Boundedness Results Research Articles

Finite-time boundedness and control of positive coupled differential-difference equations with bounded time-varying delay

This paper is concerned with the problems of finite-time boundedness and finite-time control for positive coupled differential-difference equations (CDDEs) with bounded time-varying delay. The finite-time stability of such systems is analyzed by constructing an estimate of the solutions over a finite time interval. And, sufficient conditions based on linear programming (LP) are provided for finite-time stability of positive CDDEs with bounded time-varying delay. Then, by coordinate transformation, the obtained results are extended to the finite-time bounedness of positive CDDEs with bounded time-varying delay. By the obtained result of finite-time boundedness, static output-feedback controllers and static state-feedback controllers are designed and a sufficient condition is derived to ensure the positivity and finite-time boundedness of closed-loop system. Three illustrative examples are given to show the validity of our results.

Well-posedness and asymptotic stability of solutions to the semilinear wave equation with analytic nonlinearity and time varying delay

We consider the semilinear wave equation with time varying delay and linear internal feedbackutt(x,t)−Δu(x,t)+μ1ut(x,t)+μ2ut(x,t−τ(t))+f(x,u)=g, in a regular bounded domain of Rn with Dirichlet boundary conditions. Under suitable relations between μ1 and μ2 and suitable conditions on f and τ we derive first a boundedness result and we show that any global bounded solution of the above equation has a relative compact range in the natural energy space. More importantly, when the nonlinear term f is analytic in the second variable, we show the convergence to equilibrium as well as estimates for the rate of convergence for any global bounded solution. We also discuss existence of global solutions.

Boundedness results for 2‐adic Galois images associated to hyperelliptic Jacobians

AbstractLet K be a number field, and let C be a hyperelliptic curve over K with Jacobian J. Suppose that C is defined by an equation of the form for some irreducible monic polynomial of discriminant Δ and some element . Our first main result says that if there is a prime of K dividing but not (2Δ), then the image of the natural 2‐adic Galois representation is open in and contains a certain congruence subgroup of depending on the maximal power of dividing . We also present and prove a variant of this result that applies when C is defined by an equation of the form for distinct elements . We then show that the hypothesis in the former statement holds for almost all and prove a quantitative form of a uniform boundedness result of Cadoret and Tamagawa.

Boundedness questions for Calabi–Yau threefolds

In this paper, we study boundedness questions for (simply connected) smooth Calabi–Yau threefolds. The diffeomorphism class of such a threefold is known to be determined up to finitely many possibilities by the integral middle cohomology and two integral forms on the integral second cohomology, namely the cubic cup-product form and the linear form given by cup-product with the second Chern class. The motivating question for this paper is whether knowledge of these cubic and linear forms determines the threefold up to finitely many families, that is the moduli of such threefolds is bounded. If this is true, then in particular the middle integral cohomology would be bounded by knowledge of these two forms. Crucial to this question is the study of rigid non-movable surfaces on the threefold, which are the irreducible surfaces that deform with any small deformation of the complex structure of the threefold but for which no multiple moves on the threefold. If for instance there are no such surfaces, then the answer to the motivating question is yes (Theorem 0.1). In particular, for given cubic and linear forms on the second cohomology, there must exist such surfaces for large enough third Betti number (Corollary 0.2). The paper starts by proving general results on these rigid non-movable surfaces and boundedness of the family of threefolds. The basic principle is that if the cohomology classes of these surfaces are also known, then boundedness should hold (Theorem 4.5). The second half of the paper restricts to the case of Picard number 2, where it is shown that knowledge of the cubic and linear forms does indeed bound the family of Calabi–Yau threefolds (Theorem 0.3). This appears to be the first non-trivial case where a general boundedness result for Calabi–Yau threefolds has been proved (without the assumption of a special structure).

Global bounded weak solutions for a chemotaxis-Stokes system with nonlinear diffusion and rotation

In this paper, we discuss the following chemotaxis-Stokes system with nonlinear diffusion and rotation{nt+u⋅∇n=Δnm−∇⋅(nS(x,n,c)⋅∇c),x∈Ω,t>0,ct+u⋅∇c=Δc−nc,x∈Ω,t>0,ut+∇P=Δu+n∇ϕ,x∈Ω,t>0,∇⋅u=0,x∈Ω,t>0 with m>0, where Ω is a bounded domain in R3, S(x,n,c) is a chemotactic sensitivity tensor satisfying S∈C2(Ω¯×[0,∞)2;R3×3) and |S(x,n,c)|≤(1+n)−αS0(c) for all (x,n,c)∈Ω×[0,∞)2 with α≥0 and some nondecreasing function S0:[0,∞)→[0,∞). We can show that if m+α>109, then for any reasonably smooth initial data, the corresponding initial-boundary problem possesses a globally bounded weak solution. The result extends the previous global boundedness result for m+α>109, α>0 and m+54α>98[15], and m>98 and α=0[24]. Without using the conventional free-energy inequality see [15,24], we use a new iterative bootstrap procedure to estimate the bounds on ∫Ωnεp for p>32, which is a crucial step to obtain our main result.

Local boundedness and Hölder continuity for the parabolic fractional p-Laplace equations

In this paper, we study the boundedness and Holder continuity of local weak solutions to the following nonhomogeneous equation $$\begin{aligned} \partial _tu(x,t)+\mathrm{P.V.}\int _{{\mathbb {R}}^N}K(x,y,t)|u(x,t)-u(y,t)|^{p-2}\big (u(x,t)-u(y,t)\big )dy= f(x,t,u) \end{aligned}$$ in $$Q_T=\Omega \times (0,T)$$ , where the symmetric kernel K(x, y, t) has a generalized form of the fractional p-Laplace operator of s-order. We impose some structural conditions on the function f and use the De Giorgi-Nash-Moser iteration to establish the boundedness of local weak solutions in the a priori way. Based on the boundedness result, we also obtain Holder continuity of bounded solutions in the superquadratic case. These results can be regarded as a counterpart to the elliptic case due to Di Castro et al. (Ann Inst H Poincare Anal Non Lineaire, 2016).

Stability and convergence analysis for a new phase field crystal model with a nonlocal Lagrange multiplier

In this work, an energy stable numerical scheme is proposed to solve the PFC model with a nonlocal Lagrange multiplier. The construction of the numerical scheme is based on invariant energy quadratization (IEQ) technique to transform a nonlinear system into a linear system, while the time variables are discretized by second‐order scheme. The stability term in the new scheme can balance the influence of nonlinear term. Moreover, we obtain the results of unconditional energy stability, uniqueness, and uniform boundedness of numerical solution, and the numerical scheme is convergent with order of . Several numerical tests are conducted to confirm the theoretical results.

A new result for boundedness of solutions to a higher-dimensional quasilinear chemotaxis system with a logistic source

We deal with the boundedness of solutions of a class of quasilinear chemotaxis systems generalizing the prototype(⁎){ut=∇⋅(ϕ(u)∇u)−∇⋅(u∇v)+μu(1−u),x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0,∂u∂ν=∂v∂ν=0,x∈∂Ω,t>0,u(x,0)=u0(x),v(x,0)=v0(x),x∈Ω, where RN(N≥2) is a bounded domain with zero-flux boundary condition, μ>0 is a positive parameter, and ϕ(u)=(u+1)−α. It is shown that for all reasonably regular initial data, system (⁎) admits a global existence and boundedness of solutions when α<α0 for some α0>2−NN+2, thereby proving that the value α=2−NN+2 is not critical in this regard. This extends previous results that rely on a different energy-type inequality.

Some Results for the Two-Sided Quaternionic Gabor Fourier Transform and Quaternionic Gabor Frame Operator

In this paper, we first present some properties of the two-sided quaternionic Gabor Fourier transform (QGFT) on quaternion valued function space $$L^2({\mathbb {R}}^2,{\mathbb {H}})$$ , such as Parseval’s formula and the characterization of the range of the two-sided QGFT. Then, we give the definitions of quaternionic Wiener space and quaternionic Gabor frame operator (QGFO), which are the generalizations in the quaternionic settings. Finally, we prove Walnut’s and Janssen’s representation theorems and other boundedness results of the QGFO.

Smooth and singular maximal averages over 2D hypersurfaces and associated Radon transforms

We prove Lp boundedness results, p>2, for local maximal averaging operators over a smooth 2D hypersurface S with either a C1 density function or a density function with a singularity that grows as |(x,y)|−β for β<2. Suppose one is in coordinates such that the surface is localized near some (x0,y0,z0) at which (0,0,1) is normal to the surface, and suppose the surface is represented as the graph of z0+s(x−x0,y−y0) near (x0,y0), with s(0,0)=0. It is shown that as long as the Taylor series of the Hessian determinant of s(x,y) at (0,0) is not identically zero, the maximal averaging operator is bounded on Lp for p>max⁡(2,1/g), where g is an index based on the growth rate of the distribution function of s(x,y) near the origin. Standard examples show that the exponent 1/g is best possible whenever the tangent plane to S at (x0,y0,z0) does not contain the origin. This theorem improves on the main result of [19], using different methods. We use closely related methods to prove Lp to Lαp Sobolev estimates for Radon transform operators with the same density functions, with no excluded cases. In the g<1/2 case, there is an interval I containing 2 for which Lp to Lαp boundedness is proven for α<g when p∈I, and for such p one can never gain more than g derivatives.

The existence of a compact global attractor for a class of competition model

This paper is concerned with the existence of a compact global attractor for a class of competition model in n?dimensional (n ≥ 1) domains. Using mathematical induction and more detailed interpolation estimates, especially Gagliardo-Nirenberg inequality, we obtain the existence of a compact global attractor, which implies the uniform boundedness of the global solutions. In particular, we get that the Shigesada-Kawasaki-Teramoto competition model has a compact global attractor for n < 10. The result of the S-K-T model extends the existence results of compact global attractor in [21] from n < 8 to n < 10, and extends the uniform boundedness results of the global solutions in [17] to the non-convex domain.

Unimodular multipliers on $$\alpha$$-modulation spaces: a revisit with new method

By a new method derived from Nicola–Primo–Tabacco in [J Pseudo-Differ Oper Appl 10:359–378 (2019)], we study the boundedness on $$\alpha$$ -modulation spaces of unimodular multipliers with symbol $$\mathrm {e}^{\mathrm {i}\mu (\xi )}$$ . Comparing with the previous results, the boundedness result is established for a larger family of unimodular multipliers under weaker assumptions.

Estimates of pseudo-differential operators associated with fractional Hankel transform on the space of type S

ABSTRACT In this article, we present some boundedness results of fractional Hankel transform on Gelfand–Shilov spaces of type S. Certain inequalities are obtained for pseudo-differential operators involving the fractional Hankel transform on Gelfand–Shilov spaces of type S. This article made further discussion on the continuity properties of fractional Hankel transform and pseudo-differential operator on some ultradifferentiable function spaces.

A boundedness result for Marcinkiewicz integral operator

Abstract We extend a boundedness result for Marcinkiewicz integral operator. We find a new space of radial functions for which this class of singular integral operators remains L p {L}^{p} -bounded when its kernel satisfies only the sole integrability condition.

On the Continuity and Compactness of Pseudodifferential Operators on Localizable Hardy Spaces

In this paper we establish Sobolev type compact embedding theorems for Hormander classes of pseudodifferential operators $OpS^{-\alpha }_{1,\delta }$ on localizable Hardy space. Our work include new optimal boundedness results. As application, we obtain compact embeddings for compactly supported distributions with respect to the space variables in the nonhomogeneous localizable Hardy-Sobolev spaces.

Some properties of an elliptic periodic problem with an interfacial resistance

AbstractIn the homogenization of quasilinear elliptic problems, it is crucial for the homogenized matrix to have some kind of Lipschitz continuity in order to provide the uniqueness of the solution of the limit problem. In this paper, we prove that some estimates providing the uniqueness for a class of quasilinear problems in a periodic two‐component domain, remains valid after the homogenization process. To do that, we first prove a suitable Meyers type estimate for the periodic cell problem describing the homogenized matrix, which is posed in a two‐component cell, with a jump of the solution proportional to the flux prescribed on the interface. We also complete the study by a boundedness result for this periodic solution.

Growth conditions and regularity, an optimal local boundedness result

We prove local boundedness of local minimizers of scalar integral functionals [Formula: see text], [Formula: see text] where the integrand satisfies [Formula: see text]-growth of the form [Formula: see text] under the optimal relation [Formula: see text].

Feedback control for non-stationary 3D Navier–Stokes–Voigt equations

The goal of this article is to study the feedback control for non-stationary three-dimensional Navier–Stokes–Voigt equations. Based on the existence, uniqueness, and boundedness result of the weak solutions to the equations, we obtain the existence of solutions to the feedback control system. An existence result for an optimal control problem is also given. We illustrate our main result with an evolutionary hemivariational inequality.

On the optimal numerical parameters related with two weighted estimates for commutators of classical operators and extrapolation results

We give two-weighted norm estimates for higher order commutator of classical operators such as singular integral and fractional type operators, between weighted $$L^p$$ and certain spaces that include Lipschitz, BMO and Morrey spaces. We also give the optimal parameters involved with these results, where the optimality is understood in the sense that the parameters defining the corresponding spaces belong to certain region out of which the classes of weights are satisfied by trivial weights. We also exhibit pairs of non-trivial weights in the optimal region satisfying the conditions required. Finally, we exhibit an extrapolation result that allows us to obtain boundedness results of the type described above in the variable setting and for a great variety of operators, by starting from analogous inequalities in the classical context. In order to get this result we prove a Calderon–Scott type inequality with weights that connects adequately the spaces involved.

Global weak solutions for a 3D chemotaxis–Stokes system with slow [formula omitted]-Laplacian diffusion and rotation

In this paper we study the chemotaxis–Stokes system with slow p-Laplacian diffusion and rotation: nt+u⋅∇n=∇⋅(|∇n|p−2∇n)−∇⋅(nS(x,n,c)⋅∇c), ct+u⋅∇c=Δc−nc, ut+∇P=Δu+n∇ϕ+f(x,t) and ∇⋅u=0 in a bounded domain Ω⊂R3 with p>2, subject to the Neumann–Neumann–Dirichlet boundary conditions, where ϕ:Ω̄→R, f:Ω̄×[0,∞)→R3 and S:Ω̄×[0,∞)2→R3×3 are given sufficiently smooth functions with f bounded in Ω×(0,∞), |S(x,n,c)|≤S0(c)(1+n)−α for (x,n,c)∈Ω̄×[0,∞)2 with α≥0, and nondecreasing function S0:[0,∞)→[0,∞). It is proved that the problem possesses a globally bounded weak solution provided α+43p>259 and 11p+6α+2αp>23. This extends the current global boundedness result by Tao and Li (2020), where the case of α=0 was well solved. It is mentioned that, without constructing coupled energy functionals, the technique used in the present paper is somewhat different.