Almost automorphic solutions of nonautonomous evolution equations

In this paper, we establish some new theorems about the existence of almost automorphic solutions to nonautonomous evolution equations u^′(t)=A(t)u(t)+f(t) and u^′(t)=A(t)u(t)+f(t,u(t)) in Banach spaces. As we will see, our results allow for a more general A(t) to some extent. An example is also given to illustrate our results. In addition, by means of an example, we show that one cannot ensure the existence of almost automorphic solutions to u^′(t)=A(t)u(t)+f(t) even if the evolution family U(t,s) generated by A(t) is exponentially stable and fAA(X).     

Some classes of inverse evolution problems for parabolic equations

We consider the problem of simultaneously determining coefficients of a second order nonlinear parabolic equation and a solution to this equation. The unknown coefficients occur in the main part and in the nonlinear summand as well. The overdetermination conditions are conditions of the Dirichlet type on a family of planes of arbitrary dimension. It is demonstrated that the problem in question is solvable locally in time in Hölder spaces. When the unknown functions enter the right-hand side and the equation is linear, the theorem of global unique existence (in time) is established.     

Uniform dichotomy and exponential dichotomy of evolution families on the half-line

The aim of this paper is to give characterizations for uniform and exponential dichotomies of evolution families on the half-line. We associate with a discrete evolution family Φ={Φ(m,n)}_(m,n)∈Δ the subspace . Supposing that X₁ is closed and complemented, we prove that the admissibility of the pair implies the uniform dichotomy of Φ. Under the same hypothesis on X₁, we obtain that the admissibility of the pair with p∈(1,∞] is a sufficient condition for the exponential dichotomy of Φ, which becomes necessary when Φ is with exponential growth. We apply our results in order to deduce new characterizations for exponential dichotomy of evolution families in terms of the solvability of associated difference and integral equations.     

Pseudo-almost periodicity of some nonautonomous evolution equations with delay

This paper is concerned with pseudo-almost periodicity of the solutions to the nonautonomous evolution equation with delay u^′(t)=A(t)u(t)+f(t,u(t−h))$u^{\prime }\left(t\right)=A\left(t\right)u\left(t\right)+f(t,u(t-h))$. Some sufficient conditions which ensure the existence and uniqueness of pseudo-almost periodic mild solutions to the evolution equation with delay are given. An example is shown to illustrate our results.     

Transport equations and perturbations of boundary conditions

We provide a new perturbation theorem for substochastic semigroups on abstract AL spaces extending Kato's perturbation theorem to nondensely defined operators. We show how it can be applied to piecewise deterministic Markov processes and transport equations with abstract boundary conditions. We give particular examples to illustrate our results.     

Time-dependent parabolic problems on non-cylindrical domains with inhomogeneous boundary conditions

We study the relationship between the Dirichlet problem and the Cauchy problem with inhomogeneous boundary conditions for local operators. Our results are applied to non-autonomous parabolic problems on non-cylindrical domains. Received January 10, 2001; accepted April 5, 2001.     

A global Carleman inequality and exact controllability of parabolic equations with mixed boundary conditions

This paper establishes a global Carleman inequality of parabolic equations with mixed boundary conditions and an estimate of the solution. Further, we prove exact controllability of the equation by controls acting on an arbitrarily given subdomain or subboundary.     

Problems for parabolic equations with variable exponents of nonlinearity and time delay

The initial-boundary value problems for parabolic equations with variable exponents of nonlinearity and time depended delay are considered. Existence and uniqueness of solutions of these problems are proved.     

Stability of difference schemes for two-dimensional parabolic equations with non-local boundary conditions

The stability of difference schemes for one-dimensional and two-dimensional parabolic equations, subject to non-local (Bitsadze-Samarskii type) boundary conditions is dealt with. To analyze the stability of difference schemes, the structure of the spectrum of the matrix that defines the linear system of difference equations for a respective stationary problem is studied. Depending on the values of parameters in non-local conditions, this matrix can have one zero, one negative or complex eigenvalues. The stepwise stability is proved and the domain of stability of difference schemes is found.     

The critical exponents for the quasi-linear parabolic equations with inhomogeneous terms

This paper deals with the critical exponents for the quasi-linear parabolic equations in Rn and with an inhomogeneous source, or in exterior domains and with inhomogeneous boundary conditions. For n?3, σ>−2/n and p>max{1,1+σ}, we obtain that pc=n(1+σ)/(n−2) is the critical exponent of these equations. Furthermore, we prove that if max{1,1+σ}<p?pc, then every positive solution of these equations blows up in finite time; whereas these equations admit the global positive solutions for some f(x) and some initial data u₀(x) if p>pc. Meantime, we also demonstrate that every positive solution of these equations blows up in finite time provided n=1,2, σ>−1 and p>max{1,1+σ}.     

On the spatial blow-up and decay for some nonlinear parabolic equations with nonlinear boundary conditions

In this note we investigate the spatial behavior of several nonlinear parabolic equations with nonlinear boundary conditions. Under suitable conditions on the nonlinear terms we prove that the solutions either cease to exist for a finite value of the spatial variable or else they decay algebraically. The main tool used is the weighted energy method. Our results can be applied to several situations concerning heat conduction. Received: April 4, 2004; revised: September 20, 2004     

On a mixed problem with integral conditions for one-dimensional parabolic equation

We consider an initial-boundary value problem for a one-dimensional parabolic equation with nonlocal boundary conditions. These nonlocal conditions are given in terms of integrals. Based on solution of the Dirichlet problem for the parabolic equation, we constructively establish the well-posedness for the nonlocal problem.     

Finite difference schemes for the fourth-order parabolic equations with different boundary value conditions

In this paper, the fourth-order parabolic equations with different boundary value conditions are studied. Six kinds of boundary value conditions are proposed. Several numerical differential formulae for the fourth-order derivative are established by the quartic interpolation polynomials and their truncation errors are given with the aid of the Taylor expansion with the integral remainders. Effective difference schemes are presented for the third Dirichlet boundary value problem, the first Neumann boundary value problem and the third Neumann boundary value problem, respectively. Some new embedding inequalities on the discrete function spaces are presented and proved. With the method of energy analysis, the unique solvability, unconditional stability and unconditional convergence of the difference schemes are proved. The convergence orders of derived difference schemes are all O(τ² + h²) in appropriate norms. Finally, some numerical examples are provided to confirm the theoretical results.     

Well-posedness of general nonlocal nonhomogeneous boundary value problems for pseudodifferential equations with partial derivatives

In the article we study the questions of well-posedness of general nonlocal boundary value problems for pseudodifferential equations in the Besov-type limit spaces.     

Collocation for high order differential equations with two-points Hermite boundary conditions

For the numerical solution of high even order differential equations with two-points Hermite boundary conditions a general collocation method is derived and studied. Computation of the integrals which appear in the coefficients are generated by a recurrence formula and no integrals are involved in the calculation. An application to the solution of the beam problem is given. Numerical experiments provide favorable comparisons with other existing methods.     

A class of evolution equations: Existence of solutions with functional boundary conditions

We consider the evolution equation whose right-hand side is the sum of a linear unbounded operator generating a compact strongly continuous semigroup and a continuous operator acting in function spaces. We prove the existence of a solution that stays within a given closed convex set and moreover, satisfies a functional boundary condition, particular cases of which are the Cauchy initial condition, periodicity condition, mixed condition including continuous transformations of spatial variables, etc. The main result is illustrated by using an example of the boundary-value problem for a partial operator-differential equation. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 48–60, January, 1999.