions are obtained under which every viscosity solution of the Hamilton-Jacobi equation

On <Emphasis Type="Italic">Γ</Emphasis>-convergence of quadratic functionals in the plane

In this paper, we prove that if a sequence of homeomorphisms , with bounded planar domains, of Sobolev space has uniformly equibounded distortions in EXP(Ω) and weakly converges to f in then the matrices A(x, f _j ) of the corresponding Laplace-Beltrami operators Γ-converge in the Orlicz–Sobolev space , where Q(t) = t ²log(e + t), to the matrix A(x, f) of the Laplace-Beltrami operator associated to f.      

Invariant Tori and Boundedness in Asymmetric Oscillations

In this paper,we are concerned with the boundedness of all the solutions of the equation x″ ax^ -bx- Ф(x)=p(t),where p(t) is a smooth 2π-periodic function,a and b are positive constants,and the perturbation Ф(x) is bounded.     

Affine restriction for radial surfaces

Suppose dμ is affine surface measure on a convex radial surface Γ(x) = (x, γ(|x|)), a ≤ |x| < b, in . Under appropriate smoothness and growth conditions on γ, we prove and Fourier restriction estimates for Γ.     

Virtual Eigenvalues of the High Order Schrödinger Operator II

We consider the Schr?dinger operator H_γ = ( − Δ)^l + γ V(x)· acting in the space where 2l ≥ d, V (x) ≥ 0, V (x) is continuous and is not identically zero, and We study the asymptotic behavior as of the non-bottom negative eigenvalues of H_γ, which are born at the moment γ = 0 from the lower bound λ = 0 of the spectrum σ(H₀) of the unperturbed operator H₀ = ( − Δ)^l (virtual eigenvalues). To this end we use the Puiseux-Newton diagram for a power expansion of eigenvalues of some class of polynomial matrix functions. For the groups of virtual eigenvalues, having the same rate of decay, we obtain asymptotic estimates of Lieb-Thirring type.     

Uniqueness of solutions for some nonlinear Dirichlet problems

We consider here a class of nonlinear Dirichlet problems, in a bounded domain , of the form

Best polynomial approximation in Sobolev-Laguerre and Sobolev-Legendre spaces

We investigate limiting behavior as γ tends to ∞ of the best polynomial approximations in the Sobolev-Laguerre space W^N,2([0, ∞); e^−x) and the Sobolev-Legendre space W^N,2([−1, 1]) with respect to the Sobolev-Laguerre inner product

The pressure equation for fluid flow in a stochastic medium

An equation modelling the pressurep(x) =p(x, w) atx ∈D ⊂R ^d of an incompressible fluid in a heterogeneous, isotropic medium with a stochastic permeabilityk(x, w) ≥ 0 is the stochastic partial differential equation

Some Nonlinear Elliptic Problems in Unbounded Domains

In this paper the results of some investigations concerning nonlinear elliptic problems in unbounded domains are summarized and the main difficulties and ideas related to these researches are described. The model problem