Search

In this paper, we study the asymptotic behavior of transmission system for coupled Kirchhoff plates, where one equation is conserved and the other has dissipative property, and the dissipation mechanism is given by fractional damping (- Δ) 2 θ v t with θ ∈ [ 1 2 , 1 ] . By using the semigroup method and the multiplier technique, we obtain the exact polynomial decay rates, and find that the polynomial decay rate of the system is determined by the inertia/elasticity ratios and the fractional damping order. Specifically, when the inertia/elasticity ratios are not equal and θ ∈ [ 1 2 , 3 4 ] , the polynomial decay rate of the system is t - 1 / (10 - 4 θ) . When the inertia/elasticity ratios are not equal and θ ∈ [ 3 4 , 1 ] , the polynomial decay rate of the system is t - 1 / (4 + 4 θ) . When the inertia/elasticity ratios are equal, the polynomial decay rate of the system is t - 1 / (4 + 4 θ) . Furthermore it has been proven that the obtained decay rates are all optimal. The obtained results extend the results of Oquendo and Suárez (Z Angew Math Phys 70(3):88, 2019) for the case of fractional damping exponent 2 θ from [0, 1] to [1, 2]. [ABSTRACT FROM AUTHOR]

This paper concerns the bubbling phenomena for the L2$L^2$‐critical half‐wave equation in dimension one. Given arbitrarily finitely many distinct singularities, we construct blow‐up solutions concentrating exactly at these singularities. This provides the first examples of multi‐bubble solutions for the half‐wave equation. In particular, the solutions exhibit the mass quantization property. Our proof strategy draws upon the modulation method in Krieger, Lenzmann and Raphaël [Arch. Ration. Mech. Anal. 209 (2013), no. 1, 61–129] for the single‐bubble case, and explores the localization techniques in Cao, Su and Zhang [Arch. Ration. Mech. Anal. 247 (2023), no. 1, Paper No. 4] and Röckner, Su and Zhang [Trans. Amer. Math. Soc., 377 (2024), no. 1, 517–588] for bubbling solutions to non‐linear Schrödinger equations (NLS). However, unlike the single‐bubble or NLS cases, different bubbles exhibit the strongest interactions in dimension one. In order to get sharp estimates to control these interactions, as well as non‐local effects on localization functions, we utilize the Carlderón estimate and the integration representation formula of the half‐wave operator, and find that there exists a narrow room between the orders |t|2+$|t|^{2+}$ and |t|3−$|t|^{3-}$ for the remainder in the geometrical decomposition. Based on this, a novel bootstrap scheme is introduced to address the multi‐bubble non‐local structure. [ABSTRACT FROM AUTHOR]

In this paper, we study normalized solutions of the fractional Schrödinger equation with a critical nonlinearity... we prove the existence of a second normalized solution under some conditions on a, p, s, and N. This is a continuation of our previous work (Z. Angew. Math. Phys., 73 (2022) 149) where only one solution is obtained. [ABSTRACT FROM AUTHOR]

The concept of contraction mappings plays a significant role in mathematics, particularly in the study of fixed points and the existence of solutions for various equations. In this study, we described two types of enriched contractions: enriched F-contraction and enriched F'-contraction associated with u-fold averaged mapping, which are involved with Kirk's iterative technique with order u. The contractions extracted from this paper generalized and unified many previously common super contractions. Furthermore, u-fold averaged mappings can be seen as a more general form of both averaged mappings and double averaged mappings. Moreover, we demonstrated that the ufold averaged mapping with enriched contractions has a unique fixed point. Our work examined the necessary conditions for the u-fold averaged mapping and weak enriched contractions to have equal sets of fixed points. Additionally, we illustrated that an appropriate Kirk's iterative algorithm can effectively approximate a fixed point of a u-fold averaged mapping as well as the two enriched contractions. Also, we delved into the well-posedness, limit shadowing property, and Ulam-Hyers stability of the u-fold averaged mapping. Furthermore, we established necessary conditions that guaranteed the periodic point property for each of the illustrated strengthened contractions. To underscore the generality of our findings, we presented several examples that aligned with comparable results found in the existing literature. [ABSTRACT FROM AUTHOR]

The paper is concerned with Lane–Emden and Brezis–Nirenberg problems involving the affine p-Laplace nonlocal operator Δ p 풜 , which has been introduced in [J. Haddad, C. H. Jiménez and M. Montenegro, From affine Poincaré inequalities to affine spectral inequalities, Adv. Math. 386 2021, Article ID 107808] driven by the affine L p energy ℰ p , Ω from convex geometry due to [E. Lutwak, D. Yang and G. Zhang, Sharp affine L p Sobolev inequalities, J. Differential Geom. 62 2002, 1, 17–38]. We are particularly interested in the existence and nonexistence of positive C 1 solutions of least energy type. Part of the main difficulties are caused by the absence of convexity of ℰ p , Ω and by the comparison ℰ p , Ω ⁢ (u) ≤ ∥ u ∥ W 0 1 , p ⁢ (Ω) generally strict. [ABSTRACT FROM AUTHOR]

In this paper, we consider a singular Adams' inequality with logarithmic weights in the unit ball of $ \mathbb {R}^4 $ R 4 . Our results extend the results of Zhu and Wang [Adams' inequality with logarithmic weights in $ \mathbb {R}^4 $ R 4 . Proc Amer Math Soc. 2021;149(8):3463–3472] on Adams' inequality with logarithmic weights to singular case. Then, we study the existence of solutions for some weighted mean field equations, relying on variational methods and the singular Adams' inequality with logarithmic weights we previously established. [ABSTRACT FROM AUTHOR]

This paper deals with the equation - Δ u + μ u = f on high-dimensional spaces R m , where the right-hand side f (x) = F (T x) is composed of a separable function F with an integrable Fourier transform on a space of a dimension n > m and a linear mapping given by a matrix T of full rank and μ ≥ 0 is a constant. For example, the right-hand side can explicitly depend on differences x i - x j of components of x. Following our publication (Yserentant in Numer Math 146:219–238, 2020), we show that the solution of this equation can be expanded into sums of functions of the same structure and develop in this framework an equally simple and fast iterative method for its computation. The method is based on the observation that in almost all cases and for large problem classes the expression ‖ T t y ‖ 2 deviates on the unit sphere ‖ y ‖ = 1 the less from its mean value the higher the dimension m is, a concentration of measure effect. The higher the dimension m, the faster the iteration converges. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence of multiplicity of positive solutions to a new generalized Riemann–Liouville type fractional Fisher‐like equation on a semi‐infinite interval equipped with nonlocal multipoint boundary conditions involving Riemann–Liouville fractional derivative and integral operators. The existence of at least two positive solutions for the given problem is established by using the concept of complete continuity and iterative positive solutions. We show the existence of at least three positive solutions to the problem at hand by applying the generalized Leggett–Williams fixed‐point theorem due to Bai and Ge [Z. Bai, B. Ge, Existence of three positive solutions for some second‐order boundary value problems, Comput. Math. Appl. 48 (2014) 699‐70]. Illustrative examples are constructed to demonstrate the effectiveness of the main results. It has also been indicated in Section 5 that some new results appear as special cases by choosing the parameters involved in the given problem appropriately. [ABSTRACT FROM AUTHOR]

In this paper, we focused on a one-dimensional elastic system with a double porosity structure and frictional damping acting on both porous equations. We introduce two stability numbers χ 0 {\chi_{0}} and χ 1 {\chi_{1}} and prove that the solution of the system decays exponentially provided that χ 0 = 0 {\chi_{0}=0} and χ 1 ≠ 0 {\chi_{1}\neq 0} . Otherwise, we prove the absence of exponential decay. Our results improve the results of [N. Bazarra, J. R. Fernández, M. C. Leseduarte, A. Magaña and R. Quintanilla, On the thermoelasticity with two porosities: Asymptotic behaviour, Math. Mech. Solids 24 2019, 9, 2713–2725] and [A. Nemsi and A. Fareh, Exponential decay of the solution of a double porous elastic system, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 83 2021, 1, 41–50]. [ABSTRACT FROM AUTHOR]

In this paper, benefited some ideas of Wang [J. Geom. Anal. 33, 18 (2023)] and Dufera et al. [J. Math. Anal. Appl. 509, 126001 (2022)], we investigate persistence of spatial analyticity for solution of the higher order nonlinear dispersive equation with the initial data in modified Gevrey space. More precisely, using the contraction mapping principle, the bilinear estimate as well as approximate conservation law, we establish the persistence of the radius of spatial analyticity till some time δ. Then, given initial data that is analytic with fixed radius σ0, we obtain asymptotic lower bound σ (t) ≥ c | t | − 1 2 , for large time t ≥ δ. This result improves earlier ones in the literatures, such as Zhang et al. [Discrete Contin. Dyn. Syst. B 29, 937–970 (2024)], Huang–Wang [J. Differ. Equations 266, 5278–5317 (2019)], Liu–Wang [Nonlinear Differ. Equations Appl. 29, 57 (2022)], Wang [J. Geom. Anal. 33, 18 (2023)] and Selberg–Tesfahun [Ann. Henri Poincaré 18, 3553–3564 (2017)]. [ABSTRACT FROM AUTHOR]

Many articles have recently been devoted to Mahler equations, partly because of their links with other branches of mathematics such as automata theory. Hahn series (a generalization of the Puiseux series allowing arbitrary exponents of the indeterminate as long as the set that supports them is well ordered) play a central role in the theory of Mahler equations. In this paper, we address the following fundamental question: is there an algorithm to calculate the Hahn series solutions of a given linear Mahler equation? What makes this question interesting is the fact that the Hahn series appearing in this context can have complicated supports with infinitely many accumulation points. Our (positive) answer to the above question involves among other things the construction of a computable well‐ordered receptacle for the supports of the potential Hahn series solutions. [ABSTRACT FROM AUTHOR]

In the present paper, we consider a specific class of nonautonomous wave equations on a smooth, bounded domain and their discretization in space by isoparametric finite elements and in time by the implicit Euler method. Building upon the work of Baker and Dougalis (1980, On the |${L}^{\infty }$| -convergence of Galerkin approximations for second-order hyperbolic equations. Math. Comp. , 34 , 401–424), we prove optimal error bounds in the |$W^{1,\infty } \times L^\infty $| -norm for the semidiscretization in space and the full discretization. The key tool is the gain of integrability coming from the inverse of the discretized differential operator. For this, we have to pay with (discrete) time derivatives on the error in the |$H^{1} \times L^2$| -norm, which are reduced to estimates of the differentiated initial errors. To confirm our theoretical findings, we also present numerical experiments. [ABSTRACT FROM AUTHOR]

To improve the convergence of the gradient iterative (GI) algorithm and the Jacobi‐gradient iterative (JGI) algorithm [Bayoumi, Appl Math Inf Sci, 2021], a shift‐splitting Jacobi‐gradient iterative (SSJGI) algorithm for solving the matrix equation A풱−풱‾B=C is presented in this paper, which is based on the splitting of the coefficient matrices. The proposed algorithm converges to the exact solution for any initial value with some conditions. To demonstrate the effectiveness of the SSJGI algorithm and to compare it to the GI algorithm and the JGI algorithm [Bayoumi, Appl Math Inf Sci, 2021], numerical examples are provided. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study the asymptotic monotonicity of positive solutions for fractional parabolic equation on the right half space. First, a narrow region principle for antisymmetric functions in unbounded domains is obtained, in which we remarkably weaken the decay condition u → 0 at infinity and only assume its growth rate does not exceed | x | γ (0 < γ < 2 s ) compared with (Adv. Math. 377:107463, 2021). Then we obtain asymptotic monotonicity of positive solutions of fractional parabolic equation on R + N × (0 , ∞) . [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence of small-amplitude unique solutions near the Maxwellian for the Fokker–Planck–Boltzmann equation in a finite channel with specular reflection boundary conditions. The solution space we consider is denoted as L k ̄ 1 L T ∞ L x 1 , v 2 , introduced in Duan et al. [Commun. Pure Appl. Math. 74(5), 932–1020 (2021)]. In addition, we investigate the long-time behavior of solutions for both hard and soft potentials. [ABSTRACT FROM AUTHOR]

In the present paper, we study the Hénon type parabolic equation $ \phi_t-\Delta \phi = |x|^\beta |\phi|^{\alpha-1}\phi, x\in\mathbb{R}^{N} $ with singular initial data $ \eta|x|^{-\frac{2+\beta}{\alpha-1}} $, where $ \eta, \beta>0, \alpha>1 $. We prove the existence of infinitely many sign-changing self-similar solutions by analyzing the related inverted profile equation for $ N\geq1 $. This can be seen as an extension of the recent work of T. Cazenave at al.(Amer. J. Math. 142 (2020), 1439-1495) to the Hénon type parabolic equation. [ABSTRACT FROM AUTHOR]

This paper presents some existence and uniqueness results for a solution of a system of equations. Our results extend and generalize the well-known and celebrated results of Boyd and Wong [Proc. Amer. Math. Soc. 20 (1969)], Matkowski [Dissertations Math. (Rozprawy Mat.) 127 (1975)], Proinov [Nonlinear Anal. 64 (2006)], Ri [Indag. Math. (N. S.) 27 (2016)] and many others. We also present some illustrative examples to validate our results. [ABSTRACT FROM AUTHOR]

In this paper we study the convergence of nonlinear Dirichlet problems for systems of variational elliptic PDEs defined on randomly perforated domains of Rn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {R}^n$$\end{document}. Under the assumption that the perforations are small balls whose centres and radii are generated by a stationary short-range marked point process, we obtain in the critical-scaling limit an averaged nonlinear analogue of the extra term obtained in the classical work of Cioranescu and Murat (Res Notes Math III, 1982). In analogy to the random setting recently introduced by Giunti, Höfer and Velázquez (Commun Part Differ Equ 43(9):1377–1412, 2018) to study the Poisson equation, we only require that the random radii have finite (n-q)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(n-q)$$\end{document}-moment, where 1q-capacity of the spherical holes is finite, and hence that the limit problem is well defined. On the other hand, it does not exclude the presence of balls with large radii, that can cluster up. We show however that the critical rescaling of the perforations is sufficient to ensure that no percolating-like structures appear in the limit. [ABSTRACT FROM AUTHOR]

We develop the asymptotic analysis as ε→0 for the natural gradient flow of the self-dual U(1)-Higgs energies on Hermitian line bundles over closed manifolds (Mn,g) of dimension n≥3, showing that solutions converge in a measure-theoretic sense to codimension-two mean curvature flows—i.e., integral (n−2)-Brakke flows—generalizing results of (Pigati and Stern in Invent. Math. 223:1027–1095, 2021) from the stationary case. Given any integral (n−2)-cycle Γ0 in M, these results can be used together with the convergence theory developed in (Parise et al. in Convergence of the self-dual U(1)-Yang–Mills–Higgs energies to the (n−2)-area functional, 2021, arXiv:2103.14615) to produce nontrivial integral Brakke flows starting at Γ0 with additional structure, similar to those produced via Ilmanen's elliptic regularization. [ABSTRACT FROM AUTHOR]

We apply the basic Lie symmetry method to investigate the time-dependent negative-order Calogero-Bogoyavlenskii-Schiff (vnCBS) equation. In this case, the symmetry classification problem is answered. We obtain symmetry algebra and create the optimal system of Lie sub- algebras. We obtain the symmetry reductions and invariant solutions of the considered equation using these vector fields. Finally, we determine the conservation laws of the vnCBS equation via the Bluman-Anco ho- motopy formula [ABSTRACT FROM AUTHOR]

We extend the result of Kowalczyk, Martel, and Mu\~noz [J. Eur. Math. Soc. (JEMS), 24 (2022), pp. 2133--2167] on the existence, in the context of spatially even solutions, of asymptotic stability on a center hypersurface at the soliton of the defocusing power nonlinear Klein--Gordon equation with p > 3, to the case 2 ≥ p > 5/3. . The result is attained performing new and refined estimates that allow us to close the argument for power law in the range 2 ≥ p > 5/3. . [ABSTRACT FROM AUTHOR]

We study statistical solutions of the incompressible Navier–Stokes equation and their vanishing viscosity limit. We show that a formulation using correlation measures as in [U. S. Fjordholm, S. Lanthaler, and S. Mishra, Arch. Ration. Mech. Anal., 226 (2017), pp. 809–849] and moment equations is equivalent to statistical solutions in the Foiaş–Prodi sense. Under the assumption of weak scaling, a weaker version of Kolmogorov’s self-similarity at small scales hypothesis that allows for intermittency corrections, we show that the limit is a statistical solution of the incompressible Euler equations. To pass to the limit, we derive a Kármán–Howarth–Monin relation for statistical solutions and combine it with the weak scaling assumption and a compactness theorem for correlation measures from [U. S. Fjordholm et al., Math. Models Methods Appl. Sci., 30 (2020), pp. 539–609]. [ABSTRACT FROM AUTHOR]

The adaptive technique enables us to achieve the highest efficiency index theoretically and practically. The idea of introducing an adaptive self-accelerator (via all the old information for Steffensen-type methods) is new and efficient to obtain the highest efficiency index. In this work, we have used four self-accelerating parameters and have increased the order of convergence from 8 to 16, i. e. any new function evaluations improve the convergence order up to 100%. The numerical results are compared without and with memory methods. It confirms that the proposed methods have more efficiency index. [ABSTRACT FROM AUTHOR]

In this paper, equations are established to predict how the values of the lower yield point in the stress-strain curve depending on a time of corrosion influence will change. Although this point is of theoretical importance in the theory of strength of materials, its change in corroded steel is of practical importance, since this point determines according to the theory which minimum load or stress is required to maintain the plastic behavior of material. A well-founded mathematical principle was used to process experimentally obtained data in two main directions - the stochastic method and the average method. Diagrams of the variation of values in corroded steel were drawn up and equations of the 9th degree were established using polynomial approximation.

The SOR-like iteration method for solving the system of absolute value equations of finding a vector x such that A x - | x | - b = 0 with ν = ‖ A - 1 ‖ 2 < 1 is investigated. The convergence conditions of the SOR-like iteration method proposed by Ke and Ma (Appl. Math. Comput., 311:195–202, 2017) are revisited and a new proof is given, which exhibits some insights in determining the convergent region and the optimal iteration parameter. Along this line, the optimal parameter which minimizes ‖ T ν (ω) ‖ 2 with T ν (ω) = | 1 - ω | ω 2 ν | 1 - ω | | 1 - ω | + ω 2 ν and the approximate optimal parameter which minimizes an upper bound of ‖ T ν (ω) ‖ 2 are explored. The optimal and approximate optimal parameters are iteration-independent, and the bigger value of ν is, the smaller convergent region of the iteration parameter ω is. Numerical results are presented to demonstrate that the SOR-like iteration method with the optimal parameter is superior to that with the approximate optimal parameter proposed by Guo et al. (Appl. Math. Lett., 97:107–113, 2019). [ABSTRACT FROM AUTHOR]

We investigate existence and nonexistence of global in time nonnegative solutions to the semilinear heat equation, with a reaction term of the type e μ t u p (μ ∈ R , p > 1) , posed on cones of the hyperbolic space. Under a certain assumption on µ and p, related to the bottom of the spectrum of − Δ in H n , we prove that any solution blows up in finite time, for any nontrivial nonnegative initial datum. Instead, if the parameters µ and p satisfy the opposite condition we have: (a) blow-up when the initial datum is large enough, (b) existence of global solutions when the initial datum is small enough. Hence our conditions on the parameters µ and p are optimal. We see that blow-up and global existence do not depend on the amplitude of the cone. This is very different from what happens in the Euclidean setting (Bandle and Levine 1989 Trans. Am. Math. Soc. 316 595–622), and it is essentially due to a specific geometric feature of H n . [ABSTRACT FROM AUTHOR]

We propose and study a fully discrete finite volume scheme for the linear Vlasov-Fokker-Planck equation written as an hyperbolic system using Hermite polynomials in velocity. This approach naturally preserves the stationary solution and the weighted L^2 relative entropy. Then, we adapt the arguments developed in Dolbeault, Mouhot, and Schmeiser [Trans. Amer. Math. Soc. 367 (2015), pp. 3807–3828] based on hypocoercivity methods to get quantitative estimates on the convergence to equilibrium of the discrete solution. Finally, we prove that in the diffusive limit, the scheme is asymptotic preserving with respect to both the time variable and the scaling parameter at play. [ABSTRACT FROM AUTHOR]