Math Methods Appl Sci Research Articles

Spectral Stability of Shock Profiles for Hyperbolically Regularized Systems of Conservation Laws

We report a proof that under natural assumptions shock profiles viewed as heteroclinic travelling wave solutions to a hyperbolically regularized system of conservation laws are spectrally stable if the shock amplitude is sufficiently small. This means that an associated Evans function E:Λ→C with Λ⊂C an open superset of the closed right half plane H+≡{κ∈C:Reκ≧0} has only one zero, namely, a simple zero at 0. The result is analogous to the one obtained in Freistühler and Szmolyan (Arch Ration Mech Anal 164:287–309, 2002) and Plaza and Zumbrun (Discrete Contin Dyn Syst 10(4):885–924, 2004) for parabolically regularized systems of conservation laws, and also distinctly extends findings on hyperbolic relaxation systems in Mascia and Zumbrun (Partial Differ Equ 34(1–3):119–136, 2009), Plaza and Zumbrun (2004) and Ueda (Math Methods Appl Sci 32(4):419–434, 2009).

On quasi-linear reaction diffusion systems arising from compartmental SEIR models

The global existence and boundedness of solutions to quasi-linear reaction-diffusion systems are investigated. The system arises from compartmental models describing the spread of infectious diseases proposed in Viguerie et al. (Appl Math Lett 111:106617, 2021); Viguerie et al. (Comput Mech 66(5):1131–1152, 2020), where the diffusion rate is assumed to depend on the total population, leading to quasilinear diffusion with possible degeneracy. The mathematical analysis of this model has been addressed recently in Auricchio et al. (Math Methods Appl Sci 46:12529–12548, 2023) where it was essentially assumed that all sub-populations diffuse at the same rate, which yields a positive lower bound of the total population, thus removing the degeneracy. In this work, we remove this assumption completely and show the global existence and boundedness of solutions by exploiting a recently developed Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}-energy method. Our approach is applicable to a larger class of systems and is sufficiently robust to allow model variants and different boundary conditions.

On Global Classical Limit of the (Relativistic) Vlasov–Maxwell System

It is shown that solutions of the (relativistic) Vlasov–Maxwell system converge pointwise to solutions of the Vlasov–Poisson system globally in time at the asymptotic rate of c^{-1}, as the light speed c tends to infinity, which extends the results of Asano and Ukai (Stud Math Appl 18:369–383, 1986), Degond (Math Methods Appl Sci 8:533–558, 1986) and Schaeffer (Commun Math Phys 104:403–421, 1986). The analysis relies on the method of Glassey and Strauss (Commun Math Phys 113:191–208, 1987) and Schaeffer (Commun Math Phys 104:403–421, 1986).

Combining effects ensuring boundedness in an attraction–repulsion chemotaxis model with production and consumption

This paper is framed in a series of studies on attraction–repulsion chemotaxis models combining different effects: nonlinear diffusion and sensitivities and logistic sources, for the dynamics of the cell density, and consumption and/or production impacts, for those of the chemicals. In particular, herein we focus on the situation where the signal responsible of gathering tendencies for the particles’ distribution is produced, while the opposite counterpart is consumed. In such a sense, this research complements the results in Frassu et al. (Math Methods Appl Sci 45:11067–11078, 2022) and Chiyo et al. (Commun Pure Appl Anal, 2023, doi: 10.3934/cpaa.2023047), where the chemicals evolve according to different laws.

Multiple partition Markov model for B.1.1.7, B.1.351, B.1.617.2, and P.1 variants of SARS-CoV 2 virus.

With tools originating from Markov processes, we investigate the similarities and differences between genomic sequences in FASTA format coming from four variants of the SARS-CoV 2 virus, B.1.1.7 (UK), B.1.351 (South Africa), B.1.617.2 (India), and P.1 (Brazil). We treat the virus' sequences as samples of finite memory Markov processes acting in We model each sequence, revealing some heterogeneity between sequences belonging to the same variant. We identified the five most representative sequences for each variant using a robust notion of classification, see Fernández etal. (Math Methods Appl Sci 43(13):7537-7549. 10.1002/mma.5705 ). Using a notion derived from a metric between processes, see García etal. (Appl Stoch Models Bus Ind 34(6):868-878. 10.1002/asmb.2346), we identify four groups, each group representing a variant. It is also detected, by this metric, global proximity between the variants B.1.351 and B.1.1.7. With the selected sequences, we assemble a multiple partition model, see Cordeiro etal. (Math Methods Appl Sci 43(13):7677-7691. 10.1002/mma.6079), revealing in which states of the state space the variants differ, concerning the mechanisms for choosing the next element in A. Through this model, we identify that the variants differ in their transition probabilities in eleven states out of a total of 256 states. For these eleven states, we reveal how the transition probabilities change from variant (group of variants) to variant (group of variants). In other words, we indicate precisely the stochastic reasons for the discrepancies.

Modified scattering for the nonlinear nonlocal Schr\xf6dinger equation in one-dimensional case

We study the large time asymptotics of solutions to the Cauchy problem for the nonlinear nonlocal Schrödinger equation with critical nonlinearity i∂tu-∂x2u+∂x2u-a∂x4u=λu2u,t>0,x∈R,u0,x=u0x,x∈R,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{l} i\\partial _{t}\\left( u-\\partial _{x}^{2}u\\right) +\\partial _{x}^{2}u-a\\partial _{x}^{4}u=\\lambda \\left| u\\right| ^{2}u,\\text { } t>0,{\\ }x\\in {\\mathbb {R}}\\mathbf {,} \\\\ u\\left( 0,x\\right) =u_{0}\\left( x\\right) ,{\\ }x\\in {\\mathbb {R}}\\mathbf {,} \\end{array} \\right. \\end{aligned}$$\\end{document}where a>frac{1}{5},lambda in {mathbb {R}}. We continue to develop the factorization techniques which was started in papers Hayashi and Naumkin (Z Angew Math Phys 59(6):1002–1028, 2008) for Klein–Gordon, Hayashi and Naumkin (J Math Phys 56(9):093502, 2015) for a fourth-order Schrödinger, Hayashi and Kaikina (Math Methods Appl Sci 40(5):1573–1597, 2017) for a third-order Schrödinger to show the modified scattering of solutions to the equation. The crucial points of our approach presented here are based on the {mathbf {L}}^{2}-boundedness of the pseudodifferential operators.

Quantization of fractional harmonic oscillator using creation and annihilation operators

Abstract In this article, the Hamiltonian for the conformable harmonic oscillator used in the previous study [Chung WS, Zare S, Hassanabadi H, Maghsoodi E. The effect of fractional calculus on the formation of quantum-mechanical operators. Math Method Appl Sci. 2020;43(11):6950–67.] is written in terms of fractional operators that we called α \alpha -creation and α \alpha -annihilation operators. It is found that these operators have the following influence on the energy states. For a given order α \alpha , the α \alpha -creation operator promotes the state while the α \alpha -annihilation operator demotes the state. The system is then quantized using these creation and annihilation operators and the energy eigenvalues and eigenfunctions are obtained. The eigenfunctions are expressed in terms of the conformable Hermite functions. The results for the traditional quantum harmonic oscillator are found to be recovered by setting α = 1 \alpha =1 .

Liouville-Type Theorems for Sign-Changing Solutions to Nonlocal Elliptic Inequalities and Systems with Variable-Exponent Nonlinearities

We consider the fractional elliptic inequality with variable-exponent nonlinearity $$\begin{aligned} (-\Delta )^{\frac{\alpha }{2}} u+\lambda \, \Delta u \ge |u|^{p(x)}, \quad x\in {\mathbb {R}}^N, \end{aligned}$$where \(N\ge 1\), \(\alpha \in (0,2)\), \(\lambda \in {\mathbb {R}}\) is a constant, \(p: {\mathbb {R}}^N\rightarrow (1,\infty )\) is a measurable function, and \((-\Delta )^{\frac{\alpha }{2}}\) is the fractional Laplacian operator of order \(\frac{\alpha }{2}\). A Liouville-type theorem is established for the considered problem. Namely, we obtain sufficient conditions under which the only weak solution is the trivial one. Next, we extend our study to systems of fractional elliptic inequalities with variable-exponent nonlinearities. Besides the consideration of variable-exponent nonlinearities, the novelty of this work consists in investigating sign-changing solutions to the considered problems. Namely, to the best of our knowledge, only nonexistence results of positive solutions to fractional elliptic problems were investigated previously. Our approach is based on the nonlinear capacity method combined with a pointwise estimate of the fractional Laplacian of some test functions, which was derived by Fujiwara (Math Methods Appl Sci 41:4955–4966, 2018) [see also Dao and Reissig (A blow-up result for semi-linear structurally damped \(\sigma \)-evolution equations, arXiv:1909.01181v1, 2019)]. Note that the standard nonlinear capacity method cannot be applied to the considered problems due to the change of sign of solutions.

New decay results for a viscoelastic-type Timoshenko system with infinite memory

This paper is concerned with the following memory-type Timoshenko system $$\begin{aligned} {\left\{ \begin{array}{ll} \rho _1 \varphi _{tt}-K(\varphi _x+\psi )_x =0,\\ \rho _2\psi _{tt}-b\psi _{xx}+K(\varphi _x+\psi )+\displaystyle \int \limits _0^{+\infty } g(s)\psi _{xx}(t-s){\mathrm{d}}s=0,\\ \end{array}\right. } \end{aligned}$$ with Dirichlet boundary conditions, where g is a positive nonincreasing function satisfying, for some nonnegative functions $$\xi $$ and G, $$\begin{aligned}g'(t)\le -\xi (t)G(g(t)),\qquad \forall t\ge 0.\end{aligned}$$ Under appropriate conditions on $$\xi $$ and G, we establish some new decay results that generalize and improve many earlier results in the literature such as Mustafa (Math Methods Appl Sci 41(1): 192–204, 2018), Messaoudi et al. (J Integral Equ Appl 30(1): 117–145, 2018) and Guesmia (Math Model Anal 25(3): 351–373, 2020). We consider the equal speeds of propagation case, as well as the nonequal-speed case. Moreover, we delete some assumptions on the boundedness of initial data used in many earlier papers in the literature.

Homotopy perturbation method for Fangzhu oscillator

An accurate frequency-amplitude relationship is very needed to elucidate the properties of the oldest device of Fangzhu for collecting water from the air. The Fangzhu oscillator was derived and solved approximately (He et al. in Math Methods Appl Sci, 2020, https://doi.org/10.1002/mma.6384 ), here we show that the singular Duffing-like oscillator can be more effectively solved by the homotopy perturbation method and a criterion is obtained for the existence of a periodic solution for the singular differential equation. The results obtained in this paper are helpful for the optimal design of the Fangzhu device.

Landau-Type Theorems for Certain Bounded Biharmonic Mappings

In this paper, we establish three sharp versions of the Landau-type theorems for bounded biharmonic mappings $$F(z)=|z|^2G(z)+H(z)$$, where G(z) and H(z) are harmonic in the unit disk U with $$G(0)=H(0)=0$$ and $$\lambda _{F}(0)=||F_z(0)|-|F_{{\overline{z}}}(0)||=1$$. Our results generalize (or improve) the corresponding results given in Liu et al. (Math Methods Appl Sci 40:2582–2595, 2017). Three conjectures for the sharp version of the Landau-type theorem for certain bounded biharmonic mappings are given in third section.

Notes on “stability and chaos control of regularized Prabhakar fractional dynamical systems without and with delay”

This comment is to point out some mistakes made in the paper (Shiva et al in MATH METHOD APPL SCI 42(7):2302‐2323, 2019). The stability of Prabhakar fractional dynamical systems without and with time delay is addressed. Lyapunov stability theorem is extended to these systems. But the proof process is wrong as the memory property of fractional calculus. It is necessary to point out these errors to avoid misleading. Finally, a counterexample is proposed against the proposed theorem.

The sharp decay rate of thermoelastic transmission system with infinite memories

The main contributions here are to show the lack of exponential stability and to prove that the $$t^{-1}$$ is the sharp decay rate of problem (1.1). That is to show that for this types of materials, the dissipation produced by the weak-infinite memories are not strong enough to produce an exponential decay of the solution under usual/non usual conditions on the the relaxation functions’s growth. This work extends the previous results by Bayoud et al. (Appl Sci 20:18–35, 2018), Zennir and Feng (Math Methods Appl Sci 112:6895–6906, 2018. https://doi.org/10.1002/mma.5201 ) to the weak-viscoelasticities in two parts. In order to fill this gaps, we use an appropriate estimates.

Some existence results for an elliptic equation of Kirchhoff‐type with changing sign data and a logarithmic nonlinearity

The paper deals with some existence results for an elliptic equation of Kirchhoff‐type with changing sign data and a logarithmic nonlinearity by direct variational method, Galerkin approach, and sub‐super solutions method. Our results are natural extension of Boulaaras' work in (Math Methods Appl Sci; 41(13):5203‐5210).

Asymptotic Stability for a Viscoelastic Equation with Nonlinear Damping and Very General Type of Relaxation Functions

In this paper, we consider a viscoelastic equation with a nonlinear frictional damping and a relaxation function satisfying g′(t) ≤ −ξ(t)G(g(t)). Using the Galaerkin method, we establish the existence of the solution and prove an explicit and general decay rate results, using the multiplier method and some properties of the convex functions. This work generalizes and improves earlier results in the literature. In particular, those of Messaoudi (2016) and Mustafa (Math Methods Appl Sci. 2017;V41:192–204).

Asymptotic behavior for a fourth-order parabolic equation involving the Hessian

This paper deals with a fourth-order parabolic PDE arising in the theory of epitaxial growth of crystal. For the stationary problem, we find a ground-state solution on its corresponding Nehari manifold by Lagrange multiplier method. As far as the evolution problem is concerned, we study the dynamics for both the global solution and the blow-up solution. Especially, for the global solution, we prove that the energy functional decays exponentially and we get the concretely decay rate. For the blow-up solution, the exponential growth of the solution is obtained, and we also obtain the growth rate. The behavior of the energy functional as $$t\rightarrow T$$ is also discussed, where T is the blow-up time. Moreover, we prove that there exists blow-up solution with arbitrary high initial energy. Finally, for the low initial energy case, we give some equivalent conditions for the solutions blow up in finite time and exist globally, respectively. Our results extend the results got by Escudero et al. (Eur J Appl Math 24(3):437–453, 2013; J Differ Equ 254(6):2515–2531, 2013; Math Method Appl Sci 37(6):793–807, 2014; J Math Pures Appl 103(4):924–957, 2015).

Degenerate Bernstein polynomials

Here we consider the degenerate Bernstein polynomials as a degenerate version of Bernstein polynomials, which are motivated by Simsek’s recent work ‘Generating functions for unification of the multidimensional Bernstein polynomials and their applications’ (Simsek in Filomat 30(7):1683–1689, 2016, Math Methods Appl Sci 1–12, 2018) and Carlitz’s degenerate Bernoulli polynomials. We derived their generating function, symmetric identities, recurrence relations, and some connections with generalized falling factorial polynomials, higher-order degenerate Bernoulli polynomials and degenerate Stirling numbers of the second kind.

Homogenization of an Elliptic Equation in a Domain with Oscillating Boundary with Non-homogeneous Non-linear Boundary Conditions

While considering boundary value problems with oscillating coefficients or in oscillating domains, it is important to associate an asymptotic model which accounts for the average behaviour. This model permits to obtain the average behaviour without costly numerical computations implied by the fine scale of oscillations in the original model. The asymptotic analysis of boundary value problems in oscillating domains has been extensively studied and involves some key issues such as: finding uniformly bounded extension operators for function spaces on oscillating domains, the choice of suitable sequences of test functions for passing to the limit in the variational formulation of the model equations etc. In this article, we study a boundary value problem for the Laplacian in a domain, a part of whose boundary is highly oscillating (periodically), involving non-homogeneous non-linear Neumann or Robin boundary condition on the periodically oscillating boundary. The non-homogeneous Neumann condition or the Robin boundary condition on the oscillating boundary adds a further difficulty to the limit analysis since it involves taking the limits of surface integrals where the surface changes with respect to the parameter. Previously, some model problems have been studied successfully in Gaudiello (Ricerche Mat 43(2):239–292, 1994) and in Mel’nyk (Math Methods Appl Sci 31(9):1005–1027, 2008) by converting the surface term into a volume term using auxiliary boundary value problems. Some problems of this nature have also been studied using an extension of the notion of two-scale convergence (Allaire et al. in Proceedings of the international conference on mathematical modelling of flow through porous media, Singapore, 15–25, 1996, Neuss-Radu in C R Acad Sci Paris Sr I Math 322:899–904, 1996). In this article, we use a different approach to handle of such terms based on the unfolding operator.

Boundedness of classical solutions for a chemotaxis system with general sensitivity function

ABSTRACTThis paper deals with the chemotaxis system with general sensitivity function: under homogeneous Neumann boundary conditions in a bounded domain with smooth boundary. Here, and the initial function and the sensitivity function satisfy: We prove that the classical solutions to the above system are uniformly in-time-bounded provided that there exists a smooth positive function such that for some and the following differential inequality holds where and is bounded from above. We also present our results for the special case with and These results coincide with the results obtained by Fujie et al. [Math Methods Appl Sci. 2015] in the case of and extend their results in the case of .

Lifespan for a semilinear pseudo‐parabolic equation

This paper deals with the blow‐up solution to the following semilinear pseudo‐parabolic equation urn:x-wiley:mma:media:mma4639:mma4639-math-0001 in a bounded domain , which was studied by Luo (Math Method Appl Sci 38(12):2636‐2641, 2015) with the following assumptions on p: urn:x-wiley:mma:media:mma4639:mma4639-math-0003 and the lifespan for the initial energy J(u0)<0 is considered. This paper generalizes the above results on the following two aspects: a new blow‐up condition is given, which holds for all p> 1; a new lifespan is given, which holds for all p> 1 and possible J(u0)≥0. Moreover, as a byproduct, we refine the lifespan when J(u0)<0.