70 Results Found

Let $X_0,X_1,\dots ,X_q(q<N)$ be real vector fields, which are left invariant on homogeneous group G, provided that $X_0$ is homogeneous of degree two and $X_1,\dots ,X_q$ are homogeneous of degree one. We consider the following nondivergence degenerate operator with d...

We consider a class of strictly pseudohyperbolic equations with lower order terms. The solvability of the Cauchy problem in Sobolev spaces with special weights is established. The uniqueness of the solution is proven and estimates are obtained.

In this paper, there are two aims: one is Schauder and Sobolev estimates for the one-dimensional heat equation; the other is the stabilization of differential equations by stochastic feedback control based on discrete-time state observations. The non...

In generalized inner product Sobolev spaces we investigate elliptic differential problems with additional unknown functions or distributions in boundary conditions. These spaces are parametrized with a function OR-varying at infinity. This characteri...

We consider the one dimensional initial-boundary Sobolev problem with delay. For solving this problem numerically, we construct fourth order differential- difference scheme and obtain the error estimate for its solution. Further we use the appropriat...

One problem of interest in the analysis of Navier–Stokes equations is concerned with the behavior of solutions for certain conditions in the forcing term or external force. In this work, we consider an external force of a maximum exponential gr...

A biharmonic boundary value problem with a singularity is one of the mathematical models of processes in fracture mechanics. It is necessary to have estimates of the function norms in the neighborhood of the singularity point to study the existence a...

In this paper, a finite volume element (FVE) method is proposed for the time fractional Sobolev equations with the Caputo time fractional derivative. Based on the $L1$-formula and the Crank–Nicolson scheme, a fully discrete Crank–Nicolson F...

We study a Camassa–Holm-type equation with a prescribed, time-independent bottom profile, $m_t+(u+h\left(x\right))m_x+2u_{x}m+{\displaystyle \frac{1}{2}}{h}_x\left(x\right)m=0,m=(1-{\partial }_x^2)u.$ The model is considered here as a mathematically motivated bottom-modified Camassa–Holm equatio...

A probabilistic approach is developed for the exact solution u to a deterministic partial differential equation as well as for its associated approximation $u_h^{(k)}$ performed by $P_k$ Lagrange finite element. Two limitations motivated our approach: On the...

The Marchaud fractional derivative can be obtained as a Dirichlet-to–Neumann map via an extension problem to the upper half space. In this paper we prove interior Schauder regularity estimates for a degenerate elliptic equation with mixed Dirichlet–N...

In this article, we mainly studied the Gel’fand widths of Sobolev space in the probabilistic and average settings. And, we estimated the sharp bounds of the probabilistic Gel’fand $(N,\delta )$-widths of multivariate Sobolev space $M{W}_2^r\left({\mathbb{T}}^d\right)$...

The study of small data Sobolev solutions to the Cauchy problem for weakly coupled systems of semi-linear fractional $\sigma –$evolution equations with different power nonlinearities is of interest to us in this research. These solutions must exi...

This study introduces an analysis concerning spacetime perturbations within the context of quantum foam models. Under the framework of Sobolev spaces, $H^s\left({\mathbb{R}}^n\right)$, we establish the existence and uniqueness of solutions to a linearized wave equation consid...

This work studies mixtures of probability measures on ${\mathbb{R}}^n$ and gives bounds on the Poincaré and the log–Sobolev constants of two-component mixtures provided that each component satisfies the functional inequality, and both compone...

Let D be a relatively compact domain in an n-dimensional Kähler manifold with a $C^2$ smooth boundary that satisfies some “Hartogs-pseudoconvexity” condition. Assume that $\Xi$ is a positive holomorphic line bundle over X whose curvature...

We consider Fourier multiplier systems on ${\mathbb{R}}^n$ with components belonging to the standard Hörmander class $S_{1,0}^m\left({\mathbb{R}}^n\right)$, but with limited regularity. Using a notion of parameter-ellipticity with respect to a subsector $\mathrm{\Lambda }\subset \mathbb{C}$ (introduced by Denk,...

In this paper, we investigated a three-dimensional incompressible fractional rotating magnetohydrodynamic (FrMHD) system by reformulating the Cauchy problem into its equivalent mild formulation and working in critical homogeneous Sobolev spaces. For...

The usual thermodynamic formalism is uniform in all directions and, therefore, it is not adapted to study multi-dimensional functions with various directional behaviors. It is based on a scaling function characterized in terms of isotropic Sobolev or...

In this paper, we study the probabilistic Gel’fand $\left(N,\delta \right)$-width of multivariate Sobolev spaces $M{W}_2^r\left({\mathbb{T}}^d\right)$ with mixed derivative that are equipped with Gaussian measure $\mu$ in $L_q\left({\mathbb{T}}^d\right)$. The sharp asymptotic estimates ar...

This paper deals with an inverse inequality for fractional Sobolev norms on expanding planar squares. We define a partial order by the Euclidean distance from the origin and prove that every continuous function monotone with respect to this order on...

An initial-boundary-value problem is considered for the one-dimensional diffusion equation with a general convolutional derivative in time and nonclassical boundary conditions. We are concerned with the inverse source problem of recovery of a space-d...

The paper is concerned with three-dimensional Hall-magnetohydrodynamics for partial viscosity terms. By applying basic inequality and energy estimates, we prove the global existence of classical solution for small initial data. Additionally, the loca...

Recently the identity of de Bruijn type between the relative entropy and the relative Fisher information with the reference moving has been unveiled by Verdú via MMSE in estimation theory. In this paper, we shall give another proof of this identity i...

Quasilinear parabolic boundary value problems are considered. Sharp conditions on the data ensuring the existence and uniqueness of solutions in Sobolev classes $W_p^{1,2}\left(Q\right)$$(p>(n+2)/2$ are exposed. They are smoothness and consistency conditions on the...

We consider a family of semiclassically scaled second-order elliptic differential operators on high tensor powers of a Hermitian line bundle (possibly, twisted by an auxiliary Hermitian vector bundle of arbitrary rank) on a Riemannian manifold of bou...

In this paper, we study the existence and exponential stability of solutions to a class of nonlinear delay Klein–Gordon wave type models on a bounded domain. Such models include multiple time-varying delays, frictional damping, and nonlinear lo...

In this paper, we employ the Nash–Moser iteration technique and the Saloff–Coste’s Sobolev inequality to study the local and global properties of positive solutions to the Trudinger equation ${\Delta }_{p,f}{u}^{{\displaystyle \frac{1}{p-1}}}+b{u}^q+c{u}^r=0$ on a comp...

This paper aims to fuzzify the width problem of classical approximation theory. New concepts of fuzzy Kolmogorov n-width and fuzzy linear n-width are introduced on the basis of $\alpha$-fuzzy distance which is induced by the fuzzy norm. Furthermore, t...

In this paper, the Sobolev embedding theorem, Holder inequality, the Lebesgue contrl convergence theorem, the operator norm estimation technique, and critical point theory are employed to prove the existence of nontrivial stationary solution for p-La...

The main purpose of this paper is to study the Cauchy problem of sixth order viscous Cahn–Hilliard equation with Willmore regularization. Because of the existence of the nonlinear Willmore regularization and complex structures, it is difficult...

The white-noise-driven KdV-type Boussinesq system is a class of stochastic partial differential equations (SPDEs) that describe nonlinear wave propagation under the influence of random noise—specifically white noise—and generalize feature...

We consider an equation with exponential nonlinearity under the Dirichlet boundary condition. For a one- or two-dimensional domain, a global solution has been obtained. In this paper, to extend the result to a higher dimensional case, we concentrate...

This research work investigates the existence, uniqueness, and stability of solution for a time-fractional fourth-order partial differential equation, subject to two initial conditions and four nonlocal integral boundary conditions. The equation inco...

We study inverse problems of identification of lower-order coefficients in a second-order parabolic equation. The coefficients are sought in the form of a finite series segment with unknown coefficients, depending on time. The linear case is also con...

This paper establishes the global-in-time existence and uniqueness of mild solutions to the relativistic Vlasov–Maxwell–Fokker–Planck (VMFP) system near a global Maxwellian equilibrium. We adopt a low-regularity functional framework...

We study a Dirichlet $\mu$-parametric differential problem driven by a variable competing exponent operator, given by the sum of a negative p-Laplace differential operator and a positive q-Laplace differential operator, with a multivalued reaction ter...

The basic model motivating this work is that of contaminant transport in the Earth’s subsurface, which contains layers in which analytical and semi-analytical solutions of the corresponding advection–dispersion equations could be derived....

The following semi-linear elliptic equations involving Hardy–Sobolev critical exponents $-\Delta u-\mu \frac{u}{{\left|x\right|}^2}=\frac{{\left|u\right|}^{2^{*}\left(s\right)-2}}{{\left|x\right|}^s}u+g(x,u),x\in \mathrm{\Omega }\setminus \left\{0\right\},u=0,x\in \partial \mathrm{\Omega }$ have been investigated, where $\mathrm{\Omega }$ is an open-bounde...

This paper investigates a coupled system of two-dimensional singular fractional partial differential equations within a bounded domain. The system is defined by Neumann boundary conditions, non-local weighted integral constraints, memory terms, Caput...

In 1981, Foias, Guillopé and Temam proved a priori estimates for arbitrary-order space derivatives of solutions to the Navier–Stokes equation. Such bounds are instructive in the numerical investigation of intermittency that is often observed in simul...

This paper investigates a new class of fractional integral operators, namely, the exponentially damped Riesz-type operators within the framework of variable exponent Lebesgue spaces $L^{\mathtt{p}(·)}$. To the best of our knowledge, the boundedness of such...

This paper develops novel results in the harmonic analysis of Bessel operators, extending their theory to higher-dimensional and non-Euclidean spaces. We present a refined framework for Hardy spaces associated with Bessel operators, emphasizing atomi...

The main goal of this article is to provide estimates of mild solutions of Navier–Stokes equations with arbitrary external forces in ${\mathbb{R}}^n$ for $n\ge 2$ on proposed weak Herz-type Besov–Morrey spaces. These spaces are larger than known Besov&nd...

The Steklov–Zaremba problem for the Laplace operator in a bounded domain with a strictly Lipschitz boundary is considered. A homogeneous Dirichlet condition is specified on the closed part of the boundary of the domain, and the Steklov boundary...

This article investigates the local well-posedness of a coupled system of wave equations on a half-line, with a particular emphasis on Robin boundary conditions within Sobolev spaces. We provide estimates for the solutions to linear initial-boundary-...

In this paper, we consider local Dirichlet problems driven by the $\left(r\right(u),s(u\left)\right)$-Laplacian operator in the principal part. We prove the existence of nontrivial weak solutions in the case where the variable exponents $r,s$ are real continuous functions and...

This paper is devoted to the fundamental problem of investigating the solvability of initial-boundary value problems for a quasi-linear pseudo-parabolic equation of fractional order with a sufficiently smooth boundary. The difference between the stud...

Under simply supported plate (SSP) boundary conditions, a numerical method based on the higher-order Legendre polynomial approximation was studied and developed for fourth-order problems with variable coefficients. We first divide the SSP boundary co...

In this paper, we investigate the properties of the boundedness of fractional integral operators $K_{\alpha }$ defined on general measure metric spaces. We study their action in Lebesgue spaces $L^p\left(Y\right)$, Morrey spaces $L_{\phi }^p\left(Y\right)$, and extend our analysis to f...