On nonspherical partial sums of fourier integrals of continuous functions from the Sobolev spaces

The partial integrals of the N-fold Fourier integrals connected with elliptic polynomials (not necessarily homogeneous; principal part of which has a strictly convex level surface) are considered. It is proved that if a + s > (N – 1)/2 and ap = N then the Riesz means of the nonnegative orders of the N-fold Fourier integrals of continuous finite functions from the Sobolev spaces Wpa(RN) converge uniformly on every compact set, and if a + s > (N – 1)/2 and ap = N, then for any x0∈ RN there exists a continuous finite function from the Sobolev space such, that the corresponding Riesz means of the N-fold Fourier integrals diverge to infinity at x0. AMS 2000 Mathematics Subject Classifications: Primary 42B08; Secondary 42C1

On the divergence of spectral expansions of elliptic differential operators

In this paper we consider spectral expansions of functions from Nikol'skii classes Hap (Rn), related to selfadjoint extensions of elliptic differential operators A(D) of order m in Rn. We construct a continuous function from Nikol'skii class with pa N obtained earlier by Alimov (1976) for uniform convergence of spectral expansions, related to elliptic differential operators

The diffusion equation with piecewise smooth initial conditions

In this paper we consider an initial-value problem for diffusion equation in three dimensional Euclidean space. The initial value is a piecewise smooth function. To solve this problem we apply Fourier transform method and since Fourier integrals of a piecewise smooth function do not converge everywhere, we make use of Riesz summation method

On systems of fractional nonlinear partial differential equations

The work considers a system of fractional order partial differential equations. The existence and uniqueness theorems for the classical solution of initial-boundary value problems are proved in two cases: 1) the right-hand side of the equation does not depend on the solution of the problem and 2) it depends on the solution, but at the same time satisfies the classical Lipschitz condition with respect to this variable and an additional condition which guarantees a global existence of the solution. Sufficient conditions are found (in some cases they are necessary) on the initial function and on the right-hand side of the equation, which ensure the existence of a classical solution. In previously known works, linear but more general systems of fractional pseudodifferential equations were considered and the existence of a weak solution was proven in the special classes of distributions.Comment: positiv

Backward and non-local problems for the Rayleigh-Stokes equation

The Fourier method is used to find conditions on the right-hand side and on the initial data in the Rayleigh-Stokes problem, which ensure the existence and uniqueness of the solution. Then, in the Rayleigh-Stokes problem, instead of the initial condition, consider the non-local condition: $u(x,T)=\beta u(x,0)+\varphi(x)$, where $\beta$ is either zero or one. It is well known that if $\beta=0$, then the corresponding problem, called the backward problem, is ill-posed in the sense of Hadamard, i.e. a small change in $u(x,T)$ leads to large changes in the initial data. Nevertheless, we will show that if we consider sufficiently smooth current information, then the solution exists and it is unique and stable. It will also be shown that if $\beta=1$, then the corresponding non-local problem is well-posed and coercive type inequalities are valid.Comment: 1