Showing papers in "Applicable Analysis in 2022"
TL;DR: In this article , the authors considered a class of quasilinear stationary Kirchhoff type potential systems with Neumann boundary conditions and established existence and multiplicity of solutions for the problem by using the concentration-compactness principle of Lions for variable exponents and the mountain pass theorem without the Palais-Smale condition.
Abstract: In this paper, we consider a class of quasilinear stationary Kirchhoff type potential systems with Neumann Boundary conditions, which involves a general variable exponent elliptic operator with critical growth. Under some suitable conditions on the nonlinearities, we establish existence and multiplicity of solutions for the problem by using the concentration-compactness principle of Lions for variable exponents and the mountain pass theorem without the Palais–Smale condition.
22 citations
18 citations
TL;DR: In this paper , the existence and approximation results for degenerated anisotropic (p, q)-Laplacian with weights −∑i=1N∂∂xi((a(x)|∂u∂si|pi−2+b(x),∂ u∂i|qi−2)∂ U∆∆)∆ = f(x,u,∇u),∆,u∆)=f(x u,∆),u u ∆)
Abstract: Here, the existence and approximation results for degenerated anisotropic (p,q)-Laplacian with weights −∑i=1N∂∂xi((a(x)|∂u∂xi|pi−2+b(x)|∂u∂xi|qi−2)∂u∂xi)=f(x,u,∇u)and a competing anisotropic (p,q)-Laplacian with weights −∑i=1N∂∂xi((a(x)|∂u∂xi|pi−2−b(x)|∂u∂xi|qi−2)∂u∂xi)=f(x,u,∇u)are studied with Dirichlet boundary condition and on a bounded smooth domain in RN, N≥3 where f:Ω×R×RN→R is a Carathéodory function. The proofs are based on weighted antitropic Sobolev spaces, Nemytskij operators and finite dimensional approximation.
9 citations
TL;DR: In this paper , the authors prove that the classical and modified Hilbert transform differ by a compact perturbation, when a suitable extension of a function defined on a bounded time interval is used.
Abstract: The Hilbert transform is a useful tool in the mathematical analysis of time-dependent partial differential equations in order to prove coercivity estimates in anisotropic Sobolev spaces in case of a bounded spatial domain Ω, but an infinite time interval . Instead, a modified Hilbert transform can be used if we consider a finite time interval . In this note we prove that the classical and the modified Hilbert transformations differ by a compact perturbation, when a suitable extension of a function defined on a bounded time interval onto is used. This result is important when we deal with space–time variational formulations of time-dependent partial differential equations, and for the implementation of related space–time finite and boundary element methods for the numerical solution of parabolic and hyperbolic equations with the heat and wave equations as model problems, respectively.
8 citations
TL;DR: In this paper , the Hilbert complex of elasticity involving spaces of symmetric tensor fields was investigated and closed ranges, Friedrichs/Poincaré type estimates, Helmholtz-type decompositions, regular decomposition, regular potentials, finite cohomology groups, and compact embedding results were obtained.
Abstract: We investigate the Hilbert complex of elasticity involving spaces of symmetric tensor fields. For the involved tensor fields and operators we show closed ranges, Friedrichs/Poincaré type estimates, Helmholtz-type decompositions, regular decompositions, regular potentials, finite cohomology groups, and, most importantly, new compact embedding results. Our results hold for general bounded strong Lipschitz domains of arbitrary topology and rely on a general functional analysis framework (FA-ToolBox). Moreover, we present a simple technique to prove the compact embeddings based on regular decompositions/potentials and Rellich's selection theorem, which can be easily adapted to any Hilbert complex.
7 citations
TL;DR: In this article , the authors consider the Keller-Segel-type migration-consumption system with signal-dependent motilities and show that the global existence of weak solutions with possibly quite poor regularity properties is not guaranteed.
Abstract: We consider the Keller-Segel-type migration-consumption system involving signal-dependent motilities, $$\left\{ \begin{array}{l} u_t = \Delta \big(u\phi(v)\big), \\[1mm] v_t = \Delta v-uv, \end{array} \right. \qquad \qquad$$ in smoothly bounded domains $\Omega\subset\mathbb{R}^n$, $n\ge 1$. Under the assumption that $\phi\in C^1([0,\infty))$ is positive on $[0,\infty)$, and for nonnegative initial data from $(C^0(\overline{\Omega}))^\star \times L^\infty(\Omega)$, previous literature has provided results on global existence of certain very weak solutions with possibly quite poor regularity properties, and on large time stabilization toward semitrivial equilibria with respect to the topology in $(W^{1,2}(\Omega))^\star \times L^\infty(\Omega)$. The present study reveals that solutions in fact enjoy significantly stronger regularity features when $0<\phi\in C^3([0,\infty))$ and the initial data belong to $(W^{1,\infty}(\Omega))^2$: It is firstly shown, namely, that then in the case $n\le 2$ an associated no-flux initial-boundary value problem even admits a global classical solution, and that each of these solutions smoothly stabilizes in the sense that as $t\to\infty$ we have $$ \begin{align*} u(\cdot,t) \to \frac{1}{|\Omega|}\int_\Omega u_0 \qquad \text{ and } \qquad v(\cdot,t)\to 0 \qquad \qquad (\star) \end{align*}$$ even with respect to the norm in $L^\infty(\Omega)$ in both components. In the case when $n\ge 3$, secondly, some genuine weak solutions are found to exist globally, inter alia satisfying $
abla u\in L^\frac{4}{3}_{loc}(\overline{\Omega}\times [0,\infty);\mathbb{R}^n)$. In the particular three-dimensional setting, any such solution is seen to become eventually smooth and to satisfy ($\star$).
6 citations
TL;DR: In this paper , the improved -expansion approach is utilized to extract numerous soliton solutions for NV system and the finite difference approach is successfully applied to achieve the numerical simulations of the proposed equations.
Abstract: Solutions such as symmetric, bright soliton and periodic solutions play a prominent role in the field of differential equations, and they can be used to investigate several phenomena in nonlinear sciences. Some waves such as ion and magneto-sound waves in plasma are investigated by using some partial differential equations (PDEs) such as Novikov-Veselov (NV) equations. In this work, the improved -expansion approach is utilized to extract numerous soliton solutions for NV system. Hamiltonian system is invoked to analyze the stability of some solutions. The finite difference approach is successfully applied to achieve the numerical simulations of the proposed equations. We also introduce the stability and the accuracy of the numerical scheme. In order to validate the correctness of the accomplished results, we compare the exact solutions with the numerical solutions analytically and graphically. The presented techniques are very convenient and adequate and can be employed to other types of nonlinear evolution equations.
6 citations
TL;DR: In this paper , a three-parameter family of non-linear equations with -order nonlinearities is considered and a global existence result and a scenario that prevents the wave breaking of solutions are presented.
Abstract: ABSTRACT We consider a three-parameter family of non-linear equations with -order non-linearities. Such a family includes as a particular member the well-known b-equation, which encloses the famous Camassa–Holm equation. For certain choices of the parameters, we establish a global existence result and show a scenario that prevents the wave breaking of solutions. Also, we explore unique continuation properties for some values of the parameters.
5 citations
TL;DR: In this article , the authors analyzed various classes of multi-dimensional ρ-almost periodic type functions F:I×X→Y and ρ is a binary relation on Y. The main structural properties and characterizations for the introduced classes of functions are presented.
Abstract: In this paper, we analyze various classes of multi-dimensional ρ-almost periodic type functions F:I×X→Y and multi-dimensional (ω,ρ)-almost periodic type functions F:I×X→Y, where n∈N, ∅≠I⊆Rn, X and Y are complex Banach spaces and ρ is a binary relation on Y. The proposed notion is new even in the one-dimensional setting, for the functions of the form F:I→Y. The main structural properties and characterizations for the introduced classes of functions are presented. We provide certain applications of our theoretical results to the abstract Volterra integro-differential equations, as well.
5 citations
TL;DR: In this article , a zero-flux attraction-repulsion chemotaxis model with nonlinear production rates for the chemorepellent and the chemoattractant was proposed.
Abstract: We study some zero-flux attraction-repulsion chemotaxis models, with nonlinear production rates for the chemorepellent and the chemoattractant. This paper partially improves some known results in the literature and moreover solves an open question. The research is also complemented with numerical simulations in bi-dimensional domains.
5 citations
TL;DR: In this article , the authors established a general stability result for a one-dimensional linear swelling porous-elastic system with infinite memory, irrespective of the wave speeds of the system, based on the multiplier method and some properties of convex functions.
Abstract: This paper aims to establish a general stability result for a one-dimensional linear swelling porous-elastic system with infinite memory, irrespective of the wave speeds of the system. The proof is based on the multiplier method and some properties of convex functions. The kernel in our memory term is more general and of a broader class. Our output extends and improves some of the available results on swelling porous media in the literature.
TL;DR: In this article , the authors studied local and global Cauchy problems for the semilinear parabolic equations with initial data in Herz spaces, which unify and generalize many classical function spaces such as Lebesgue spaces of power weights.
Abstract: In this paper, we will study local and global Cauchy problems for the semilinear parabolic equations with initial data in Herz spaces. These spaces unify and generalize many classical function spaces such as Lebesgue spaces of power weights. Our results cover the results obtained with initial data in Lebesgue spaces. Moreover, the results in Herz spaces are a little different from the results in Lebesgue spaces.
TL;DR: In this paper , the authors considered the scattering of a time-harmonic electromagnetic wave by a penetrable chiral obstacle in an achiral environment, and the method of fundamental solutions was employed to obtain numerically the solution of the problem using fundamental solutions in dyadic form.
Abstract: The scattering of a time-harmonic electromagnetic wave by a penetrable chiral obstacle in an achiral environment is considered. The method of fundamental solutions is employed in order to obtain numerically the solution of the problem using fundamental solutions in dyadic form. Surface vector potentials in terms of dyadic fundamental solutions together with the associated boundary integral operators are defined and their regularity properties are presented. Based on the dependence of the solution to the boundary data, appropriate systems of functions containing elements of dyadic fundamental solutions on the surface of the scatterer are constructed. Completeness and linear independence for these systems are proved with the usage of surface vector potentials. Using the transmission conditions, the scattering problem is transformed into a linear algebraic system with a coefficient matrix which consists of chiral and achiral blocks.
TL;DR: In this article , a discrete model governing the deformation of a convex regular polygon subjected not to cross a given flat rigid surface, on which it initially lies in correspondence of one point only, is presented.
Abstract: In this paper we present a discrete model governing the deformation of a convex regular polygon subjected not to cross a given flat rigid surface, on which it initially lies in correspondence of one point only. First, we set up the model in the form of a set of variational inequalities posed over a non-empty, closed and convex subset of a suitable Euclidean space. Secondly, we show the existence and uniqueness of the solution. The model provides a simplified illustration of processes involved in virus imaging by atomic force microscopy: adhesion to a surface, distributed strain, relaxation to a shape that balances adhesion and elastic forces. The analysis of numerical simulations results based on this model opens a new way of estimating the contact area and elastic parameters in virus contact mechanics studies.
TL;DR: In this article , the stability of a new coupling to a thermoelastic Timoshenko-type system with memory was investigated, and the stability result was obtained without imposing the condition of equal-wave speed.
Abstract: This paper investigates the asymptotic stability of a new coupling to a thermoelastic Timoshenko-type system with memory. The memory term acts on the shear force in the evolution equations. We establish the well-posedness result for the system. Furthermore, we prove a general decay result for the solution energy. Interestingly, our stability result is obtained without imposing the condition of equal-wave speed.
TL;DR: In this paper , some complicate dynamical behaviors are formulated for a discrete predator-prey model with group defense and nonlinear harvesting in prey, and conditions for the occurrences of transcritical bifurcation, saddle-node bifurocation, and Neimark-Sacker bifurbation, respectively.
Abstract: In this paper, some complicate dynamical behaviors are formulated for a discrete predator–prey model with group defense and nonlinear harvesting in prey. After considering the existence and stability for all of its nonnegative fixed points, our main work is to present those conditions for the occurrences of transcritical bifurcation, saddle-node bifurcation and Neimark–Sacker bifurcation, respectively. Numerical simulations not only verify the theoretical results for saddle-node bifurcation and Neimark–Sacker bifurcation but also display more interesting dynamical properties of the model.
TL;DR: In this paper , the mathematical justification of a macroscopic Baer-Nunziato PDE bifluid system with a physical relaxation term that is linked to the two viscosities and the two pressure laws of the two compressible phases of the fluid which may be different is discussed.
Abstract: ABSTRACT This paper concerns the mathematical justification of a macroscopic Baer–Nunziato PDE bifluid system with a physical relaxation term that is linked to the two viscosities and the two pressure laws of the two compressible phases of the fluid which may be different. This is achieved using an homogenization approach in a periodic framework from a mesoscopic PDE description of two immiscible compressible viscous fluids with interfaces and no mass transfer. Our result extends the work in Bresch D, Hillairet M. [Note on the derivation of multi-component flow systems. Proc Am Math Soc. 2015;143:3429–3443] by allowing to consider different pressure laws for each component introducing an order parameter. This paper is complementary to the recent work [Bresch D, Burtea C, Lagoutière F. Mathematical justification of a compressible bi-fluid system with different pressure laws: a semi-discrete approach and numerical illustrations. Submitted 2021] which focuses on a semi-discretized approach and numerical illustrations. These two papers correspond to the extended versions of the document arXiv:2012.06497.
TL;DR: In this paper , the mountain pass theorem and the concentration compactness principles for fractional Sobolev spaces with variable exponents were used to obtain the existence and multiplicity of nontrivial solutions for non-degenerate and degenerate cases.
Abstract: In this paper, we are interested in a class of critical nonlocal problems with variable exponents of the form: {M(Tp(⋅,⋅)(u))[(−Δ)p(x,y)su+|u|p~(x)−2u]=λf(x,u)+|u|q(x)−2u,in RN,u∈Ws,p(⋅,⋅)(RN),p~(x)=p(x,x),where Tp(⋅,⋅)(u):=∬R2N|u(x)−u(y)|p(x,y)p(x,y)|x−y|N+sp(x,y)dxdy+∫RN1p~(x)|u|p~(x)dx,M is the Kirchhoff function, λ is a real parameter and f is a continuous function. We also assume that {x∈RN:q(x)=ps∗(x)}≠∅, where ps∗(x)=Np~(x)/(N−sp~(x)) is the critical Sobolev exponent for variable exponents. The strategy of the proof for these results is to approach the problem variationally by using the mountain pass theorem and the concentration-compactness principles for fractional Sobolev spaces with variable exponents. In addition, we obtain the existence and multiplicity of nontrivial solutions for the above problem in non-degenerate and degenerate cases.
TL;DR: In this article , the authors solved time fractional reaction diffusion equations with the Caputo fractional derivative using the classical L1-formula and the finite volume element (FVE) methods on triangular grids.
Abstract: In this paper, time fractional reaction–diffusion equations with the Caputo fractional derivative are solved by using the classical L1-formula and the finite volume element (FVE) methods on triangular grids. The existence and uniqueness for the fully discrete FVE scheme are given. The stability results and optimal a priori error estimate in L2(Ω)-norm are derived, but it is difficult to obtain the corresponding results in H1(Ω)-norm, so another analysis technique is introduced and used to achieve our goal. Finally, some numerical results are given to verify the feasibility and effectiveness.
TL;DR: In this article , the asymptotic behavior of the periodically mixed boundary value problem was studied and an approximation was obtained for the Dirichlet and Neumann boundary conditions with small period ε.
Abstract: We study the asymptotic behavior of the periodically mixed boundary value problem. The Dirichlet and Neumann boundary conditions are non-homogeneous and periodically mixed with small period ε. Using asymptotic analysis with respect to , we derive an asymptotic approximation that has boundary condition of the Robin type. We justify the obtained condition by proving an appropriate error estimate.
TL;DR: In this article , a Pohožaev manifold is defined via analyzing the term , by using Moser iteration, and the existence of positive ground state solutions is obtained.
Abstract: This paper is concerned with the following quasilinear Schrödinger system in the entire space : where , , k>0 is a parameter. The main strategy is to define a Pohožaev manifold via analyzing the term , by using Moser iteration, we obtain the existence of positive ground state solutions. Our main contribution is related to the fact that we are able to deal with the case k>0 for quasilinear system.
TL;DR: In this paper , approximate solutions of a nonsmooth semi-infinite programming problem with multiple interval-valued objective functions are studied. But the authors focus on the lower-upper interval order relation and apply some advanced tools of variational analysis and generalized differentiation to establish necessary optimality conditions.
Abstract: This paper deals with approximate solutions of a nonsmooth semi-infinite programming with multiple interval-valued objective functions. We first introduce four types of approximate quasi Pareto solutions of the considered problem by considering the lower-upper interval order relation and then apply some advanced tools of variational analysis and generalized differentiation to establish necessary optimality conditions for these approximate solutions. Sufficient conditions for approximate quasi Pareto solutions of such a problem are also provided by means of introducing the concepts of approximate (strictly) pseudo-quasi generalized convex functions defined in terms of the limiting subdifferential of locally Lipschitz functions. Finally, a Mond–Weir type dual model in approximate form is formulated, and weak, strong and converse-like duality relations are proposed.
TL;DR: In this article , the authors further investigated the classical problem of the wind in the steady atmospheric Ekman layer and constructed an explicit solution for the case of eddy viscosity being subjected to a quadratic function, where the used approach is different from Delia.
Abstract: This paper further investigates the classical problem of the wind in the steady atmospheric Ekman layer. For the case of eddy viscosity being subjected to a quadratic function, we construct an explicit solution, where the used approach is different from Delia [Analytical atmospheric Ekman-type solutions with height-dependent eddy viscosities. J Math Fluid Mech. 2021;23:Article ID 18]. For the case of eddy viscosity being related to two-value piecewise-constant, we give a formula to compute the surface deflection angle by the constructed solution.
TL;DR: In this paper , a product integration approach based on the first kind of Chebyshev polynomials is presented to solve the weakly singular integral-algebraic equations of index-1.
Abstract: This paper aims to present a product integration approach based on the first kind of Chebyshev polynomials to solve the weakly singular integral-algebraic equations of index-1. These are mixed systems of weakly singular Volterra integral equations of the second and first kind. The solutions of these equations have a singularity at the lower bound of the domain of integration. To deal with this non-smooth behavior of solutions, we apply the suitable transformations to get a new system with regular solutions. We use the differentiation index in the theoretical part. One of the advantages of selecting the weights of the presented numerical method is that when we determine the error bounds, some expressions will be equal to zero speeding up the process of achieving the error bounds. Finally, by providing some numerical examples, we will observe that the numerical results confirm the validity of convergence analysis.
TL;DR: In this paper , it was shown that every signal can be recovered from a windowed linear canonical transform with a univariate integral, and that the integral involved is convergent almost everywhere on as well as in for all.
Abstract: We study the inversion formula for recovering a signal from its windowed linear canonical transform. Different from the known inversion formula, where a double integral is invoked, we show that every signal can be recovered from its windowed linear canonical transform with a univariate integral. Moreover, we show that the integral involved is convergent almost everywhere on as well as in for all . Furthermore, we also obtain an inversion formula for functions in with the method of Cesàro summability.
TL;DR: A more comprehensive mathematical model for diabetic atherosclerosis which include more variables, in particular it includes the variable for Advanced Glycation End-Products (AGEs) concentration, which may accelerate the progression of vascular disease in diabetic patients.
Abstract: Atherosclerosis is a leading cause of death worldwide; it emerges as a result of multiple dynamical cell processes including hemodynamics, endothelial damage, innate immunity and sterol biochemistry. Making matters worse, nearly 463 million people have diabetes, which increases atherosclerosis-related inflammation, diabetic patients are twice as likely to have a heart attack or stroke. The pathophysiology of diabetic vascular disease is generally understood. Dyslipidemia with increased levels of atherogenic LDL, hyperglycemia, oxidative stress and increased inflammation are factors that increase the risk and accelerate development of atherosclerosis. In a recent paper [53], we have developed mathematical model that includes the effect of hyperglycemia and insulin resistance on plaque growth. In this paper, we propose a more comprehensive mathematical model for diabetic atherosclerosis which include more variables; in particular it includes the variable for Advanced Glycation End-Products (AGEs)concentration. Hyperglycemia trigger vascular damage by forming AGEs, which are not easily metabolized and may accelerate the progression of vascular disease in diabetic patients. The model is given by a system of partial differential equations with a free boundary. We also establish local existence and uniqueness of solution to the model. The methodology is to use Hanzawa transformation to reduce the free boundary to a fixed boundary and reduce the system of partial differential equations to an abstract evolution equation in Banach spaces, and apply the theory of analytic semigroup.
TL;DR: In this paper , the authors considered the wave equation on an unbounded domain for highly oscillatory coefficients with the scaling and considered settings in which the homogenization process for this equation is well understood.
Abstract: We consider the wave equation on an unbounded domain for highly oscillatory coefficients with the scaling . We consider settings in which the homogenization process for this equation is well understood, which means that holds for the solution of the homogenized problem . In this context, domain truncation methods are studied. The goal is to calculate an approximate solution on a subdomain, say . We are ready to solve the ε-problem on , but we want to solve only homogenized problems on the unbounded domains or . The main task is to define transmission conditions at the interface to have small differences . We present different methods and corresponding error estimates.
TL;DR: In this article , an analytical study of the transmission eigenvalue problem with two conductivity parameters is presented, where the underlying physical model is given by the scattering of a plane wave for an isotropic scatterer.
Abstract: In this paper, we provide an analytical study of the transmission eigenvalue problem with two conductivity parameters. We will assume that the underlying physical model is given by the scattering of a plane wave for an isotropic scatterer. In previous studies, this eigenvalue problem was analyzed with one conductive boundary parameter whereas we will consider the case of two parameters. We will prove the existence and discreteness of the transmission eigenvalues as well as study the dependence on the physical parameters. We are able to prove monotonicity of the first transmission eigenvalue with respect to the parameters and consider the limiting procedure as the second boundary parameter vanishes. Lastly, we provide extensive numerical experiments to validate the theoretical work.