Variable Exponent Research Articles

Stabilization of quasilinear systems with variable exponents: A modified approach for energy decay rate determination

This article investigates the stabilization of quasilinear systems governed by elliptic partial differential equations (PDEs) with variable exponents using the modified Tikhonov approach. Our main focus is to determine the energy decay rate of the solution. The findings have significant potential for practical applications in various fields of science and engineering, where the energy decay rate plays a vital role in system stability and control.

Some variable exponent boundedness and commutators estimates for fractional Rough Hardy operators on central Morrey space

In this article, we study the boundedness of the fractional Rough Hardy operator and its adjoint operators on the central Morrey space with a variable exponent. We also establish the same boundedness for their commutators when the symbol functions are on the λ-central BMO space with a variable exponent.

On Weighted Cauchy-Type Problem of Riemann-Liouville Fractional Differential Equations in Lebesgue Spaces with Variable Exponent

This paper aims to investigate the existence, uniqueness, and stability properties for a class of fractional weighted Cauchy-type problem in the variable exponent Lebesgue space $L^{p(.)}$. The obtained results are set up by employing generalized intervals and piece-wise constant functions so that the $L^{p(.)}$ is transformed into the classical Lebesgue spaces. Moreover, the usual Banach Contraction Principle is utilized, and the Ulam-Hyers (UH) stability is studied. At the final stage, we provide an example to support the accuracy of the obtained results.

Multiple Solutions for a Class of Biharmonic Nonlocal Elliptic Systems

We prove the existence of three distinct solutions for a biharmonic nonlocal elliptic system with singular terms under the Navier boundary conditions, by using variational methods and the theory of the variable exponent Sobolev space.

Multiplicity of Solutions for the Noncooperative Kirchhoff-Type Variable Exponent Elliptic System with Nonlinear Boundary Conditions

Considering the solutions of a class of noncooperative Kirchhoff-type p(x)-Laplacian elliptic systems with nonlinear boundary conditions, we derive a sequence of solutions utilizing both the variational method and limit index theory under certain underlying assumptions. The novelty of this study is that we verify the (PS)c* condition using another method, diverging from the approaches cited in the previous literature.

On a class of a coupled nonlinear viscoelastic Kirchhoff equations variable-exponents: global existence, blow up, growth and decay of solutions

In this work, we consider a quasilinear system of viscoelastic equations with dispersion, source, and variable exponents. Under suitable assumptions on the initial data and the relaxation functions, we obtained that the solution of the system is global and bounded. Next, the blow-up is proved with negative initial energy. After that, the exponential growth of solutions is showed with positive initial energy, and by using an integral inequality due to Komornik, the general decay result is obtained in the case of absence of the source term.

Two solutions for Dirichlet double phase problems with variable exponents

Abstract This paper is devoted to the study of a double phase problem with variable exponents and Dirichlet boundary condition. Based on an abstract critical point theorem, we establish existence results under very general assumptions on the nonlinear term, such as a subcritical growth and a superlinear condition. In particular, we prove the existence of two bounded weak solutions with opposite energy sign and we state some special cases in which they turn out to be nonnegative.

Nonlinear parabolic double phase variable exponent systems with applications in image noise removal

In this paper, a novel parabolic system involving nonlinear and nonhomogeneous partial differential operator with variable growth structure is introduced for investigating the image denoising and restoration. More precisely, our model is based on regularizing the classical models involving variable exponent operators by considering a nonlinear operator with double phase flux. We begin by investigating theoretically the solvability to the parabolic system under consideration. Under the setting of Musielak-Orlicz spaces, we build a suitable functional framework to study the proposed system. Therefore, we develop the Faedo-Galerkin approach to demonstrate the existence and uniqueness of a weak solution to our model. To illustrate our theoretical results in the context of image noise removal, we present various numerical implementations on some grayscale images. To enrich these simulations, we test the robustness efficiency of the proposed model in the so-called Magnetic Resonance Images (MRI). The obtained numerical results claim that our model is more efficient and robust against noise, in comparison (visually and quantitatively) to some existing state-of-the-art methods.

On a biharmonic coupled system with non-standard nonlinearity: Existence, blow up and numerics

In this paper, we consider a coupled system of two biharmonic equations with damping and source terms of variable-exponent nonlinearities, supplemented with initial and mixed boundary conditions. We establish an existence and uniqueness result of a weak solution, under suitable assumptions on the variable exponents. Then, we show that solutions with negative-initial energy blow up in finite time. To illustrate our theoritical findings, we present two numerical examples.

Existence of solutions for elliptic systems involving the fractional p(x)-Laplacian

In this paper, we consider a class of non-homogeneous fractional (p (.), q (.))−Laplacian systems. By using variational methods and combining with the theory of Lebesgue and fractional Sobolev spaces with variable exponents, we prove the existence of solutions to the system.

Power-law frequency-dependent Q simulations in viscoacoustic media using decoupled fractional Laplacians

Quantifying the seismic attenuation of wave propagation in the earth’s interior is essential for studying subsurface structures. Previous approaches for attenuation simulations (e.g., the standard linear solid and the fractional derivative model) are mainly based on the frequency-independent quality factor Q assumption. However, seismic attenuation in high-temperature and high-pressure regions usually exhibits power-law frequency-dependent Q characteristics. To simulate this Q effect in attenuative media, we derive a new viscoacoustic wave equation with decoupled fractional Laplacians in the time domain. Unlike the existing methods using relaxation functions to fit the power-law relationship in a specific frequency band, our equation is directly derived from the approximated complex modulus, which explicitly involves the reference quality factor and fractional exponent parameters. Furthermore, this equation contains two fractional Laplacians, which can easily simulate decoupled amplitude dissipation and phase distortion effects, making it amenable to Q-compensated reverse time migration. In the implementation, a Taylor-series expansion and a pseudospectral method are introduced to solve the fractional Laplacians with variable fractional exponents. Numerical experiments demonstrate the effectiveness of our method for power-law frequency-dependent Q simulations. As a forward-modeling engine, our derived viscoacoustic wave equation is a good supplement to the current Q simulation methods and it could be applied in many seismic applications, such as Q-compensated reverse time migration and full-waveform inversion.

Sobolev embeddings in Musielak-Orlicz spaces

An embedding theorem for Sobolev spaces built upon general Musielak-Orlicz norms is offered. These norms are defined in terms of generalized Young functions which also depend on the x variable. Under minimal conditions on the latter dependence, a Sobolev conjugate is associated with any function of this type. Such a conjugate is sharp, in the sense that, for each fixed x, it agrees with the sharp Sobolev conjugate in classical Orlicz spaces. Both Sobolev inequalities in the whole Rn and Sobolev-Poincaré inequalities in domains are established. Compact Sobolev embeddings are also presented. In particular, optimal embeddings for standard Orlicz-Sobolev spaces, variable exponent Sobolev spaces, and double-phase Sobolev spaces are recovered and complemented in borderline cases. A key tool, of independent interest, in our approach is a new weak type inequality for Riesz potentials in Musielak-Orlicz spaces involving a sharp fractional-order Sobolev conjugate.

Analyticity for the fractional Navier-Stokes equations in critical Fourier-Besov-Morrey Spaces with variable exponents

In this paper, by using the Littlewood-Paley theory and the Fourier localization argument, we obtained the analyticity of the solution to the fractional Navier-Stokes equations in variable exponents Fourier-Besov-Morrey spaces when the initial data are small.

Direct Estimates of the Rate of Approximation by the Kantorovich Operator in Variable Exponent Lebesgue Spaces

Direct Estimates of the Rate of Approximation by the Kantorovich Operator in Variable Exponent Lebesgue Spaces

Extrapolation to two-weighted Herz spaces with three variable exponents

Extrapolation to two-weighted Herz spaces with three variable exponents

Multiplicity of solutions for a nonlinear nonlocal problem with variable exponent

This work deals with a class of value problem involving the $p(x)$-biharmonic elliptic equation \begin{equation*} \begin{gathered} M_1(\int_{\Omega}\frac{1}{p(x)}|\Delta u|^{p(x)}dx)\Delta^2_{p(x)}u-M_2(\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx)\Delta_{p(x)}u=\lambda f(x,u)+\mu g(x,u) \quad\text{in }\Omega,\\ u=\Delta u=0,\quad x\in \partial \Omega, \end{gathered} \end{equation*} where $\Omega$ is a bounded domain in $\mathbb{R}^N,~N\geq1.$ with smooth boundary $\partial \Omega.$ Our technical method is based on a theorem obtained by B. Ricceri.

Sequences of nodal solutions for critical double phase problems with variable exponents

In this paper, we study a double phase problem with both variable exponents. Such problem has a reaction consisting of a Carathéodory perturbation defined only locally and of a critical term. The presence of the critical term does not permit to use results of the critical point theory for the corresponding energy functional. Consequently, using suitable cut-off functions and truncation techniques we focus on an auxiliary coercive problem on which, differently from our main problem, we can act with variational tools. In this way, we are able to produce a sequence of sign-changing solutions to our main problem converging to 0 in L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^{\\infty }$$\\end{document} and in the Musielak–Orlicz Sobolev space.

Infinitely many solutions for a p(x)-triharmonic equation with Navier boundary conditions

Abstract In this work, we will study the existence of an infinity of solutions of a Navier problem governed by the p ⁢ ( x ) {p(x)} -triharmonic operator using the theory of Ljusternick–Shrilemann and the theory of the variable exponent Sobolev spaces.

Bilinear Calderón-Zygmund operator and its commutator on some variable exponent spaces of homogeneous type

Let $(X,d,\mu)$ be a space of homogeneous type in the sense of Coifman and and Weiss. In this setting, the author proves that a bilinear Calderon-Zygmund operator is bounded from the product of variable exponent Lebesgue spaces $L^{p_{1}(\cdot)}(X)\times L^{p_{2}(\cdot)}(X)$ into spaces $L^{p(\cdot)}(X)$, and it is bounded from the product of variable exponent generalized Morrey spaces $\mathcal{L}^{p_{1}(\cdot),\varphi_{1}}(X)\times \mathcal{L}^{p_{2}(\cdot),\varphi_{2}}(X)$ into spaces $\mathcal{L}^{p(\cdot),\varphi}(X)$, where the Lebesgue measure functions $\varphi(\cdot,\cdot), \varphi_{1}(\cdot,\cdot)$ and $\varphi_{2}(\cdot,\cdot)$ satisfy $\varphi_{1}\times\varphi_{2}=\varphi$, and $\frac{1}{p(\cdot)}=\frac{1}{p_{1}(\cdot)}+\frac{1}{p_{2}(\cdot)}$. Furthermore, by establishing sharp maximal estimate for the commutator $[b_{1},b_{2},BT]$ generated by $b_{1}, b_{2}\in\mathrm{BMO}(X)$ and $BT$, the author shows that the $[b_{1},b_{2},BT]$ is bounded from the product of spaces $L^{p_{1}(\cdot)}(X)\times L^{p_{2}(\cdot)}(X)$ into spaces $L^{p(\cdot)}(X)$, and it is also bounded from product of spaces $\mathcal{L}^{p_{1}(\cdot),\varphi_{1}}(X)\times \mathcal{L}^{p_{2}(\cdot),\varphi_{2}}(X)$ into spaces $L^{p(\cdot),\varphi}(X)$.

Author Correction: On the concentration-compactness principle for anisotropic variable exponent Sobolev spaces and its applications

Author Correction: On the concentration-compactness principle for anisotropic variable exponent Sobolev spaces and its applications