Sobolev Type Research Articles

ОБ ОБОБЩЕННОЙ КРАЕВОЙ ЗАДАЧЕ ДЛЯ ЛИНЕЙНЫХ УРАВНЕНИЙ СОБОЛЕВСКОГО ТИПА НА ГРАФЕ

On the geometric graph, where in addition to the continuity conditions and balance flow, condition of immobility is first introduced into the vertices of the graph, which is converted to a Dirichlet condition when the graph has one edge with two vertices. To solve this problem we first consider the corresponding Sturm–Liouville problem, and the results are then used to solve the Cauchy problem for two linear models, defined on the graph: Hoff equation and Barenblatt–Zheltov–Kochina equation. A feature of the work is the fact that on each edge of the graph given by the equation with different coefficients, which coupled with the introduction of vertices, is fixed for the first time in this problem. Both models relate to Sobolev type equations, the study of which is experiencing an era of its heyday. Reduction of these equations to an abstract Sobolev type equation makes it possible to apply the method of degenerate semigroups of operators. The phase space of solutions is determined by the phase space method, which consists in reducing the singular equation to a regular equation defined on some subspace of the original space. The obtained results of theorems can be used in consideration of inverse problems, optimal control problems, the initial-end and multipoint problems, and also in consideration of stochastic equations for the models set in a geometric graph.

Faedo-Galerkin approximate solutions for nonlocal fractional differential equation of Sobolev type

Faedo-Galerkin approximate solutions for nonlocal fractional differential equation of Sobolev type

The Cauchy Problem for the Sobolev Type Equation of Higher Order

The Cauchy Problem for the Sobolev Type Equation of Higher Order

Sharp Sobolev type embeddings on the entire Euclidean space

A comprehensive approach to Sobolev type embeddings, involving arbitrary rearrangement-invariant norms on the entire Euclidean space \begin{document}${\mathbb R^n}$\end{document} , is offered. In particular, the optimal target space in any such embedding is exhibited. Crucial in our analysis is a new reduction principle for the relevant embeddings, showing their equivalence to a couple of considerably simpler one-dimensional inequalities. Applications to the classes of the Orlicz-Sobolev and the Lorentz-Sobolev spaces are also presented. These contributions fill in a gap in the existing literature, where sharp results in such a general setting are only available for domains of finite measure.

Multipoint Initial-final Problem for One Class of Sobolev Type Models of Higher Order with Additive "White Noise"

Multipoint Initial-final Problem for One Class of Sobolev Type Models of Higher Order with Additive "White Noise"

ПОЛИНОМЫ, ОРТОГОНАЛЬНЫЕ ПО СОБОЛЕВУ, ПОРОЖДЕННЫЕ ПОЛИНОМАМИ ШАРЛЬЕ

ПОЛИНОМЫ, ОРТОГОНАЛЬНЫЕ ПО СОБОЛЕВУ, ПОРОЖДЕННЫЕ ПОЛИНОМАМИ ШАРЛЬЕ

Numerical Study of a Mathematical Model of an Autocatalytic Reaction with Diffusion in a Tubular Reactor

The article analyzes analytically and numerically the model of the autocatalytic reaction with diffusion in the degenerate case on a finite connected directed graph G with the Showalter -- Sidorov condition. The mathematical model of the autocatalytic reaction with diffusion is based on the system of distributed Brusselator equations. The system of degenerate equations of a distributed Brusselator whose functions satisfy the conditions of continuity and flow balance belongs to a wide class of semilinear Sobolev type equations. To investigate the existence of a solution of this system of equations, the phase space method which was developed by G.A.Sviridyuk and his students to study the solvability of Sobolev type equations will be used. We will show the simplicity of the phase space and the existence of a unique local solution of the given Showalter -- Sidorov problem. The theoretical results of this article served as the basis for developing an algorithm for numerical study of the model in a Maple environment. The algorithm of numerical investigation is based on the Galerkin method, which allows us to take into account the phenomenon of degeneracy of the equation. The article gives several examples illustrating the results of the computational experiment obtained on the three-ribbed and five-ribbed graphs.

Nonlocal operators of order near zero

We study Dirichlet forms defined by nonintegrable Lévy kernels whose singularity at the origin can be weaker than that of any fractional Laplacian. We show some properties of the associated Sobolev type spaces in a bounded domain, such as symmetrization estimates, Hardy inequalities, compact inclusion in L2 or the inclusion in some Lorentz space. We then apply those properties to study the associated nonlocal operator L and the Dirichlet and Neumann problems related to the equations Lu=f(x) and Lu=f(u) in Ω.

On the Sobolev boundedness results of the product of pseudo-differential operators involving a couple of fractional Hankel transforms

This article is concerned with the study of pseudo-differential operators associated with fractional Hankel transform. The product of two fractional pseudo-differential operators is defined and investigated its basic properties on some function space. It is shown that the pseudo-differential operators and their products are bounded in Sobolev type spaces. Particular cases are discussed.

Properties of solution sets for Sobolev type fractional differential inclusions via resolvent family of operators

In this manuscript, by properties on some corresponding resolvent operators and techniques in multivalued analysis, we establish some results for solution sets of Sobolev type fractional differential inclusions in the Caputo and Riemann-Liouville fractional derivatives with order 1 < α < 2, respectively. We show that the solution sets are nonempty, compact, contractible and thus arcwise connected under some suitable conditions. We remark that our results are directly established through resolvent operators instead of subordination formulas usually applied, and the existence and compactness of E−1 is not necessarily needed. Some applications are also given in the final.

The Haar System as a Schauder Basis in Spaces of Hardy–Sobolev Type

We show that, for suitable enumerations, the Haar system is a Schauder basis in the classical Sobolev spaces in $${\mathbb R}^d$$ with integrability $$1<p<\infty $$ and smoothness $$1/p-1<s<1/p$$ . This complements earlier work by the last two authors on the unconditionality of the Haar system and implies that it is a conditional Schauder basis for a nonempty open subset of the (1 / p, s)-diagram. The results extend to (quasi-)Banach spaces of Hardy–Sobolev and Triebel–Lizorkin type in the range of parameters $$\frac{d}{d+1}<p<\infty $$ and $$\max \{d(1/p-1),1/p-1\}<s<\min \{1,1/p\}$$ , which is optimal except perhaps at the end-points.

A Study of Sobolev Type Fractional Impulsive Differential Systems with Fractional Nonlocal Conditions

In this paper, we study the existence and uniqueness of mild solutions for Sobolev type fractional impulsive differential systems with fractional order nonlocal conditions by applying monotone iterative technique coupled with the method of lower and upper solutions. The sufficient conditions are obtained by measure of noncompactness and generalized Gronwall inequality. Finally, an application is given to illustrate the obtained results.

Existence and Stability of Solutions for Bilinear Control System with Rieman-Leovel Initial Condition

In this paper, we shall study the existence of a new class called fractional Caputo type of order sobolev type fractional order differential Equations motion in separable Banach spaces. The class of impulsive nonlinear fractional order bilinear control differential Equations with Riemann Leovel differential initial value studied and discussed also given the important results for the almost periodic mild solution to be sTable by using Menards function and granwal fractional inequality.

Characterisation of zero trace functions in higher-order spaces of Sobolev type

Let Ω be a bounded open subset of Rn with a mild regularity property, let m∈N and p∈(1,∞), and let Wm,p(Ω) be the usual Sobolev space of order m based on Lp(Ω); the closure in Wm,p(Ω) of the smooth functions with compact support is denoted by W0m,p(Ω). A special case of the results given below is that u∈W0m,p(Ω) if and only if all distributional derivatives of u of order m belong to Lp(Ω) and u/dm∈L1(Ω), where d(x)=dist(x,∂Ω). In fact what is proved is the analogous result when the Sobolev space is based on a member of a class of Banach function spaces that includes both Lp(Ω) and Lp(⋅)(Ω), the Lebesgue space with variable exponent p(⋅) satisfying natural conditions.

Stochastic fractional perturbed control systems with fractional Brownian motion and Sobolev stochastic non local conditions

This paper investigates the approximate controllability for Sobolev type stochastic perturbed control systems of fractional order with fractional Brownian motion and Sobolev fractional stochastic nonlocal conditions in a Hilbert space, A new set of sufficient conditions are established by using semigroup theory, fractional calculus, stochastic integrals for fractional Brownian motion, Banach’s fixed point theorem. The results are obtained under the assumption that the associated linear system is approximately controllable. Finally, an example is also given to illustrate the obtained theory.

The Kontorovich–Lebedev Transform and Sobolev Type Space

In this paper boundedness of convolution operator in $$L^p(\mathbb {R}_+; ~xdx)$$ is given and continuity of pseudo-differential operator $$\mathcal {P}_a$$ associated with the Kontorovich–Lebedev transform (KL-transform) from $$H(\mathbb {R}_+)$$ into itself is discussed. The Kontorovich–Lebedev potential (KL-potential) $$\mathcal {P}_a^s$$ is defined on $$H(\mathbb {R}_+)$$ and then it is extended to the space of distributions also some of its properties are studied. A Sobolev type space $$W^{s,p}(\mathbb {R}_+)$$ associated with the KL-transform is discussed and proved as a Banach space. Moreover, it is shown that the KL-potential is an isometry of $$W^{s,p}$$ . An $$L^p$$ -boundedness for the KL-potential is obtained and at the end applications of the KL-potential in pseudo-differential equation are discussed.

Existence of mild solutions for fractional nonautonomous evolution equations of Sobolev type with delay

In this paper, we deal with a class of nonlinear fractional nonautonomous evolution equations with delay by using Hilfer fractional derivative, which generalizes the famous Riemann-Liouville fractional derivative. The definition of mild solutions for the studied problem was given based on an operator family generated by the operator pair (A,B) and probability density function. Combining the techniques of fractional calculus, measure of noncompactness, and fixed point theorem with respect to k-set-contractive, we obtain a new existence result of mild solutions. The results obtained improve and extend some related conclusions on this topic. At last, we present an application that illustrates the abstract results.

A multi-species chemotaxis system: Lyapunov functionals, duality, critical mass

We introduce a multi-species chemotaxis type system admitting an arbitrarily large number of population species, all of which are attracted versus repelled by a single chemical substance. The production versus destruction rates of the chemotactic substance by the species is described by a probability measure. For such a model, we investigate the variational structures, in particular, we prove the existence of Lyapunov functionals, we establish duality properties as well as a logarithmic Hardy–Littlewood–Sobolev type inequality for the associated free energy. The latter inequality provides the optimal critical value for the conserved total population mass.

Homotopy classes of Newtonian maps

In this paper we study notions of homotopy in the Newtonian space N^{1,p}(X;Y) of Sobolev type maps between metric spaces. After studying the properties and relations of two different notions we prove a compactness result for sequences in homotopy classes with controlled homotopies.

A generalization of the Paley–Wiener theorem for Mellin transforms and metric characterization of function spaces

Abstract We characterize the function space whose elements have a Mellin transform with exponential decay at infinity. This result can be seen as a generalization of the Paley–Wiener theorem for Mellin transforms. As a byproduct in a similar spirit, we also characterize spaces of functions whose distances from Mellin–Paley–Wiener spaces have a prescribed asymptotic behavior. This leads to Mellin–Sobolev type spaces of fractional order.