Asymptotic expansions for elliptic-like regularizations of semilinear evolution equations

Consider in a real Hilbert space H the Cauchy problem (_P0$P_0$): u^′(t)+Au(t)+Bu(t)=f(t)$u^{\prime }(t)+Au(t)+Bu(t)=f(t)$, 0≤t≤T$0\le t\le T$; u(0)=_u0$u(0)=u_0$, where −A   is the infinitesimal generator of a _C0$C_0$-semigroup of contractions, B is a nonlinear monotone operator, and f is a given H-valued function. Inspired by the excellent book on singular perturbations by J.L. Lions, we associate with problem (_P0$P_0$) the following regularization (_Pε$P_{\epsilon }$): −εu^″(t)+u^′(t)+Au(t)+Bu(t)=f(t)$-\epsilon u^{″}(t)+u^{\prime }(t)+Au(t)+Bu(t)=f(t)$, 0≤t≤T$0\le t\le T$; u(0)=_u0$u(0)=u_0$, u^′(T)=_uT$u^{\prime }(T)=u_T$, where ε>0$\epsilon >0$ is a small parameter. We investigate existence, uniqueness and higher regularity for problem (_Pε$P_{\epsilon }$). Then we establish asymptotic expansions of order zero, and of order one, for the solution of (_Pε$P_{\epsilon }$). Problem (_Pε$P_{\epsilon }$) turns out to be regularly perturbed of order zero, and singularly perturbed of order one, with respect to the norm of C([0,T];H)$C([0,T];H)$. However, the boundary layer of order one is not visible through the norm of L²(0,T;H)$L^2(0,T;H)$.     

Regularity properties of the solutions to the 3D Maxwell–Landau–Lifshitz system in weighted Sobolev spaces

This paper is concerned with special regularity properties of the solutions to the Maxwell–Landau–Lifshitz (MLL) system describing ferromagnetic medium. Besides the classical results on the boundedness of _∂tm,_∂tE${\mathrm{\partial }}_t\mathbf{m},{\mathrm{\partial }}_t\mathbf{E}$ and _∂tH${\mathrm{\partial }}_t\mathbf{H}$ in the spaces L^∞(I,L²(Ω))$L^{\infty }(I,L^2(\Omega ))$ and L²(I,W^1,2(Ω))$L^2(I,W^{1,2}(\Omega ))$ we derive also estimates in weighted Sobolev spaces. This kind of estimates can be used to control the Taylor remainder when estimating the error of a numerical scheme.     

On fractal properties of non-normal numbers with respect to Rényi f-expansions generated by piecewise linear functions

The paper is devoted to the study of fractal properties of subsets of the set of non-normal numbers with respect to Rényi f  -expansions generated by continuous increasing piecewise linear functions defined on [0,+∞)$[0,+\infty )$. All such expansions are expansions for real numbers generated by infinite linear IFS f={_f0,_f1,…,_fn,…}$f=\{f_0,f_1,\dots ,f_n,\dots \}$ with the following list of ratios _Q∞=(_q0,_q1,…,_qn,…)$Q_{\infty }=(q_0,q_1,\dots ,q_n,\dots )$.     

Oscillation and nonoscillation of second order differential equations with delay depending on the unknown function

Oscillation and nonoscillation of the second order differential equation with delay depending on the unknown function

(r(t)x^′(t))^′+f(t,x(t),x(?(t,x(t))))=0$(r(t)x^{\prime }(t){)}^{\prime }+f(t,x(t),x(?(t,x(t))))=0$

Minimal time of controllability of two parabolic equations with disjoint control and coupling domains

We consider two parabolic equations coupled by a matrix A(x)=q(x)A₀$A(x)=q(x)A_0$, where A₀$A_0$ is a Jordan block of order 1, and controlled by a single localized function, or by a single boundary control. The support of the coupling coefficient, q  , and the control domain may be disjoint. We exhibit an explicit minimal time of null-controllability, T₀(q)∈[0,+∞]$T_0(q)\in [0,+\infty ]$.     

Exact finite-difference schemes for two-dimensional linear systems with constant coefficients

An exact finite-difference scheme for a system of two linear differential equations with constant coefficients, (d/dt)x(t)=Ax(t)$(\mathrm{d}/\mathrm{d}t)\mathbf{x}(t)=A\mathbf{x}(t)$, is proposed. The scheme is different from what was proposed by Mickens [Nonstandard Finite Difference Models of Differential Equations, World Scientific, New Jersey, 1994, p. 147], in which the derivatives of the two equations are formed differently. Our exact scheme is in the form of (1/φ(h))(_xk+1-_xk)=A[θ_xk+1+(1-θ)_xk]$(1/\phi (h))({\mathbf{x}}_{k+1}-{\mathbf{x}}_k)=A[\theta {\mathbf{x}}_{k+1}+(1-\theta ){\mathbf{x}}_k]$; both derivatives are in the same form of (_xk+1-_xk)/φ(h)$(x_{k+1}-x_k)/\phi (h)$.     

A linearized compact difference scheme for an one-dimensional parabolic inverse problem

The parabolic equation with the control parameter is a class of parabolic inverse problems and is nonlinear. While determining the solution of the problems, we shall determinate some unknown control parameter. These problems play a very important role in many branches of science and engineering. The article is devoted to the following parabolic initial-boundary value problem with the control parameter: ∂u/∂t=∂²u/∂x²+p(t)u+?(x,t),0<x<1,0<t?T$\mathrm{\partial }u/\mathrm{\partial }t={\mathrm{\partial }}^{2}u/\mathrm{\partial }{x}^2+p(t)u+?(x\text{,}t)\text{,}0<x<1\text{,}0<t?T$ satisfying u(x,0)=f(x),0<x<1$u(x\text{,}0)=f(x)\text{,}0<x<1$; u(0,t)=_g0(t)$u(0\text{,}t)=g_0(t)$, u(1,t)=_g1(t)$u(1\text{,}t)=g_1(t)$, u(x^∗,t)=E(t),0?t?T$u(x^{\ast }\text{,}t)=E(t)\text{,}0?t?T$ where ?(x,t),f(x),_g0(t),_g1(t)$?(x\text{,}t)\text{,}f(x)\text{,}{g}_0(t)\text{,}{g}_1(t)$ and E(t)$E(t)$ are known functions, u(x,t)$u(x\text{,}t)$ and p(t)$p(t)$ are unknown functions. A linearized compact difference scheme is constructed. The discretization accuracy of the difference scheme is two order in time and four order in space. The solvability of the difference scheme is proved. Some numerical results and comparisons with the difference scheme given by Dehghan are presented. The numerical results show that the linearized difference scheme of this article improve the accuracy of the space and time direction and shorten computation time largely. The method in this article is also applicable to the two-dimensional inverse problem.     

How rich is the class of multifractional Brownian motions?

The multifractional Brownian motion (MBM) processes are locally self-similar Gaussian processes. They extend the classical fractional Brownian motion processes _BH={_BH(t)_}t∈R$B_H=\{B_H(t){\}}_{t\in \mathbb{R}}$ by allowing their self-similarity parameter H∈(0,1)$H\in (0,1)$ to depend on time.     

Fredholm operators,evolution semigroups,and periodic solutions of nonlinear periodic systems

Let X be a Banach space and L the generator of the evolution semigroup associated with the τ  -periodic evolutionary process _{{U(t,s)}t≥s}${\{U(t,s)\}}_{t\ge s}$ on the space _Pτ(X)$P_{\tau }(X)$ of all τ-periodic continuous X  -valued functions. We give criteria for the existence of periodic solutions to nonlinear systems of the form Lp=−?F(p,?)$Lp=-?F(p,?)$ under the condition that 1 is a normal eigenvalue of the monodromy operator U(τ,0)$U(\tau ,0)$. The proof is based on a new decomposition of the space _Pτ(X)$P_{\tau }(X)$ by constructing a right inverse of L.