Differ Equ Research Articles

Normalized solutions for a fractional Schrödinger–Poisson system with critical growth

In this paper, we study the fractional critical Schrödinger–Poisson system (-Δ)su+λϕu=αu+μ|u|q-2u+|u|2s∗-2u,inR3,(-Δ)tϕ=u2,inR3,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned}{\\left\\{ \\begin{array}{ll} (-\\Delta )^su +\\lambda \\phi u= \\alpha u+\\mu |u|^{q-2}u+|u|^{2^*_s-2}u,&{}~~ \\hbox {in}~{\\mathbb {R}}^3,\\\\ (-\\Delta )^t\\phi =u^2,&{}~~ \\hbox {in}~{\\mathbb {R}}^3,\\end{array}\\right. } \\end{aligned}$$\\end{document}having prescribed mass ∫R3|u|2dx=a2,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\int _{{\\mathbb {R}}^3} |u|^2dx=a^2,\\end{aligned}$$\\end{document}where s,t∈(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ s, t \\in (0, 1)$$\\end{document} satisfy 2s+2t>3,q∈(2,2s∗),a>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2\\,s+2t> 3, q\\in (2,2^*_s), a>0$$\\end{document} and λ,μ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda ,\\mu >0$$\\end{document} parameters and α∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in {\\mathbb {R}}$$\\end{document} is an undetermined parameter. For this problem, under the L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-subcritical perturbation μ|u|q-2u,q∈(2,2+4s3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu |u|^{q-2}u, q\\in (2,2+\\frac{4\\,s}{3})$$\\end{document}, we derive the existence of multiple normalized solutions by means of the truncation technique, concentration-compactness principle and the genus theory. In the L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-supercritical perturbation μ|u|q-2u,q∈(2+4s3,2s∗)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu |u|^{q-2}u,q\\in (2+\\frac{4\\,s}{3}, 2^*_s)$$\\end{document}, we prove two different results of normalized solutions when parameters λ,μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda ,\\mu $$\\end{document} satisfy different assumptions, by applying the constrained variational methods and the mountain pass theorem. Our results extend and improve some previous ones of Zhang et al. (Adv Nonlinear Stud 16:15–30, 2016); and of Teng (J Differ Equ 261:3061–3106, 2016), since we are concerned with normalized solutions.

Non-uniform dependence of the data-to-solution map for the rotation b-family system

In this paper, we consider the Cauchy problem for a generalized rotation b-family system on the real line and prove that the data-to-solution map of this problem is not uniformly continuous in B p , r s × B p , r s − 1 . We have removed the restriction of μ ( b + 1 ) = Aσ in Holmes et al. [Nonuniform dependence of the R-b-family system in Besov spaces. Z Angew Math Mech. 2021;101(8):18] and Yang [Non-uniform continuity of the solution map to the rotation-two-component Camassa–Holm system. J Differ Equ. 2020;268:4423–4463].

A Priori Estimates for Solutions to Landau Equation Under Prodi–Serrin Like Criteria

In this paper, we introduce Prodi–Serrin like criteria which enable us to provide a priori estimates for the solutions to the spatially homogeneous Landau equation for all classical soft potentials and dimensions d≧3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d \\geqq 3$$\\end{document}. The physical case of Coulomb interaction in dimension d=3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=3$$\\end{document} is included in our analysis; this generalizes the work of Silvestre (J Differ Equ 262:3034–3055, 2017). Our approach is quantitative and does not require a preliminary knowledge of elaborate tools for nonlinear parabolic equations.

On the Existence and Partial Stability of Standing Waves for a Nematic Liquid Crystal Director Field Equations

AbstractIn this paper, following the studies in Amorim et al. (Partial Differ Equ Appl 4, 36, 2023), we consider some new aspects of the motion of the director field of a nematic liquid crystal submitted to a magnetic field and to a laser beam. In particular, we study the existence and partial orbital stability of special standing waves, in the spirit of Cazenave and Lions (Commun Math Phys 85:549–561, 1982) and Hadj Selem et al. (Milan J Math 82:273–295, 2014) and we present some numerical simulations.

A logarithmic Choquard equation with critical exponential growth in ℝ2

ABSTRACT In this paper, we are paid attention to the following logarithmic Choquard equation − Δ u + u = [ I 2 ∗ F ( u ) ] f ( u ) , x ∈ R 2 , where I 2 = 1 2 π ln ⁡ 1 | x | is the Newtonian kernel in dimension 2, F ( u ) = ∫ 0 u f ( t ) d t and f ∈ C 1 ( R , R ) is critical exponential growth with respect to Trudinger–Moser. Based on a variational framework proposed in Liu et al. (J Differ Equ 2022;328), we obtain a positive finite energy solution of the above logarithmic Choquard equation via the variational methods. Furthermore, we use another approach to investigate the planar logarithmic Choquard equations involving critical exponential growth.

Optimal Gevrey Regularity for Supercritical Quasi-Geostrophic Equations

We consider the two dimensional surface quasi-geostrophic equations with super-critical dissipation. For large initial data in critical Sobolev and Besov spaces, we prove optimal Gevrey regularity endowed with the same decay exponent as the linear part. This settles several open problems in Biswas (J Differ Equ 257(6):1753–1772, 2014), Biswas et al. (J Funct Anal 269(10):3083–3119, 2015).

Classification of traveling waves to the generalized Camassa–Holm equation with dual-power nonlinearities

In this paper, we mainly devote to investigate the classification of traveling waves to the generalized Camassa–Holm equation with dual-power nonlinearities. Utilizing the celebrated approach of classification of traveling wave solution which was proposed by Jonatan Lenells [Traveling wave solutions of the Camassa-Holm equation. J Diff Equ. 2005;217(2):393–430]. We show a result concerns the regularity of traveling waves and then classify all traveling wave solutions.

On anisotropic parabolic equation with nonstandard growth order

ABSTRACT In this paper, the existence and the uniqueness of an evolutionary anisotropic p i ( x ) -Laplacian equation with a damping term are studied. If the damping term is with a subcritical index, by the Di Giorgi iteration technique, the L ∞ -estimate of the weak solutions can be obtained. The existence of weak solution is proved by the renormalized solution method, and how the anisotropic characteristic of the considered equation affect the L ∞ -estimate of the weak solutions is revealed. The uniqueness is true strongly depending on subcritical index of the damping term, and this result goes beyond previous efforts in the literature (Bertsch M, Dal Passo R, Ughi M: Discontinuous viscosity solutions of a degenerate parabolic equation. Trans Amer Math Soc. 1990;320:779–798; Li Z, Yan B, Gao W. Existence of solutions to a parabolic p ( x ) − Laplace equation with convection term via L ∞ − Estimates. Electron J Differ Equ. 2015;46:1–21; Zhang Q, Shi P. Global solutions and self-similar solutions of semilinear parabolic equations with nonlinear gradient terms. Nonlinear Anal. 2010;72:2744–2752; Zhou W, Cai S. The continuity of the viscosity of the Cauchy problem of a degenerate parabolic equation not in divergence form. J Jilin University (Natural Sci.). 2004;42:341–345), etc.

Refined Decay Estimate and Analyticity of Solutions to the Linear Heat Equation in Homogeneous Besov Spaces

The heat semigroup {T(t)}_{t ge 0} defined on homogeneous Besov spaces dot{B}_{p,q}^s(mathbb {R}^n) is considered. We show the decay estimate of T(t)f in dot{B}_{p,1}^{s+sigma }(mathbb {R}^n) for f in dot{B}_{p,infty }^s(mathbb {R}^n) with an explicit bound depending only on the regularity index sigma >0 and space dimension n. It may be regarded as a refined result compared with that of the second author (Takeuchi in Partial Differ Equ Appl Math 4:100174, 2021). As a result of the refined decay estimate, we also improve a lower bound estimate of the radius of convergence of the Taylor expansion of T(cdot )f in space and time. To refine the previous results, we show explicit pointwise estimates of higher order derivatives of the power function |xi |^{sigma } for sigma in mathbb {R}. In addition, we also refine the L^1-estimate of the derivatives of the heat kernel.

A Speed Restart Scheme for a Dynamics with Hessian-Driven Damping

AbstractIn this paper, we analyze a speed restarting scheme for the inertial dynamics with Hessian-driven damping, introduced by Attouch et al. (J Differ Equ 261(10):5734–5783, 2016). We establish a linear convergence rate for the function values along the restarted trajectories. Numerical experiments suggest that the Hessian-driven damping and the restarting scheme together improve the performance of the dynamics and corresponding iterative algorithms in practice.

Zabczyk Type Criteria for Asymptotic Behavior of Dynamical Systems and Applications

AbstractWe obtain new characterizations for nonuniform and uniform asymptotic behaviors of variational systems in infinite dimensional spaces, following the line of studies recently developed in Dragičević et al. (J Differ Equ 268:4786–4829, 2020, J Dyn Differ Equ 34:1107–1137, 2022, Appl Math Comput 414:1–22, 2022, J Math Anal Appl 515:1–37, 2022). First, we give complete descriptions for both nonuniform and uniform stability by means of some nonuniform conditions of convergence for series of suitable nonlinear trajectories. We apply our criteria to present a novel method of exploring the robustness of the nonuniform exponential stability under additive and multiplicative perturbations and we also deduce some consequences for the robustness of the uniform exponential stability. After that, we obtain characterizations for nonuniform and uniform instability in terms of certain nonuniform conditions imposed to the sums of some well-chosen series of nonlinear trajectories. We apply our results to provide a new technique of exploring the robustness of the nonuniform exponential instability under additive and multiplicative perturbations and discuss the consequences for uniform exponential instability. Hence, for both stability and instability we generalize the previous Zabczyk type results and extend their applicability. Next, as another class of applications, we provide nonuniform criteria of Rolewicz type for stability and instability of skew-product semiflows. Thus, we extend the Zabczyk–Rolewicz type methods in two directions: by giving local conditions for global behaviors and by employing specific nonuniform boundedness conditions. Our criteria can be applied to broad classes of discrete and continuous variational dynamical systems.

On the fractional P–Q laplace operator with weights

We exploit the existence and non-existence of positive solutions to the eigenvalue problem driven by the nonhomogeneous fractional p &q Laplacian operator with indefinite weights ( − Δ p ) α u + ( − Δ q ) β u = λ [ a | u | p − 2 u + b | u | q − 2 u ] in Ω , where Ω ⊆ R N is a smooth bounded domain that has been extended by zero. We further show the existence of a continuous family of eigenvalues in the case Ω = R N and b ≡ 0 a.e. Our approach relies strongly on variational Analysis, in which the Mountain pass theorem plays the key role. Due to the lack of spatial compactness and the embedding W α , p ( R N ) ↪ W β , q ( R N ) in R N , we employ the concentration-compactness principle of P.L. Lions [The concentration-compactness principle in the calculus of variations. The limit case. II, Rev Mat Iberoamericana. 1985;1(2):45–121]. to overcome the difficulty. Our paper can be considered as a counterpart to the important works [Alves et al. Existence, multiplicity and concentration for a class of fractional p &q Laplacian problems in R N , Commun Pure Appl Anal, 2019;18(4):2009–2045], [Benci et al. An eigenvalue problem for a quasilinear elliptic field equation. J Differ Equ, 2002;184(2):299–320], [Bobkov et al. On positive solutions for ( p , q ) -Laplace equations with two parameters, Calc Var Partial Differ Equ, 2015;54(3):3277–3301], [Colasuonno and Squassina. Eigenvalues for double phase variational integrals, Ann Mat Pura Appl (4), 2016;195(6):1917–1956], [Papageorgiou et al. Positive solutions for nonlinear Neumann problems with singular terms and convection, J Math Pures Appl (9), 2020;136:1–21], [Papageorgiou et al. Ground state and nodal solutions for a class of double phase problems, Z Angew Math Phys, 2020;71:1–15], and may have further applications to deal with other problems.

Blow-up solutions of damped Klein–Gordon equation on the Heisenberg group

We prove the blow-up of solutions of the semilinear damped Klein–Gordon equation in a finite time for arbitrary positive initial energy on the Heisenberg group. This work complements the paper Ruzhansky and Tokmagambetov (J Differ Equ 265(10):5212–5236, 2018), where the global in time well-posedness was proved for the small energy solutions.

One-dimensional viscoelastic von Kármán theories derived from nonlinear thin-walled beams

We derive an effective one-dimensional limit from a three-dimensional Kelvin–Voigt model for viscoelastic thin-walled beams, in which the elastic and the viscous stress tensor comply with a frame-indifference principle. The limiting system of equations comprises stretching, bending, and twisting both in the elastic and the viscous stress. It coincides with the model already identified via Friedrich and Kružík (Arch Ration Mech Anal 238:489–540, 2020) and Friedrich and Machill (Nonlinear Differ Equ Appl NoDEA 29, Article number: 11, 2022) by a successive dimension reduction, first from 3D to a 2D theory for von Kármán plates and then from 2D to a 1D theory for ribbons. In the present paper, we complement the previous analysis by showing that the limit can also be obtained by sending the height and width of the beam to zero simultaneously. Our arguments rely on the static Gamma -convergence in Freddi et al. (Math Models Methods Appl Sci 23:743–775, 2013), on the abstract theory of metric gradient flows (Ambrosio et al. in Gradient flows in metric spaces and in the space of probability measures. Lectures mathematics, ETH Zürich, Birkhäuser, Basel, 2005), and on evolutionary Gamma -convergence (Sandier and Serfaty in Commun Pure Appl Math 57:1627–1672, 2004).

Spectral stability of the critical front in the extended Fisher-KPP equation

We revisit the existence and stability of the critical front in the extended Fisher-KPP equation, refining earlier results of Rottschäfer and Wayne (J Differ Equ 176(2):532–560, 2001) which establish stability of fronts without identifying a precise decay rate. Our main result states that the critical front is marginally spectrally stable, with essential spectrum touching the imaginary axis but with no unstable point spectrum. Together with the recent work of Avery and Scheel (SIAM J Math Anal 53(2):2206–2242, 2021; Commun Am Math Soc, 2:172–231, 2022), this establishes both sharp stability criteria for localized perturbations to the critical front, as well as propagation at the linear spreading speed from steep initial data, thereby extending front selection results beyond systems with a comparison principle. Our proofs are based on far-field/core decompositions which have broader use in establishing robustness properties and bifurcations of invasion fronts.

A variant of symmetric mountain pass theorem and its applications for generalized quasi-linear elliptic equations

By using a variant of the variational perturbation techniques introduced by Rabinowitz [Multiple critical points of perturbed symmetric functionals. Tran Amer Math Soc. 1982;272:753–769], we build a new non-smooth symmetric Mountain Pass Theorem without symmetric condition. This new abstract result can be applied to generalized quasi-linear elliptic equations with small perturbations. Our results extend the results concerning a semi-linear elliptic equation made by Li-Liu [Perturbations from symmetric elliptic boundary value problems. J Differ Equ. 2002;185:271–280].

Sobolev estimates for fractional parabolic equations with space-time non-local operators

We obtain $$L_p$$ estimates for fractional parabolic equations with space-time non-local operators $$\begin{aligned} \partial _t^\alpha u - Lu + \lambda u= f \quad \text {in } (0,T) \times {\mathbb {R}}^d , \end{aligned}$$ where $$\partial _t^\alpha u$$ is the Caputo fractional derivative of order $$\alpha \in (0,1]$$ , $$T\in (0,\infty )$$ , and $$\begin{aligned} Lu(t,x) {:}{=} \int _{{\mathbb {R}}^d} \bigg ( u(t,x+y)-u(t,x) - y\cdot \nabla _xu(t,x)\chi ^{(\sigma )}(y)\bigg )K(t,x,y)\,dy \end{aligned}$$ is an integro-differential operator in the spatial variables. Here we do not impose any regularity assumption on the kernel K with respect to t and y. We also derive a weighted mixed-norm estimate for the equations with operators that are local in time, i.e., $$\alpha = 1$$ , which extend the previous results in Mikulevičius and Pragarauskas (J Differ Equ 256(4):1581–1626, 2014) and Zhang (Annales l’IHPAnalyse Nonlinéaire 30:573–614, 2013) by using a quite different method.

Construction of Multi-solitons for a Generalized Derivative Nonlinear Schrödinger Equation

We consider a derivative nonlinear Schrödinger equation with general nonlinearlity: $$\begin{aligned} i\partial _tu+\partial _x^2u+i|u|^{2\sigma }\partial _xu=0, \end{aligned}$$ In Tang and Xu (J Differ Equ 264(6):4094–4135, 2018), the authors prove the stability of two solitary waves in energy space for $$\sigma \in (1,2)$$ . As a consequence, there exists a solution of the above equation which is close arbitrary to sum of two solitons in energy space when $$\sigma \in (1,2)$$ . Our goal in this paper is proving the existence of multi-solitons in energy space for $$\sigma \geqslant \frac{3}{2}$$ . Our proofs proceed by fixed point arguments around the desired profile, using Strichartz estimates.

Blow-up versus global existence of solutions for reaction–diffusion equations on classes of Riemannian manifolds

It is well known from the work of Bandle et al. (J Differ Equ 251:2143–2163, 2011) that the Fujita phenomenon for reaction–diffusion evolution equations with power nonlinearities does not occur on the hyperbolic space {mathbb {H}}^N, thus marking a striking difference with the Euclidean situation. We show that, on classes of manifolds in which the bottom Lambda of the L^2 spectrum of -Delta is strictly positive (the hyperbolic space being thus included), a different version of the Fujita phenomenon occurs for other kinds of nonlinearities, in which the role of the critical Fujita exponent in the Euclidean case is taken by Lambda . Such nonlinearities are time-independent, in contrast to the ones studied in Bandle et al. (2011). As a consequence of our results we show that, on a class of manifolds much larger than the case M={mathbb {H}}^N considered in Bandle et al. (2011), solutions to (1.1) with power nonlinearity f(u)=u^p, p>1, and corresponding to sufficiently small data, are global in time. Though qualitative similarities with similar problems in bounded, Euclidean domains can be seen in the results, the methods are significantly different because of noncompact setting dealt with.

On Non-linear Characterizations of Classical Orthogonal Polynomials

Classical orthogonal polynomials are known to satisfy seven equivalent properties, namely the Pearson equation for the linear functional, the second-order differential/difference/q-differential/ divided-difference equation, the orthogonality of the derivatives, the Rodrigues formula, two types of structure relations, and the Riccati equation for the formal Stieltjes function. In this work, following previous work by Kil et al. (J Differ Equ Appl 4:145–162, 1998a; Kyungpook Math J 38:259–281, 1998b), we state and prove a non-linear characterization result for classical orthogonal polynomials on non-uniform lattices. Next, we give explicit relations for some families of these classes.