This paper is concerned with the sideways problem for parabolic equations in a bounded domain. The Dirichlet data on the remaining part of the boundary is reconstructed from the Dirichlet and Neumann measurements on a portion of the boundary. Based on existing theories, the uniqueness of the inverse problem can be established, and its ill-posedness is analyzed. Then, by introducing two auxiliary problems, the inverse problem is reformulated as an optimal control problem with Kohn-Vogelius regularization. We further prove the existence and stability of solutions to the optimal control problem. Finally, a physics-informed neural network framework based on the Kohn-Vogelius type functional is applied to several numerical examples.
In this paper, we investigate the existence of square-mean S-asymptotically (omega,c) -periodic solutions for a class of second-order neutral stochastic impulsive differential equations driven by fractional Brownian motion. The model under our consideration incorporates long-range dependence, neutral delays, impulsive perturbations, and arises naturally in stochastic control problems. The established results offer a unified analytical framework for neutral impulsive systems influenced by fractional noise, extending classical periodicity methods to settings with memory, scaling modulation and long-range correlations.
We investigate the asymptotic behavior of incompressible thermally coupled Bingham flows in a three-dimensional thin domain Omega epsilon , subject to Tresca slip conditions on part of the boundary and Dirichlet conditions elsewhere. Using a variational inequality framework and a rescaling of the thin domain onto a fixed reference domain, we establish the existence and uniqueness of weak solutions together with uniform epsilon-independent estimates. By combining compactness arguments and asymptotic analysis as epsilon -> 0 , we derive the effective limiting model and identify the asymptotic form of the Tresca slip condition. In particular, we prove that the pressure becomes independent of the transverse variable and obtain a Reynolds-type equation governing the limit flow.
This paper discusses the quasi-exponential stability and ultimate boundedness of nonautonomous impulsive delayed systems with conformable derivatives. First, a novel conformable differential inequality is established using the definition of conformable derivatives and proof-by-contradiction techniques. Subsequently, by employing the mathematical induction technique along with the established inequality and Lyapunov function theory, several sufficient conditions are obtained to ensure the quasi-exponential stability and exponential ultimate boundedness of the systems. Furthermore, more relaxed criteria for exponential ultimate boundedness are provided, which do not require the derivative of the Lyapunov function to be negative definite. As an application, the synchronization problem for impulsive delayed multilayer networks with conformable derivatives is addressed. Finally, several numerical examples are provided to demonstrate the effectiveness and reduced conservatism of the proposed approach compared to existing results.
We construct a tubular surface generated by a great circular surface in the three-sphere. This tubular surface is the great circular surface when the radius function is equal to 1. We then compute the conditions for the regular tubular surface to be extrinsic flat or minimal. We also study the geodesic curves on both the regular tubular surface. Additionally, we investigate the singular points of the tubular surfaces and have the conditions for a tubular surface and great circular surface be cross-cap.
Method of asymptotic partial decomposition of a domain (MAPDD) proposed and justified earlier for thin domains (rod structures, tube structures consisting of a set of thin cylinders) generates some specific interface conditions between three-dimensional and one-dimensional parts. In the case of the wave equation, these conditions ensure the continuity of the solution and continuity in average of the normal flux. However, some numerical approaches prefer to have more degrees of liberty for the approximate solution. For example, the finite elements method (FEM) may use discontinuous elements. That is why the strong continuity condition for the solution may be replaced by more flexible condition of continuity in average over the interface cross section. In the present paper, we introduce and justify this alternative junction condition allowing such discontinuity of the approximate solution of the wave equation with Neumann's boundary condition. The closeness of solutions of these hybrid dimension problems to the solution of the fully three-dimensional setting is proved. A finite volume scheme is used to solve the wave equation as a hybrid dimension problem. The convergence of this scheme, when the wave equation is formulated as a second-order partial differential equation in space and time, is proved using energy techniques.
We consider the recovery of small inclusions in multi-layer medium. We obtain an asymptotic expansion for the perturbed electric field in multi-layer inhomogeneous conductivity problem. Furthermore, we recover the positions and conductivities of small inclusions through a single measurement.
This paper is concerned with the designing, analyzing and implementing linear and nonlinear discretization scheme for the distributed optimal control problem (OCP) with the Cahn-Hilliard (CH) equation as constrained. We propose three difference schemes to approximate and investigate the solution behavior of the OCP for the CH equation. We present the convergence analysis of the proposed discretization. We verify our findings by presenting numerical experiments.
This paper addresses the inverse problem of simultaneously identifying the unknown source term and initial value in a stochastic Caputo-Hadamard time-fractional diffusion equation, where each term consists of a deterministic function and a stochastic process. The ill-posedness is analyzed through the expectation and variance of the measured data at two distinct time instants. To stabilize the solution, a fractional Landweber iterative regularization method is employed, and both a priori and a posteriori error estimates are derived. Numerical experiments demonstrate the feasibility and effectiveness of the employed method.
In this paper, we investigate the following Schrodinger-Choquard equation in R-n (I - Delta)(log)u(x) + omega u(x) = C-n,C-s(|x|(2s-n) * u(p))u(q), x is an element of R-n, where 0 < s < 1, 2 < p < infinity, 1 < q <= p - 1, omega > 0, (I - Delta)(log) denotes the logarithmic Schrodinger operator appearing as a formal derivative d/ds |(s=0)(I - Delta)(s) of the pseudo-relativistic Schrodinger operator at s = 0. By using the direct method of moving planes, we prove that positive solutions of the equation above must be radially symmetric and monotone decreasing about some point in R-n. To achieve our objects, we need to develop the narrow region principle and decay at infinity theorem.
To investigate the influence of vaccination and spatial diffusion on the spatial transmission behavior of diseases, in this paper, an SVIR epidemic model with general incidence rates and population dynamics factors under spatial diffusion influences is proposed. The weak dissipation and uniform persistence are proved by the comparison principle and strong maximum principle. The existence of nontrivial traveling wave solutions connecting the disease-free equilibrium and endemic equilibrium are discussed utilizing the fixed point theorem, some limit techniques and Lyapunov functionals. The results show that the existence and nonexistence of traveling waves depends on the reproduction number $ \mathcal {R}_0 $ R0 and the critical wave speed $ c<^>* $ c & lowast;. Finally, the validity of the theoretical findings was verified through numerical simulations, and the influence of vaccination and spatial diffusion on disease transmission and spread speed was further investigated.
This paper considers the mathematical framework for the two-dimensional generalized Boussinesq equations driven by strictly anisotropic fractional dissipation in highly singular endpoint Besov spaces. We investigate the critical fluid regime where the velocity equation exclusively possesses horizontal fractional dissipation, while the active scalar temperature equation exclusively possesses vertical fractional dissipation, subject precisely to the critical singular condition where the sum of their fractional derivative orders equals one. To surmount the profound degeneracy caused by the complete lack of full Laplacian smoothing, we systematically develop a novel, sharp anisotropic commutator estimate within the refined Littlewood-Paley framework. We strictly prove the local well-posedness via a delicate contraction mapping argument exclusively in critical Besov spaces, simultaneously incorporating Calderon-Zygmund singular integral estimates to rigorously isolate the internal pressure field. Furthermore, we establish a sharp Beale-Kato-Majda type blow-up criterion, demonstrating that any finite-time singularity is fundamentally governed solely by the accumulation of the maximum spatial gradient of the velocity field. Ultimately, by meticulously balancing the cross-directional derivatives and employing a generalized Cordoba-Cordoba pointwise inequality, we extract global a priori bounds, upgrading the local solutions to unconditional global well-posedness. The global uniqueness is proven via a severe perturbation analysis in negative-index function spaces.
This paper studies the initial-boundary value problem of the three-dimensional incompressible magnetohydrodynamics (MHD) system in the strip domain $ \Omega = \mathbb {R}<^>2 imes [0,1] $ Omega=R2 & times;[0,1]. The system includes partial dissipation, with horizontal viscosity $ \Delta _h u $ Delta hu and vertical magnetic diffusion $ \partial _3<^>2 B $ partial derivative 32B. We apply the slip boundary condition to the velocity field and the Dirichlet boundary condition to the magnetic field. We establish the global stability of small perturbations around a steady state driven by the background magnetic field $ e_1 $ e1. For sufficiently small initial perturbations in the Sobolev space $ H<^>3(\Omega ) $ H3(Omega), we rigorously prove the existence and uniqueness of global-in-time solutions, which remain uniformly bounded in the $ H<^>3 $ H3-norm for all time. This work extends the results of Wu and Zhu to the case with physical boundaries. Moreover, we analyze the convergence behavior in the singular limit where the vertical viscosity coefficient mu and horizontal magnetic diffusion coefficient nu tend to zero. We show that solutions of the viscous MHD system converge to those of the corresponding limit system, with an explicit convergence rate obtained.
Considered herein is the blow-up mechanism to the periodic generalized Camassa-Holm equations with dual-power nonlinearities. We first establish the local well-posedness by applying the Kato's semigroup theory.The precise blow-up scenarios of strong solutions and several sufficient conditions on the initial data to guarantee blow-up of the induced solutions are described in detail according to the different real-valued intervals in which the dispersive parameter s is located. The challenge we need to meet is to readjust the corresponding spatial estimates in order to balance the relationship between the various dispersive parameters and overcome the difficulty caused by complicated mixed dual-power nonlinear structure like the non-periodic case.
To examine the effect of evolving domain on the spread and control of disease, we study an SIS reaction-diffusion model with Dirichlet boundary conditions in an asymptotically bounded domain. Firstly, the basic reproduction number R-0 is defined, which relies on the evolution rate of the domain, the diffusion rate of infected individuals and spatial heterogeneity. We analyze the asymptotic stability of the disease-free equilibrium by the sub-solution and super-solution method and energy-estimates. Secondly, the existence and uniqueness, as well as the global stability of the endemic equilibrium in a special case are investigated. Finally, we discuss the asymptotic profiles of the endemic equilibrium for small and large diffusion rates of the susceptible individuals and the infected individuals. Numerical simulations are performed to illustrate the analytical results.
We analyze the existence of traveling waves for a degenerate reaction diffusion equation featuring flux limitation effects together with time delay. The approach adopted is the upper and lower solutions method. The main technical issue for the proof is to overcome the obstacle caused by the flux-limited nonlinear degenerate diffusion and the time delay.
