The Cauchy problem for inhomogeneous parabolic Shilov equations

In this paper, we consider the Cauchy problem for parabolic Shilov equations with continuous bounded coefficients. In these equations, the inhomogeneities are continuous exponentially decreasing functions, which have a certain degree of smoothness by the spatial variable. The properties of the fundamental solution of this problem are described without using the kind of equation. The corresponding volume potential, which is a partial solution of the original equation, is investigated. For this Cauchy problem the correct solvability in the class of generalized initial data, which are the Gelfand and Shilov distributions, is determined.

Modified Cauchy Problem with Impulse Action for Parabolic Shilov Equations

For parabolic Shilov equations with continuous coefficients, the problem of finding classical solutions that satisfy a modified initial condition with generalized data such as the Gelfand and Shilov distributions is considered. This condition arises in the approximate solution of parabolic problems inverse in time. It linearly combines the meaning of the solution at the initial and some intermediate points in time. The conditions for the correct solvability of this problem are clarified and the formula for its solution is found. Using the results obtained, the corresponding problems with impulse action were solved.

NONLOCAL BY TIME PROBLEM FOR SOME DIFFERENTIAL-OPERATOR EQUATION IN SPACES OF S AND S TYPES

In the theory of fractional integro-differentiation the operator $A := \displaystyle \Big(I-\frac{\partial^2}{\partial x^2}\Big)$ is often used. This operator called the Bessel operator of fractional differentiation of the order of $ 1/2 $. This paper investigates the properties of the operator $B := \displaystyle \Big(I-\frac{\partial^2}{\partial x^2}+\frac{\partial^4}{\partial x^4}\Big)$, which can be understood as a certain analogue of the operator $A$. It is established that $B$ is a self-adjoint operator in Hilbert space $L_2(\mathbb{R})$, the narrowing of which to a certain space of $S$ type (such spaces are introduced in \cite{lit_bodn_2}) matches the pseudodifferential operator $F_{\sigma \to x}^{-1}[a(\sigma) F_{x\to \sigma}]$ constructed by the function-symbol $a(\sigma) = (1+\sigma^2+\sigma^4)^{1/4}$, $\sigma \in \mathbb{R}$ (here $F$, $F^{-1}$ are the Fourier transforms). This approach allows us to apply effectively the Fourier transform method in the study of the correct solvability of a nonlocal by time problem for the evolution equation with the specified operator. The correct solvability for the specified equation is established in the case when the initial function, by means of which the nonlocal condition is given, is an element of the space of the generalized function of the Gevrey ultradistribution type. The properties of the fundamental solution of the problem was studied, the representation of the solution in the form of a convolution of the fundamental solution of the initial function is given.

A Nonlocal in Time Problem for Evolutionary Singular Equations in Generalized Spaces of Type S∘

In this paper, we establish the correct solvability of a nonlocal multipoint in time problem for the evolutionary equation of a parabolic type with the Bessel operator of infinite order in the case where the initial function is an element of the space of generalized functions of type S∘′.

CONJUGATION PROBLEM WITH MULTIPOINT CONDITIONS FOR MIXED HYPERBOLIC TYPE EQUATIONS

The conditions of correct solvability in the Sobolev spaces of the conjugation problem with local multipoint conditions and periodic conditions for higher order mixed hyperbolic type equations is obtained. It has been proved that these conditions fulfill for almost all (with respect to the Lebesgue measure) vectors made up of the nodes of multipoint conditions.