Faedo Galerkin Approximation Research Articles

Boundary Stability Criterion for a Nonlinear Axially Moving Beam

This article deals with, in the framework of absolute stability, boundary stabilization for a nonlinear axially moving beam under boundary velocity feedback controls. The nonlinear boundary control that satisfies a slope-sector condition covering many types of nonlinear control schemes is a negative feedback of the transverse velocity at the right eyelet of the moving beam. Under the nonlinear control scheme, the well-posedness of the nonlinear partial differential equation, which depends continuously on the initial value is investigated by means of the Faedo–Galerkin approximation and priori estimates. By exploiting the integral-type multiplier method, the exponential stability of the closed-loop system is established, where a novel energy like function is constructed. The numerical simulation examples using the finite element method are presented to illustrate the effectiveness of the established criterion of the controller.

On a singular parabolic p-biharmonic equation with logarithmic nonlinearity

This paper is concerned with the well-posedness and asymptotic behavior of Dirichlet initial boundary value problem for a singular parabolic p-biharmonic equation with logarithmic nonlinearity. We establish the local solvability by the technique of cut-off combining with the methods of Faedo–Galerkin approximation and multiplier. Meantime, by virtue of the family of potential wells, we use the technique of modified differential inequality and improved logarithmic Sobolev inequality to obtain the global solvability, infinite and finite time blow-up phenomena, and derive the upper bound of blow-up time as well as the estimate of blow-up rate. Furthermore, the results of blow-up with arbitrary initial energy and extinction phenomena are presented.

Weak solutions to the time-fractional g-Bénard equations

The Bénard problem consists in a system that couples the well-known Navier–Stokes equations and an advection-diffusion equation. In thin varying domains this leads to the g-Bénard problem, which turns out to be the classical Bénard problem when g is constant. The main goal of this paper is to, first of all, introduce the g-Bénard problem with time-fractional derivative of order alpha in (0,1). This formulation is new even in the classical Bénard problem, that is with constant g. The second goal of this paper is to prove the existence and uniqueness of a weak solution by means of the Faedo–Galerkin approximation method. Some recent works on time-fractional Navier–Stokes equations have opened new perspectives in studying variational aspects in problems involving time-fractional derivatives.

High‐gain stabilization for an axially moving beam under boundary feedback control

This paper establishes the global existence and high‐gain stabilization of a nonlinear axially moving beam with control input at the free boundary. A high‐gain controller based on the transverse velocity feedbacks of the moving beam at the free end is designed. The existence and uniqueness of the solution depending on the initial values continuously for the resulting closed‐loop system are established by invoking the Faedo–Galerkin approximation approach. Then constructing a novel energy‐like function, the explicit exponential decay rate of the closed‐loop system is obtained via a generalized Gronwall‐type integral inequality.

New existence results on nonlocal neutral fractional differential equation in concepts of Caputo derivative with impulsive conditions

In this article, we examine a nonlinear impulsive neutral fractional differential equation with nonlocal condition in an arbitrary Hilbert space. We obtain an associated integral equation and then consider a sequence of approximate integral equations by the projection of considered associated nonlocal neutral fractional integral equation onto finite dimensional space. The existence and uniqueness of an approximate solution by using analytic semigroup theory and the Fixed-point method are demonstrated. Convergence of the solutions of the approximate integral equation is proved. The Faedo-Galerkin approximation of the solution is studied and demonstrated some convergence results. Lastly, the application is presented to illustrate the theory of the main results.

Wong-Zakai approximation and support theorem for 2D and 3D stochastic convective Brinkman-Forchheimer equations

In this work, we demonstrate the Wong-Zakai approximation results for two- and three-dimensional stochastic convective Brinkman-Forchheimer (SCBF) equations forced by Hilbert space valued Wiener noise on bounded domains. Even though the existence and uniqueness of a pathwise strong solution to SCBF equations is known, the existence of a unique solution to the approximating system is not immediate from the solvability results of SCBF equations, and we prove it by using the Faedo-Galerkin approximation technique and monotonicity arguments. Moreover, as an application of the Wong-Zakai approximation, we obtain the support of the distribution of solutions to SCBF equations.

Exponential stability for wave equation with delay and dynamic boundary conditions

In this paper, we consider a multi-dimensional wave equation with delay and dynamic boundary conditions, related to the Kelvin–Voigt damping. By using the Faedo–Galerkin approximations together with some priori estimates, we prove the local existence of solution. Since the damping may stabilize the system while the delay may destabilize it, we discuss the interaction between the damping and the delay, and obtain that the system is uniformly stable when the effect of damping is greater than that of time delay. Exponential stability result of system is also established by constructing suitable Lyapunov functionals.

Martingale solutions of stochastic nonlocal cross-diffusion systems

<p style='text-indent:20px;'>We establish the existence of solutions for a class of stochastic reaction-diffusion systems with cross-diffusion terms modeling interspecific competition between two populations. More precisely, we prove the existence of weak martingale solutions employing appropriate Faedo-Galerkin approximations and the stochastic compactness method. The nonnegativity of solutions is proved by a stochastic adaptation of the well-known Stampacchia approach.</p>

Exponential stability of axially moving Kirchhoff-beam systems with nonlinear boundary damping and disturbance

<p style='text-indent:20px;'>This paper examines the stabilization problem of the axially moving Kirchhoff beam. Under the nonlinear damping criterion established by the slope-restricted condition, the existence and uniqueness of solutions of the closed-loop system equipped with nonlinear time-delay disturbance at the boundary is investigated via the Faedo-Galerkin approximation method. Furthermore, the solution is continuously dependent on initial conditions. Then the exponential stability of the closed-loop system is established by the direct Lyapunov method, where a novel energy function is constructed.</p>

Well-posedness and energy decay of a transmission problem of Kirchhoff type wave equations with damping and delay terms

We consider a transmission problem of Kirchhoff type wave equations with delay and damping terms, subject to a memory condition on one part of the boundary. Under appropriate hypotheses on the relaxation function and the relationship between weights of damping and delay terms, we establish well-posedness of the problem. Using the Faedo-Galerkin approximation technique, and introducing suitable energyand Lyapunov functionals, we obtain estimates for exponential, polynomial, and logarithmic decay. For more information see https://ejde.math.txstate.edu/Volumes/2021/95/abstr.html

Existence of tumor invasion model with mesenchymal and epithelial transition processes in variable exponent spaces

This work illustrate the solvability of the system consist of tumor cells of two types along with epithelial cells and mesenchymal densities. The role of proteins matrix metalloproteinases and extra cellular matrix are considered. Along with the square integrability of the mesenchymal epithelial transition function, the solvability is illustrated under the Faedo–Galerkin approximation method.

Existence and blow up in a system of wave equations with nonstandard nonlinearities

In this article, we consider a coupled system of two nonlinear hyperbolic equations, where the exponents in the damping and source terms are variables. First, we prove a theorem of existence and uniqueness of weak solution, by using the Faedo Galerkin approximations and the Banach fixed oint theorem. Then, using the energy method, we show that certain solutions with positive initial energy blow up in finite time. We also give some numerical applications to illustrate our theoretical results. For more information see https://ejde.math.txstate.edu/Volumes/2021/91/abstr.html

Weighted Hardy–Sobolev inequality and global existence result of thermoelastic system on manifolds with corner‐edge singularities

This article concerns with the thermoelastic corner‐edge‐type system with singular potential function on a wedge manifold with corner singularities. First, we introduce weighted p‐Sobolev spaces on manifolds with corner‐edge singularities. Then, we prove the corner‐edge‐type Sobolev inequality, Poincar inequality and Hardy inequality and obtain some results about the compactness of embedding maps on the weighted corner‐edge Sobolev spaces. Finally, as an application of these results, we apply the potential well theory and the Faedo–Galerkin approximations to obtain the global weak solutions for the thermoelastic corner‐edge‐type system 1.1.

The initial-nonlinear nonlocal solutions for a parabolic system in a weighted Sobolev space

In this paper, we prove the existence of the initial-nonlinear nonlocal solutions for a parabolic system in a weighted Sobolev space. The methods applied are the Faedo–Galerkin approximation and the general theory of weak compactness in appropriate weighted Sobolev spaces together with using the Poincaré-type operator for dealing nonlinear nonlocal conditions. Furthermore, the boundedness and positivity of solutions depending on the boundedness and positivity of given data are also discussed by using suitable test functions.

Bresse-Timoshenko type systems with thermodiffusion effects: well-possedness, stability and numerical results

Bresse-Timoshenko beam model with thermal, mass diffusion and theormoelastic effects is studied. We state and prove the well-posedness of problem. The global existence and uniqueness of the solution is proved by using the classical Faedo-Galerkin approximations along with two a priori estimates. We prove an exponential stability estimate for problem under an unusual assumption, and by using a multiplier technique with frictional damping in the vertical displacement. Numerically, we construct a numerical scheme based on the \(P_1\)-finite element method for space discretization and implicit Euler scheme for time discretization. Then, we showed that the discrete energy decays, later a priori error estimates are established. Finally, some numerical simulations are presented.

Well-posedness and large deviations for 2D stochastic constrained Navier-Stokes equations driven by Lévy noise in the Marcus canonical form

We consider stochastic two-dimensional constrained Navier-Stokes equations driven by Lévy noise in the Marcus canonical form. The aim of this work is two-fold. At first, we prove the existence of a martingale solution based on the construction relying on classical Faedo-Galerkin approximations, compactness method and the Jakubowski's version of Skorokhod representation theorem for non-metric spaces. We further prove that the martingale solution is pathwise unique and deduces the existence of a strong solution. In the second part of the paper, we establish a Wentzell-Freidlin type large deviations principle for the small noise asymptotic of solutions using weak convergence method.

Absolute boundary stabilization for an axially moving Kirchhoff beam

In this paper, we consider absolute stability for an axially moving Kirchhoff beam under nonlinear boundary feedback controls. The nonlinear boundary control which satisfies a slope-restricted condition is a negative feedback of the transverse velocity at the right eyelet of the moving beam. First, the well-posedness of the resulting closed-loop system is established by means of the Faedo–Galerkin approximation combined with some priori estimates. By invoking the integral-type multiplier method, the exponential stability of the closed-loop is developed. The numerical simulations by the finite element method are presented to demonstrate the effectiveness of the proposed controller.

Multivalued nonmonotone dynamic boundary condition

In this paper, we introduce a new class of hemivariational inequalities, called dynamic boundary hemivariational inequalities, reflecting the fact that the governing operator is also active on the boundary. In our context, it concerns the Laplace operator with Wentzell (dynamic) boundary conditions perturbed by a multivalued nonmonotone operator expressed in terms of Clarke subdifferentials. We show that one can reformulate the problem so that standard techniques can be applied. We use the well-established theory of boundary hemivariational inequalities to prove that under growth and general sign conditions, the dynamic boundary hemivariational inequality admits a weak solution. Moreover, in the situation where the functionals are expressed in terms of locally bounded integrands, a “filling in the gaps” procedure at the discontinuity points is used to characterize the subdifferential on the product space. Finally, we prove that, under a growth condition and eventually smallness conditions, the Faedo–Galerkin approximation sequence converges to a desired solution.

Local existence and blow up of solutions for a logarithmic nonlinear viscoelastic wave equation with delay

In this work, we consider a logarithmic nonlinear viscoelastic wave equation with a delay term in a bounded domain. We obtain the local existence of the solution by using the Faedo-Galerkin approximation. Then, under suitable conditions, we prove the blow up of solutions in finite time.

An extremum problem for a linear integro-differential system describing creeping flows of a viscoelastic fluid

We consider an optimal control problem for an integro-differential system (with a quadratic cost functional) modeling a three-dimensional creeping flow of an incompressible viscoelastic fluid in a bounded domain with impermeable solid walls. The fluid flow is controlled by the time-dependent external force. The concept of the control operator is proposed. We prove a theorem on the existence of a unique optimal control under the assumption that the set of admissible controls is convex and that it is closed in a suitable function space. Moreover, we obtain a variational ine-quality for the optimal control. The proof of this theorem is based on the application of the Faedo–Galerkin approximation scheme taking into account energy estimates of approximate solutions and using the lemma on the existence and uniqueness of the metric projection of a point onto a closed convex set in a real Hilbert space.