Continuous Dependence Research Articles

Well-posedness, monotonicity, asymptotic translation of the Cauchy problem of delayed reaction–diffusion systems

In this paper, we establish a new basic theory on the Cauchy problem of delayed reaction–diffusion systems on the whole Euclidean space, where the initial function space is equipped with the compact open topology (also called coarse topology). Generally, under this coarse topology, the reaction terms are not locally Lipschitz continuous. Even they are, it is difficult to boil down the basic theory on the Cauchy problem to the (parameterized) contraction mapping problem. To overcome this difficulty, we explain and prove the uniqueness of solutions from a new perspective. This provides a unified way to obtain the basic theory on the Cauchy problem of delayed reaction–diffusion systems on the whole Euclidean space, which includes the local existence, uniqueness, and continuous dependence of solutions. Moreover, we show that, with respect to the compact open topology, the solution semiflow is monotone and possesses the property of asymptotic translation.

Kolmogorov’s theorem for degenerate Hamiltonian systems with Hölder continuous parameters

Abstract In this paper, we investigate Kolmogorov-type theorems for small perturbations of degenerate Hamiltonian systems. These systems are index by a parameter ξ as $ H(y,x,\xi) = \langle\omega(\xi),y\rangle {+ \bar h(y,\xi)}+\varepsilon P(y,x,\xi,\varepsilon) $ , where ɛ > 0. We assume that the frequency mapping $\omega(\cdot)$ , $\bar h(y,\cdot)=O(|y|^2)$ and the perturbation $\varepsilon P(y,x,\cdot, \varepsilon)$ maintain Hölder continuity about ξ. We prove that the persistent invariant tori retain the same frequency as those of the unperturbed tori, under a certain topological degree condition and a weak convexity condition for the frequency mapping. Notably, this paper presents, to our understanding, pioneering results on the KAM theorem under such conditions with only assumption of Hölder continuous dependence of frequency mapping ω on the parameter.

Well‐posedness and inviscid limits for the Keller–Segel–Navier–Stokes system of the parabolic–elliptic type

AbstractWe show the local well‐posedness of the Keller–Segel system of the parabolic–elliptic type coupled with the Navier–Stokes system for arbitrary initial data with Sobolev regularities, where the solution is uniformly bounded with respect to the viscosity. We also show the continuous dependence of the solutions with respect to the initial data. As a result of the uniform boundedness of the solutions, we obtain inviscid limits of the system. The proof is mainly based on a priori estimates in the Sobolev spaces.

On the Cahn–Hilliard equation with kinetic rate dependent dynamic boundary conditions and non-smooth potentials: Well-posedness and asymptotic limits

We analyze a class of Cahn–Hilliard equations with kinetic rate dependent dynamic boundary conditions that describe possible short-range interactions between the binary mixture and the solid boundary. In the presence of surface diffusion on the boundary, the initial boundary value problem can be viewed as a transmission problem consisting of Cahn–Hilliard-type equations both in the bulk and on the boundary. We first establish the existence, uniqueness, and continuous dependence of global weak solutions. In the construction of weak solutions, an explicit convergence rate in terms of the parameter for the Yosida approximation is obtained. Under some additional assumptions, we further prove the existence and uniqueness of global strong solutions. Next, we study the asymptotic limit as the coefficient of the boundary diffusion goes to zero and show that the limit problem with a forward-backward dynamic boundary condition is well posed in a suitable weak formulation. At last, we investigate the asymptotic limits as the kinetic rate tends to zero and infinity, respectively. Our results are valid for a general class of bulk and boundary potentials with double-well structure, including the physically relevant logarithmic potential and the non-smooth double obstacle potential.

Structural Stability of Pseudo-Parabolic Equations for Basic Data

This article investigates the spatial decay properties and continuous dependence on the basic geometric structure. Assuming that the total potential energy is bounded and the homogeneous Dirichlet condition is satisfied on the side of the solution within the cylindrical domain, we establish an auxiliary function related to the solution. By extending the data at the finite end forward, we can establish the continuous dependence on the perturbation of base data.

Well-Posedness and Convergence Results for History-Dependent Inclusions

We consider an inclusion in a real Hilbert space governed by a time-dependent set of constraints and a history-dependent operator. We introduce the concept of T - well-posedness for this inclusion, associated to a given Tykhonov triple T . Next, we provide a T -well-posedness result that we use in order to deduce the continuous dependence of the solution with respect to the data. Then, we state and prove a convergence criterion to the solution of the inclusion that we use to prove a convergence result for an associate penalty problem. Moreover, we show that this criterion allows us to construct a Tykhonov triple T -which give rise to an optimal well-posedness concept for the corresponding inclusion. Finally, we use these abstract results in the study of a nonlinear viscoelastic constitutive law with long memory term and unilateral constraints.

On a class of evolution equations in Lipschitz subdomains of Riemannian manifolds

We study the existence, uniqueness, analyticity, and continuous dependence of mild solutions for a class of evolutionary integrodifferential equations with free boundary conditions on L p -spaces in Lipschitz subdomains of Riemannian manifolds. Our strategy is to use the Laplace transform theory to ensure the existence of an analytic resolvent to the Hodge–Laplacian in the space of all ℓ-differential forms with p-th power integrable coefficients.

Global well-posedness and exponential decay estimates for semilinear Newell–Whitehead–Segel equation

Abstract This article presents the application of the Faedo–Galerkin compactness method to establish the local well-posedness of the Newell–Whitehead–Segel equation. By analyzing a finite-dimensional approximate problem, the existence and uniqueness of a local solution were demonstrated. A priori estimates were derived, enabling the transition to the limit and the recovery of the original problem’s local solution. The study further proves the uniqueness and continuous dependence of the solution on initial data. Additionally, under certain conditions, it is shown that the energy norm of the solution decays exponentially over time, and the L 2 {L}^{2} -norm of the time derivative of the solution approaches zero asymptotically.

Factors determining economic growth in Syria in 2014 – 2023 : macroeconomic analysis

The economy of Syria has been experiencing very strong and diverse pressure from Western countries on the national economy for a long period. Despite this, the country was able to survive and continues its development, although very unstable. In this regard, the issue of thefactors of the growth of the Syrian economy in the last decade, the problems and prospects for the development of the national economy and the foreign trade sector of the country, as well as the prospects for its cooperation with foreign partners, including at the regional level, remains relevant. The article shows that in the last decade, the Syrian economy has faced serious problems, including two new (external) factors – the global COVID-19 pandemic and the escalation of military tensions in the Middle East region, affecting both the territory (and the economy) of Syria. At the domestic socio-economic level, market volatility affects the dynamics of macroeconomic indicators (high instability of development, including in the field of per capita income, a significant degree of closure of the national economy – low indicators of the foreign trade quota with continued import dependence, a significant share of agriculture and low urbanization relative to other countries in the region), while high final consumption costs do not allow the country to ensure the necessary level of fixed capital accumulation for the development and diversification of the national economy. At the same time, Syria's economy remains one of the most diversified in the region, and the country does not face a high degree of food dependence like its regional neighbors. From this, the authors conclude that it is necessary and possible at this stage to intensify and escalate intraregional inter-Arab economic cooperation and trade as one of the tools to reduce existing risks.

Structural stability for double-diffusion Brinkman fluid with Soret effect

This paper considers the double diffusive Brinkman flow in a semi-infinite pipe. By establishing a priori estimates of the solutions and setting an appropriate “energy” function, we not only obtain the continuous dependence and convergence of the solution on the Soret coefficient but also prove that the solutions decay exponentially with the distance from the finite end of the cylinder.

Cement and Alternatives in the Anthropocene

Globally, the production of concrete is responsible for 5% to 8% of anthropogenic CO2 emissions. Cement, a primary ingredient in concrete, forms a glue that holds concrete together when combined with water. Cement embodies approximately 90% of the greenhouse gas emissions associated with concrete production, and decarbonization methods focus primarily on cement production. But mitigation strategies can accrue throughout the concrete life cycle. Decarbonization strategies in cement manufacture, use, and disposal can be rapidly implemented to address the global challenge of equitably meeting societal needs and climate goals. This review describes (a) the development of our reliance on cement and concrete and the consequent environmental impacts, (b) pathways to decarbonization throughout the concrete value chain, and (c) alternative resources that can be leveraged to further reduce emissions while meeting global demands. We close by highlighting a research agenda to mitigate the climate damages from our continued dependence on cement.

Granular fuzzy calculus on time scales and its applications to fuzzy dynamic equations

This paper introduces the foundational theory of fuzzy calculus on time scales, utilizing granular arithmetic operations between fuzzy intervals. These operations are developed based on the concept of the horizontal membership function (HMF), which is applied in multidimensional fuzzy arithmetic (MFA). Furthermore, the paper explores the existence of a unique solution and the continuous dependence of the solution to fuzzy dynamic equations on initial data, employing the Banach fixed-point theorem under a new metric for fuzzy functions in time scales involving the generalized exponential function. Finally, to highlight the practical significance of these results and their potential applications, the paper presents mathematical models relevant to nuclear physics and biology.

A Comprehensive View of the Solvability and Stability of a Feedback Control Problem with a State-Dependent Delay Implicit Pantograph Equation

In this paper, we examine the existence of a unique solution of a feedback control problem with an implicit state-dependent pantograph equation. Additionally, the study implements the problem’s Hyers-Ulam stability and the continuous dependence of the unique solution on the initial data and the parameters. Furthermore, we investigate this problem in the absence of feedback control. We also provide some examples to illustrate our results.

Delayed Interval-Valued Symmetric Stochastic Integral Equations

In this paper, delayed stochastic integral equations with an initial condition and a drift coefficient given as interval-valued mappings are considered. These equations have a certain symmetric form that distinguishes them from classical single-valued stochastic integral equations and has implications for the properties of the diameter of the values of the solutions of the equations. The main result of the paper is the theorem that there is a unique solution to the equation considered. It was obtained under the assumptions of continuity of the kernels and Lipschitz continuity of the drift and diffusion coefficients. The proof of the existence of the solution is carried out by the method of iterating successive approximations. The paper ends with theorems about the continuous dependence of the solution on the initial function, kernels and nonlinearities.

An advection–diffusion equation with a generalized advection term: Well-posedness analysis and examples

We consider a Cauchy–Dirichlet problem for a semilinear advection–diffusion equation with a generalized advection term. Specific examples include an incision–diffusion landscape evolution model and a viscous Hamilton–Jacobi equation with an absorbing gradient term. We establish existence and uniqueness of a weak solution and its continuous dependence on initial data and a source term.

On implicit fuzzy fractional differential equations with $\phi$-Caputo derivative: Existence, uniqueness and continuous dependence results

On implicit fuzzy fractional differential equations with $\phi$-Caputo derivative: Existence, uniqueness and continuous dependence results

Dependence on initial data for a stochastic modified two-component Camassa-Holm system

We study a stochastic modified two-component Camassa-Holm equation on R. We establish a local well-posedness result in the sense of Hadamard, i.e. existence, uniqueness and continuous dependence on initial data, as well as blow-up criteria for pathwise solutions in the Sobolev spaces Hs with s>3/2. Motivated by the work of Miao et al. (2024) [29], we show that the solution map y0↦y(t) defined by the corresponding Cauchy problem is weakly unstable, due to either a lack of strong stability in the exiting time or the absence of uniformly continuous dependence on the initial data.

A qualitative analysis on the double porous thermoelastic bodies with microtemperature

This study examines a mixed initial-boundary value problem in thermoelastic materials with a double porosity structure, taking into account the effects of microtemperature. The existence of a solution is established by converting the problem into a Cauchy-type problem. Given the complexity of the equations, unknowns, and conditions, we apply contraction semigroup theory within a specific Hilbert space. We prove the existence of a solution using the Lax-Milgram theorem. Additionally, the uniqueness of the solution is demonstrated based on the Lumer-Phillips corollary, which corresponds to the Hille-Yosida theorem. In the final section, we show the continuous dependence of the solution on the mixed initial-boundary value problem for double porous thermoelasticity with microtemperature.

Strong Solutions of Brusselator System

The study involves a mathematical analysis of the Brusselator system on a convex bounded three-dimensional open domain, considering Neumann boundary conditions. We establish the global existence and uniqueness of the strong solution for this system. Achieving high regularity for the strong solution requires stringent conditions on the initial data. The study demonstrates the continuous dependence of the solution on the initial conditions.

Quantum entanglement in a mixed-spin trimer: Effects of a magnetic field and heterogeneous g factors.

Mixed spin-(1/2,1/2,1) trimer with two different Landé gfactors and two different exchange couplings is considered. The main feature of the model is nonconserving magnetization. The Hamiltonian of the system is diagonalized analytically. We presented a detailed analysis of the ground-state properties, revealing several possible ground-state phase diagrams and magnetization profiles. The main focus is on how nonconserving magnetization affects quantum entanglement. We have found that nonconserving magnetization can bring a continuous dependence of the entanglement quantifying parameter (negativity) on themagnetic field within the same eigenstate, while for the case of uniform gfactors it is a constant. The main result is an essential enhancement of the entanglement in thecase of uniform couplings for one pair of spins caused by an arbitrary small difference in the values of gfactors. This enhancement is robust and brings almosta sevenfold increase of the negativity. We have also found aweakening of entanglement for other cases. Thus, nonconserving magnetization offers a broad opportunity to manipulate the entanglement by means of amagnetic field.