Faedo Galerkin Approximation Research Articles

Existence of Solutions for a Viscoelastic Plate Equation with Variable Exponents and a General Source Term

The subject of this study is a nonlinear viscoelastic plate equation with variable exponents and a general source term. Through the application of the Faedo–Galerkin approximation method and a fixed point theorem under appropriate assumptions, we proved the existence of weak solutions.

Analytical and numerical studies of a cancer invasion model with nonlocal diffusion

Abstract This article investigates the implicit Euler–Galerkin finite element method applied to a cancer invasion model with nonlocal diffusion. This model describes the normal cell density, tumor cell density, excess H + ion concentration, extracellular matrix, and extracellular matrix active metalloproteinases. The primary goal of this work is to utilize numerical solutions to comprehend cancer invasion and transmission within the patient. The Faedo–Galerkin approximation method is initially used to establish the existence of a weak solution, followed by deriving the a priori error estimates of optimal order. Additionally, the implicit Euler–Galerkin finite element method is employed as a numerical tool for solving the given cancer invasion parabolic system. Furthermore, a priori error bounds and convergence estimates for the fully discrete problem are derived. Finally, numerical simulations are conducted to validate the theoretical conclusions and enhance the understanding of how cancer spreads within the patient.

Global Existence and Blow-up of Solutions for a Class of Singular Parabolic Equations with Viscoelastic Term

In this paper, we consider the initial boundary value problem for a class of singular parabolic equations with viscoelastic term and logarithmic term. By using the technique of cut-off and the method of Faedo-Galerkin approximation, the local existence of the weak solution is established. Based on the potential well method, the global existence of the weak solution is derived. Furthermore, we prove that the weak solution blows up in finite time by taking the concavity analysis method.

Hybrid variable exponent model for image denoising: A nonstandard high-order PDE approach with local and nonlocal coupling

In this paper, we propose a novel hybrid model combining local and nonlocal methods for effective image denoising. The model utilizes variable exponents p(x) to achieve adaptive diffusion behavior and preserve image features. Nevertheless, the coupling structure of our proposed process, along with the spatial dependence of p(x), poses challenges in theoretical analysis. To address this, we investigate the existence and uniqueness of the solution using the Faedo-Galerkin approximation, a priori estimates, and compactness considerations. Through these theoretical investigations, we demonstrate the existence of a weak solution for the coupled parabolic system. Overall, our model outperforms existing methods in noise reduction and image quality preservation.

Energy decay analysis for Porous elastic system with thermoelasticity of type III: A second spectrum approach

Numerous studies have been conducted to investigate porous systems under different damping effects. Recent research has consistently achieved the expected exponential decay of energy solutions when employing stabilization techniques that involve non-physical assumptions of equal wave velocities. In this study, we examine a one-dimensional thermoelastic porous system within the framework of the second frequency spectrum. Remarkably, we demonstrate that exponential decay can be achieved without relying on the assumption of equal wave speeds. We consider the porous system, and we incorporated thermoelastic damping based on the Green–Naghdi law of heat conduction into our study. To begin with, we use the Faedo–Galerkin approximation method to validate the global well-posedness of the system. By utilizing a Lyapunov functional, we establish exponential stability without relying on the assumption of equal wave speed. We then introduce and analyze a numerical scheme. Finally, by assuming additional regularity of the solution, we derive a priori error estimates.

On a singular parabolic $ p $-Laplacian equation with logarithmic nonlinearity

<p>In this paper, we considered a singular parabolic $ p $-Laplacian equation with logarithmic nonlinearity in a bounded domain with homogeneous Dirichlet boundary conditions. We established the local solvability by the technique of cut-off combining with the method of Faedo-Galerkin approximation. Based on the potential well method and Hardy-Sobolev inequality, the global existence of solutions was derived. In addition, we obtained the results of the decay. The blow-up phenomenon of solutions with different indicator ranges was also given. Moreover, we discussed the blow-up of solutions with arbitrary initial energy and the conditions of extinction.</p>

New general decay results for a fully dynamic piezoelectric beam system with fractional delays under memory boundary controls

In this paper, we investigate the general stability results of a fully dynamic piezoelectric beam system with internal fractional delays under memory boundary controls. Firstly, by introducing two new equations and using two Volterra's inverse operators to deal with fractional delay terms and boundary memory terms, respectively, a new equivalent system is obtained. Secondly, by combining Faedo–Galerkin approximations with the compactness argument, we prove the local existence and uniqueness of weak solution. Finally, under a broader class of kernel functions, by constructing Lyapunov functionals and new equations and applying some technical lemmas, we establish the optimal and explicit energy decay results for two cases, namely, initial values and without imposing initial values equal to 0, that is, . It is worth pointing out that we focus more on the analysis of , because the stability results of are special cases of . This is the first time to analyze the stability of strong coupling problem by combining internal fractional delay and boundary viscoelastic damping. The obtained general decay results are not necessarily exponential or polynomial, which generalize the relevant results in the existing literature.

Global Solution and Blow-up for a Thermoelastic System of $p$-Laplacian Type with Logarithmic Source

This manuscript deals with global solution, polynomial stability and blow-up behavior at a finite time for the nonlinear system $$ \left\{ \begin{array}{rcl} & u'' - \Delta_{p} u + \theta + \alpha u' = \left\vert u\right\vert ^{p-2}u\ln \left\vert u\right\vert \\ &\theta' - \Delta \theta = u' \end{array} \right. $$ where $\Delta_{p}$ is the nonlinear $p$-Laplacian operator, $ 2 \leq p < \infty$. Taking into account that the initial data is in a suitable stability set created from the Nehari manifold, the global solution is constructed by means of the Faedo-Galerkin approximations. Polynomial decay is proven for a subcritical level of initial energy. The blow-up behavior is shown on an instability set with negative energy values.

A stochastic Allen–Cahn–Navier–Stokes model with inertial effects driven by multiplicative noise of jump type

AbstractWe consider a stochastic Allen–Cahn–Navier–Stokes equations with inertial effects and multiplicative noise of jump type in a bounded domain , , driven by a multiplicative noise. The existence of a global weak martingale solution is proved under non‐Lipschitz assumptions on the coefficients. The construction of the solution is based on the Faedo–Galerkin approximation, compactness method and the Skorokhod representation theorem.

Well‐posedness and asymptotic behavior for a p‐biharmonic pseudo‐parabolic equation with logarithmic nonlinearity of the gradient type

AbstractThis paper is concerned with the well‐posedness and asymptotic behavior for an initial boundary value problem of a pseudo‐parabolic equation with p‐biharmonic operator and logarithmic nonlinearity of the gradient type. The existence of the global weak solution is established by combining the technique of potential‐well and the method of Faedo–Galerkin approximation. Meantime, by virtue of the improved logarithmic Sobolev inequality and modified differential inequality, we obtain the results on infinite and finite time blow‐up and derive the lifespan of blow‐up solutions in various energy levels. Furthermore, the extinction phenomenon with extinction time is presented.

Approximate solutions for neutral stochastic fractional differential equations

This study focuses on a class of neutral stochastic fractional differential equations of order α∈(1,2] in a separable Hilbert space. The existence and uniqueness of approximate solutions are demonstrated using semigroup theory of bounded linear operators, stochastic analysis techniques, and the Banach contraction principle. The convergence of approximate solutions is illustrated using Faedo–Galerkin approximations. Finally, we give an example to illustrate the abstract results.

Exponential Stability for Damped Shear Beam Model and New Facts Related to the Classical Timoshenko System With a Distributed Delay Term

We consider in this manuscript  a Timoshenko type beam model with a distributed delay term.  If the distributed delay term is small enough, we prove the global existence of solutions by using the Faedo-Galerkin approximations together with some energy estimates. Under suitable assumptions, we prove exponential stability of the solution. This result is obtained by introducing a suitable Lyapounov function.

WELL-POSEDNESS AND ASYMPTOTIC BEHAVIOR FOR THE DISSIPATIVE P-BIHARMONIC WAVE EQUATION WITH LOGARITHMIC NONLINEARITY AND DAMPING TERMS

This paper concerns with the initial and boundary value problem for a p-biharmonic wave equation with logarithmic nonlinearity and damping terms. We establish the well-posedness of the global solution by combining Faedo–Galerkin approximation and the potential well method, and derive both the polynomial and exponential energy decay by introducing an appropriate Lyapunov functional. Moreover, we use the technique of differential inequality to obtain the blow-up conditions and deduce the life-span of the blow-up solution.

Do equal speed condition and exponential stability relate for the truncated thermoelastic Timoshenko system under Green Naghdi law?

Over the years, the stabilization Timoshenko systems with dissipative features have piqued the interest of researchers. The study of Timoshenko systems under various damping effects has resulted in a significant number of studies. When nonphysical assumptions of equal wave velocities are used in stabilization, the expected exponential decay of the energy solution is attained in all recent research. In this study, we analyze a one-dimensional thermooelastic Timoshenko type system in the setting of the second frequency spectrum, where the assumption of equal wave speed is not required for exponential decay to occur. In fact, According to Elishakoff’s studies [Elishakoff, Advances in Mathematical Modeling and Experimental Methods for Materials and Structures: The Jacob Aboudi Volume, Dordrecht, The Netherlands: Springer, pp. 249–254, 2009.], we consider the so-called truncated version of the Timoshenko system, and we added a thermoelastic damping according to Green Naghdi law of heat conduction. We first use Faedo–Galerkin approximation to verify the system’s global well-posedness. Using a Lyapunov functional we establish an exponential stability without assuming the condition of equal wave speed. A numerical scheme is introduced and analyzed. Finally, assuming extra regularity on the solution, we get some a priori error estimates and we present some numerical results which demonstrate the exponential behavior of the solution. This result significantly improves the previous results in the literature in which equal wave velocities are used to obtain exponential stability.

Exponential and polynomial decay results for a swelling porous elastic system with a single nonlinear variable exponent damping: theory and numerics

We consider a swelling porous elastic system with a single nonlinear variable exponent damping. We establish the existence result using the Faedo–Galerkin approximations method, and then, we prove that the system is stable under a natural condition on the parameters of the system and the variable exponent. We obtain exponential and polynomial decay results by using the multiplier method, and these results generalize the existing results in the literature. In addition, we end our paper with some numerical illustrations.

Faedo-Galerkin Approximations for nonlinear Heat equation on Hilbert Manifold

In this work, we aim to study the well-posedness of a deterministic problem consisting of the non-linear heat equation of gradient type. The evolution equation emerges by projecting the Laplace operator with Dirichlet boundary conditions and polynomial nonlinearity of degree $2n-1$, onto the tangent space of the sphere $\mathcal{M}$ in a Hilbert space $\mathcal{H}$. We are going to deal with the question of existence and uniqueness based on the Faedo-Galerkin compactness method.

Blow‐up results for a viscoelastic beam equation of p‐Laplacian type with strong damping and logarithmic source

In this paper, we investigate the existence, uniqueness, exponential decay, and blow‐up behavior of the viscoelastic beam equation involving the ‐Laplacian operator, strong damping, and a logarithmic source term, given by where is a bounded domain of and is a memory kernel. Using the Faedo–Galerkin approximation, we establish the existence and uniqueness result for the global solutions, taking into account that the initial data must belong to an appropriate stability set created from the Nehari manifold. The study of the exponential decay of our problem is based on Nakao's method. Finally, the blow‐up behavior on the instability set is proved.

Existence, uniqueness and approximation of nonlocal fractional differential equation of sobolev type with impulses

<abstract><p>This paper is concerned with the study of nonlocal fractional differential equation of sobolev type with impulsive conditions. An associated integral equation is obtained and then considered a sequence of approximate integral equations. By utilizing the techniques of Banach fixed point approach and analytic semigroup, we obtain the existence and uniqueness of mild solutions to every approximate solution. Then, Faedo-Galerkin approximation is used to establish certain convergence outcome for approximate solutions. In order to illustrate the abstract results, we present an application as a conclusion.</p></abstract>

GLOBAL WELL-POSEDNESS AND EXPONENTIAL DECAY OF SHEAR BEAM MODEL SUBJECT TO A NEUTRAL DELAY

In this paper, we consider a one-dimensional system known as Shear beam model (no rotary inertia) where the transverse displacement equation is subject to a distributed delay of neutral type. Under some assumptions on the kernel h, we first achieved the global well- posedness of the system by using the classical Faedo-Galerkin approximations along with two a priori estimates. Next, we find the energy expression and by using technique of Lyapunov functional we demonstrate, although delays are known to be of a destructive nature in the general case, that this system is exponentially stable regardless any relationship between coefficients of the system.

On the strong convergence of the Faedo-Galerkin approximations to a strong T-periodic solution of the torso-coupled bidomain model

In this paper, we investigate the convergence of the Faedo-Galerkin approximations, in a strong sense, to a strong T-periodic solution of the torso-coupled bidomain model where T is the period of activation of the inner wall of the heart. First, we define the torso-coupled bidomain operator and prove some of its more important properties for our work. After, we define the abstract evolution system of the equations that are associated with torso-coupled bidomain model and give the definition of a strong solution. We prove that the Faedo-Galerkin’s approximations have the regularity of a strong solution, and we find that some restrictions can be imposed over the initial conditions, so that this sequence of Faedo-Galerkin fully converges to a strong solution of the Cauchy problem. Finally, these results are used for showing the existence a strong T-periodic solution.