Sobolev Inequality Research Articles

Rotating waves in nonlinear media and critical degenerate Sobolev inequalities

Rotating waves in nonlinear media and critical degenerate Sobolev inequalities

Blow-up in finite time for a pseudo-parabolic equation with variable exponents

This paper investigates the blow-up phenomenon in a class of pseudo-parabolic equations with variable exponents. We demonstrate that the solutions to the corresponding initial boundary value problem blow up in finite time when the initial energy is positive and the initial data is sufficiently large. Recently, a blow-up result for this equation with non-positive initial energy was established in Applied Mathematics Letters (2023). Notably, we have eliminated the conditions typically imposed by Sobolev's inequality, this process is applicable to a variety of equations.

Reversed Hardy–Littlewood–Sobolev inequalities with vertical weights on the upper half space

Reversed Hardy–Littlewood–Sobolev inequalities with vertical weights on the upper half space

The spectrum for the p-Laplacian systems with rotating periodic boundary value conditions and applications

In this paper, we study the spectrum problem for the p-Laplacian system (ϕp(u′))′+λϕp(u)=θ with rotating periodic boundary value conditions u(T)=Qu(0),u′(T)=Qu′(0), where Q is an N×N orthogonal real matrix and θ represents the zero vector. We give the definitions of spectrum and eigenvalue for the above problem, prove the relationship between them, discuss the basic properties of the eigenvalues and obtain the variational characteristic of the first positive eigenvalue, which deduce the generalized Wirtinger inequality and Sobolev inequality. These results provide a theoretical basis for subsequent research. In order to use the variational method, we establish a Sobolev space corresponding to rotating periodic boundary value conditions and give some useful properties. As applications of the spectrum of the above problem, we prove the existence of solutions for the p-Laplacian system rotating periodic boundary value problem with growth of order p and the rotating periodic Liénard type system involving p-Laplacian. Our results extend and enrich the existing relative work.

Existence of periodic measures of fractional stochastic delay FitzHugh-Nagumo systems on ℝ n

. This article is concerned with periodic measures of fractional stochastic Fitzhugh-Nagumo systems with variable time delay defined on unbounded domains. First, we derive uniform estimates of solutions. Second, we establish the equicontinuity of solutions in probability that is used for proving the tightness of distributions of solutions. Then, we deduce the uniform estimates on the tails of solutions in probility in order to overcome the issue of non-compactness of Sobolev embeddings on unbounded domains. Finally, we demonstrate the existence of periodic measures by utilizing the Ascoli-Arzelà theorem and Krylov-Bogolyubov′s method.

Existence and stability for the travelling waves of the Benjamin equation

Abstract In the seminal work of Benjamin (1974 Nonlinear Wave Motion (American Mathematical Society)), in the late 70s, he has derived the ubiquitous Benjamin model, which is a reduced model in the theory of water waves. Notably, it contains two parameters in its dispersion part and under some special circumstances, it turns into the celebrated KdV or the Benjamin–Ono equation, During the 90s, there was renewed interest in it. Benjamin (1992 J. Fluid Mech. 245 401–11; 1996 Phil. Trans. R. Soc. A 354 1775–806) studied the problem for existence of solitary waves, followed by works of Bona–Chen (1998 Adv. Differ. Equ. 3 51–84), Albert–Bona–Restrepo (1999 SIAM J. Appl. Math. 59 2139–61), Pava (1999 J. Differ. Equ. 152 136–59), who have showed the existence of travelling waves, mostly by variational, but also bifurcation methods. Some results about the stability became available, but unfortunately, those were restricted to either small waves or Benjamin model, close to a distinguished (i.e. KdV or BO) limit. Quite recently, in 2024 (arXiv:2404.04711 [math.AP]), Abdallah et al, proved existence, orbital stability and uniqueness results for these waves, but only for large values of c γ 2 ≫ 1 . In this article, we present an alternative constrained maximization procedure for the construction of these waves, for the full range of the parameters, which allows us to ascertain their spectral stability. Moreover, we extend this construction to all L 2 subcritical cases (i.e. power nonlinearities ( | u | p − 2 u ) x , 2 < p ⩽ 6 ). Finally, we propose a different procedure, based on a specific form of the Sobolev embedding inequality, which works for all powers 2 < p < ∞ , but produces some unstable waves, for large p. Some open questions and a conjecture regarding this last result are proposed for further investigation.

Improved Sobolev inequalities on CR sphere

Improved Sobolev inequalities on CR sphere

Multivalued Random Dynamics of Fractional Reaction–Diffusion–AdvectionEquations Driven by Superlinear Colored Noise on Unbounded Domains

This article investigates the multivalued random dynamics generated by solution operators of fractional random reaction–diffusion–advection equations driven by superlinear colored noise on unbounded domains. The equations under consideration exhibit non-Lipschitz continuous nonlinear drift and diffusion terms, which lead to the non-uniqueness of solutions and hence generate multivalued random dynamical systems. Our goals are to demonstrate: (i) the existence of solutions for fractional reaction–diffusion–advection equations driven by superlinear colored noise on unbounded domains; and (ii) the existence and uniqueness of pullback random attractors for these systems. The measurability of the random attractors is demonstrated using a method that relies on the weak upper semicontinuity of the solutions. The asymptotic compactness of solution operators is obtained by employing Ball’s approach to energy equations in order to address the non-compactness of Sobolev embeddings on the entire space.

Pointwise estimates for rough operators with applications to Sobolev inequalities

Pointwise estimates for rough operators with applications to Sobolev inequalities

Gagliardo–Nirenberg–Sobolev inequalities in John domains

We build up a Gagliardo–Nirenberg–Sobolev inequality in John domains and, conversely, under an extra separation property, we show that a bounded domain supporting such a Gagliardo–Nirenberg–Sobolev inequality should be a John domain.

Quantitative Stability of Sobolev Inequalities on Compact Riemannian Manifolds

Abstract We study quantitative stability results for different classes of Sobolev inequalities on general compact Riemannian manifolds. We prove that, up to constants depending on the manifold, a function that nearly saturates a critical Sobolev inequality is quantitatively $W^{1,2}$-close to a non-empty set of extremal functions, provided that the corresponding optimal Sobolev constant satisfies a suitable strict bound. The case of sub-critical Sobolev inequalities is also covered. Finally, we discuss degenerate phenomena in our quantitative controls.

Doubly Exponential Growth and Decay for a Semilinear Heat Equation With Logarithmic Nonlinearity

ABSTRACTIn this note, we consider the initial boundary value problem for a parabolic equation with logarithmic nonlinearity, which has been studied by Chen et al. and Han. We firstly estimate the upper bound function with doubly exponential property for the weak solutions and establish the doubly exponential decay and at most doubly exponential growth criteria for the weak solutions by the logarithmic Sobolev inequality, the derivative formula for the product, the Newton–Leibniz formula, and the logarithmic properties. We secondly establish a newly infinity blow‐up criterion such that the lower bound function of weak solutions is at least doubly exponential growth by the nonconcavity method and the nonincreasing property for the energy functional. We finally establish a sufficient condition such that the upper bound function with doubly exponential growth is greater than or equal to the lower bound function with doubly exponential growth by the quotient comparison principle. In addition, we also find the threshold of the doubly exponential growth and the doubly exponential decay and obtain for any as These research results improve previous results from both decay and growth.

Sobolev spaces on locally finite graphs

In this paper, we focus on the theory of Sobolev spaces on locally finite graphs, including completeness, reflexivity, separability and Sobolev inequalities. We introduce a linear space composed of vector-valued functions with variable dimensions such that the gradients of functions on graphs happen to fit into such a space and we can get the desired properties of various Sobolev spaces along this line. Moreover, we also derive several Sobolev inequalities under certain assumptions on measures or weights of graphs. Although these results are within the framework of functional analysis, the key is that we provide an appropriate perspective for applying variational methods on graphs. As fundamental analytical tools, all these results are highly applicable and useful for partial differential equations on locally finite graphs.

Inverse problems for the fractional diffusion equation driven by fractional Brownian sheet

Abstract This paper explores the inverse problems of the space-time fractional diffusion equation driven by a space-time fractional Brownian sheet, including spatial stochastic source term, spatial deterministic source term, potential, and fractional order. In order to deal with the stochastic integral of fractional Brownian sheet with space-time Hurst indexes $\mathcal{H}_1 , \mathcal{H}_2 \in (0,1)$, we construct a unified Itô isometry framework for the first time, upon which we establish the regularity of the solution to the direct problem. Then, we ingeniously employ the generalized fractional Grönwall inequality, Sobolev embedding, and combine the statistical characteristics of the outward normal derivative of the weak solution on open subdomain at the boundary to achieve accurate recovery of the stochastic space-time source terms and prove the uniqueness of the space deterministic source term. Furthermore, by constructing a contraction mapping and exploiting the Banach fixed point theorem, we establish the uniqueness of the potential. Afterward, using the statistical characteristics of the solution at the terminal time, we establish the stability of the potential. Finally, leveraging the representation of the mild solution and the asymptotic properties of the Mittag-Leffler function, we examine the uniqueness and stability of fractional order at known terminal time. In scenario involving an unknown terminal time, we derive the uniqueness and stability of the terminal time.

Sharp quantitative stability of Poincaré-Sobolev inequality in the hyperbolic space and applications to fast diffusion flows

Consider the Poincaré-Sobolev inequality on the hyperbolic space: for every n≥3 and 1<p≤n+2n-2, there exists a best constant Sn,p,λ(Bn)>00$$\\end{document}]]> such that Sn,p,λ(Bn)∫Bn|u|p+1dvBn2p+1≤∫Bn|∇Bnu|2-λu2dvBn,holds for all u∈Cc∞(Bn), and λ≤(n-1)24, where (n-1)24 is the bottom of the L2-spectrum of -ΔBn. It is known from the results of Mancini and Sandeep (Ann. Sc. Norm. Super. Pisa Cl. Sci. 7 (4): 635–671, 2008) that under appropriate assumptions on n, p and λ there exists an optimizer, unique up to the hyperbolic isometries, attaining the best constant Sn,p,λ(Bn). In this article, we investigate the quantitative gradient stability of the above inequality and the corresponding Euler-Lagrange equation locally around a bubble. Our result generalizes the sharp quantitative stability of Sobolev inequality in Rn by Bianchi and Egnell (J. Funct. Anal. 100 (1): 18–24. 1991) and Ciraolo, Figalli and Maggi (Int. Math. Res. Not. IMRN (21): 6780–6797, 2018) to the Poincaré-Sobolev inequality on the hyperbolic space. Furthermore, combining our stability results and implementing a novel and refined smoothing estimates in spirit of Bonforte and Figalli (Comm. Pure Appl. Math. 74 (4): 744–789, 2021), we prove a quantitative extinction rate towards its basin of attraction of the solutions of the sub-critical fast diffusion flow for radial initial data. In another application, we derive sharp quantitative stability of the Hardy-Sobolev-Maz’ya inequalities for the class of functions which are symmetric in the component of singularity.

A Logarithmic Sobolev Inequality for Closed Submanifolds with Constant Length of Mean Curvature Vector

A Logarithmic Sobolev Inequality for Closed Submanifolds with Constant Length of Mean Curvature Vector

Almost sharp Sobolev trace inequalities in the unit ball under constraints

We establish three families of Sobolev trace inequalities of orders two and four in the unit ball under higher order moments constraint, and are able to construct smooth test functions to show all such inequalities are almost optimal. Some distinct feature in almost sharp examples between the fourth order and second order Sobolev trace inequalities is discovered. This has been neglected in higher order Sobolev inequality case in [21]. As a byproduct, the method of our construction can be used to show the sharpness of the generalized Lebedev-Milin inequality under constraints.

SOBOLEV SPACES ON HYPERGROUP GELFAND PAIRS

This paper introduces Sobolev spaces over Gelfand pairs in the framework of hypergroups. The Sobolev spaces in question are constructed from the Fourier transform on hypergroup Gelfand pairs. Mainly, the paper focuses on the investigation of Sobolev embedding results.

Quantitative profile decomposition and stability for a nonlocal Sobolev inequality

In this paper, we focus on studying the quantitative stability of the nonlocal Sobolev inequality given bySHL(∫RN(|x|−μ⁎|u|2μ⁎)|u|2μ⁎dx)12μ⁎≤∫RN|∇u|2dx,∀u∈D1,2(RN), where ⁎ denotes the convolution of functions, 2μ⁎:=2N−μN−2 and SHL are positive constants that depends solely on N and μ. For N≥3 and 0<μ<N, it is well-known that, up to translation and scaling, the nonlocal Sobolev inequality possesses a unique extremal function W[ξ,λ] that is positive and radially symmetric.Our research consists of three main parts. Firstly, we prove a result that provides quantitative stability of the nonlocal Sobolev inequality with the level of gradients. Secondly, we establish the stability of profile decomposition in relation to the Euler-Lagrange equation of the aforementioned inequality for nonnegative functions. Lastly, we investigate the quantitative stability of the nonlocal Sobolev inequality in the following form:‖∇u−∑i=1κ∇W[ξi,λi]‖L2≤C‖Δu+(1|x|μ⁎|u|2μ⁎)|u|2μ⁎−2u‖(D1,2(RN))−1, where the parameter region satisfies κ≥2, 3≤N<6−μ, μ∈(0,N) with 0<μ≤4, or in the case of dimension N≥3 and κ=1, μ∈(0,N) with 0<μ≤4.

Maximal noncompactness of limiting Sobolev embeddings

Abstract We develop a new method suitable for establishing lower bounds on the ball measure of noncompactness of operators acting between considerably general quasinormed function spaces. This new method removes some of the restrictions oft-presented in the previous work. Most notably, the target function space need not be disjointly superadditive nor equipped with a norm. Instead, a property that is far more often at our disposal is exploited—namely the absolute continuity of the target quasinorm. We use this new method to prove that limiting Sobolev embeddings into spaces of Brezis–Wainger type are so-called maximally noncompact, i.e. their ball measure of noncompactness is the worst possible.