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This paper considers the Cauchy problem for a second-order hyperbolic equation with periodic coefficients and zero initial data, and the right-hand side of the equation is the function , . For this problem, the limiting amplitude principle and the asymptotic behavior as are established. [ABSTRACT FROM AUTHOR]

Following the methodology of Brasco (Adv Math 394:108029, 2022), we study the long-time behavior for the signed fractional porous medium equation in open bounded sets with smooth boundary. Homogeneous exterior Dirichlet boundary conditions are considered. We prove that if the initial datum has sufficiently small energy, then the solution, once suitably rescaled, converges to a nontrivial constant sign solution of a sublinear fractional Lane–Emden equation. Furthermore, we give a nonlocal sufficient energetic criterion on the initial datum, which is important to identify the exact limit profile, namely the positive solution or the negative one. [ABSTRACT FROM AUTHOR]

We study the asymptotic behavior of solutions for nn-component Ginzburg-Landau equations as ε→0\varepsilon \to 0. We prove that the minimizers converge locally in any Ck{C}^{k}-norm to a solution of a system of generalized harmonic map equations.

In this paper, the large time behavior of globally smooth solutions of the Cauchy problem for the three dimensional Euler-Poisson system of compressible fluids coupled to a magnetic field is studied. We prove that the smooth solutions (near a given constant equilibrium state) of the problem converge asymptotically to a stationary solution exponentially fast as t goes to ∞.

We establish the stabilization of the strong solution (u ⁢ (t , x) , k ⁢ (t)) to the inverse problem for a pseudoparabolic equation (u + η ⁢ M ⁢ u) t + M ⁢ u + k ⁢ (t) ⁢ u = f with an unknown coefficient k ⁢ (t) to the solution (u ∞ , k ∞) of the appropriate stationary inverse problem. The operator M = - div ⁡ (ℳ ⁢ (x) ⁢ ∇) + m ⁢ (x) ⁢ I is supposed to be elliptic and self-adjoint. The asymptotic behavior of the solution (u ⁢ (t , x) , k ⁢ (t)) is investigated as t → + ∞ . [ABSTRACT FROM AUTHOR]

Solutions of boundary value problems for linear partial differential equations are known to exhibit singular behaviors near the boundary of nonsmooth domains. In this case, one is usually interested on the asymptotic behavior of the solutions near the geometric singularities. However, for both mathematical and engineering purposes, it is important to have extraction formulas for the coefficients in the asymptotic expansion. In this paper, we consider initial boundary value problems for the wave equation in two‐dimensional domains with corners and derive, by means of Fourier analysis, explicit computable expressions for the coefficients in the asymptotic expansion of the solution near the corners. The explicit structure of the singular solutions presented herein makes it fairly easy to compute numerically. Thus, the formulas can be used to construct postprocessing procedures for improving the accuracy of standard numerical techniques of approximating the solution and for improving the rates of convergence in the error estimates. The analysis presented herein would also apply to more general hyperbolic or parabolic equations with constant coefficients. [ABSTRACT FROM AUTHOR]

We introduce a mathematically rigorous formulation for a nonlocal interface problem with jumps and propose an asymptotically compatible finite element discretization for the weak form of the interface problem. After proving the well-posedness of the weak form, we demonstrate that solutions to the nonlocal interface problem converge to the corresponding local counterpart when the nonlocal data are appropriately prescribed. Several numerical tests in one and two dimensions show the applicability of our technique, its numerical convergence to exact nonlocal solutions, its convergence to the local limit when the horizons vanish, and its robustness with respect to the patch test. [ABSTRACT FROM AUTHOR]

Motivated by the vanishing contact problem, we study in the present paper the convergence of solutions of Hamilton–Jacobi equations depending nonlinearly on the unknown function. Let H ⁢ (x , p , u) be a continuous Hamiltonian which is strictly increasing in u, and is convex and coercive in p. For each parameter λ > 0 , we denote by u λ the unique viscosity solution of the Hamilton–Jacobi equation H ⁢ (x , D ⁢ u ⁢ (x) , λ ⁢ u ⁢ (x)) = c. Under quite general assumptions, we prove that u λ converges uniformly, as λ tends to zero, to a specific solution of the critical Hamilton–Jacobi equation H ⁢ (x , D ⁢ u ⁢ (x) , 0) = c . We also characterize the limit solution in terms of Peierls barrier and Mather measures. [ABSTRACT FROM AUTHOR]

This paper addresses the qualitative theory of mixed-order positive linear coupled systems with bounded or unbounded delays. First, we introduce a general result on the existence and uniqueness of solutions to mixed-order linear systems with time-varying delays. Next, we obtain necessary and sufficient criteria which characterize the positivity of mixed-order delay linear coupled systems. Our main contributions are in Section 5. More precisely, by using a smoothness property of solutions to fractional differential equations and developing a new appropriated comparison principle for solutions to mixed-order delay positive systems, we prove the attractivity of mixed-order non-homogeneous linear positive coupled systems under the impact of bounded or unbounded delays. We also establish a necessary and sufficient condition to| ensure the stability of homogeneous systems. As a consequence of these results, we show the smallest asymptotic bound of solutions to mixed-order delay positive non-homogeneous linear coupled systems where disturbances are continuous and bounded. Finally, we provide numerical simulations to illustrate the proposed theoretical results. [ABSTRACT FROM AUTHOR]

The goal of this paper is to discuss the asymptotic behavior of weak solutions to a class of parabolic equations involving fractional Laplacian in cylindrical domains becoming unbounded in one direction. The results presented in this paper are new and extend some main results in the literature for local and nonlocal elliptic problems with Dirichlet boundary condition. [ABSTRACT FROM AUTHOR]

In this paper, we design efficient quadrature rules for finite element (FE) discretizations of nonlocal diffusion problems with compactly supported kernel functions. Two of the main challenges in nonlocal modeling and simulations are the prohibitive computational cost and the nontrivial implementation of discretization schemes, especially in three‐dimensional settings. In this work, we circumvent both challenges by introducing a parametrized mollifying function that improves the regularity of the integrand, utilizing an adaptive integration technique, and exploiting parallelization. We first show that the "mollified" solution converges to the exact one as the mollifying parameter vanishes, then we illustrate the consistency and accuracy of the proposed method on several two‐ and three‐dimensional test cases. Furthermore, we demonstrate the good scaling properties of the parallel implementation of the adaptive algorithm and we compare the proposed method with recently developed techniques for efficient FE assembly. [ABSTRACT FROM AUTHOR]

In this paper, we consider a Timoshenko type system coupled with parabolic equation that represents the thermal effect given by the Gurtin-Pipkin law while taking into account that the temperature influences on the shear force. We establish the global existence of solution the behavior of the solution. We also prove the lack of the exponential stability when ξ ≠ 0 , we obtain the exponential stability in the case where ξ = 0 and the parameter p = 1 and also the polynomial stability in the case where ξ ≠ 0 and 1 < p < 3 2 such that p is a parameter related to the thermal memory. One of the novel contribution here is that there is a new number ξ that is different from the number of stability that has been proven previously. [ABSTRACT FROM AUTHOR]

The paper is devoted to studying the behavior of solutions of the Cauchy problem for large values of time—more precisely, obtaining an asymptotic expansion characterizing the behavior of the solution of the Cauchy problem for a one-dimensional second-order hyperbolic equation with periodic coefficients for large values of the time parameter t. To obtain this asymptotic expansion as t→∞, methods of the spectral theory of differential operators are used, as well as the properties of the spectrum of a non-positive Hill operator with periodic coefficients. [ABSTRACT FROM AUTHOR]

The main goal of this article is to study the behavior of solutions of non-stationary problems at large timescales, namely, to obtain an asymptotic expansion characterizing the behavior of the solution of the Cauchy problem for a one-dimensional second-order hyperbolic equation with periodic coefficients at large values of the time parameter t. To obtain an asymptotic expansion as t → ∞ , the basic methods of the spectral theory of differential operators are used, as well as the properties of the spectrum of the Hill operator with periodic coefficients in the case when the operator is positive: H 0 > 0 . [ABSTRACT FROM AUTHOR]

In this paper, we consider the large time asymptotic behavior of solutions to systems of two cubic nonlinear Klein-Gordon equations in one space dimension. We classify the systems by studying the quotient set of a suitable subset of systems by the equivalence relation naturally induced by the linear transformation of the unknowns. It is revealed that the equivalence relation is well described by an identification with a matrix. In particular, we characterize some known systems in terms of the matrix and specify all systems equivalent to them. An explicit reduction procedure from a given system in the suitable subset to a model system, i.e., to a representative, is also established. The classification also draws our attention to some model systems which admit solutions with a new kind of asymptotic behavior. Especially, we find new systems which admit a solution of which decay rate is worse than that of a solution to the linear Klein-Gordon equation by logarithmic order. [ABSTRACT FROM AUTHOR]

We develop mathematical methods which allow us to study asymptotic properties of solutions to the three dimensional Navier-Stokes system for incompressible fluid in the whole three dimensional space. We deal either with the Cauchy problem or with the stationary problem where solutions may be singular due to singular external forces which are either singular finite measures or more general tempered distributions with bounded Fourier transforms. We present results on asymptotic properties of such solutions either for large values of the space variables (so called the far-field asymptotics) or for large values of time. [ABSTRACT FROM AUTHOR]

In this work, we consider a diffusive tumor-CD4 + + -cytokine interactions model with immunotherapy under homogeneous Neumann boundary conditions. We first investigate the large-time behavior of nonnegative equilibria, including the system persistence and the stability conditions. We also give the existence of nonconstant positive steady states (i.e., a stationary pattern), which indicate that this stationary pattern is driven by diffusion effects. For this study, we employ the comparison principle for parabolic systems, linearization method, the method of energy integral and the Leray-Schauder degree. [ABSTRACT FROM AUTHOR]

This article concerns the asymptotic behavior of solutions of nonautonomous Bresse systems. We establish the existence of pullback attractor and upper semicontinuity of attractors as a non-autonomous perturbations tend to zero. In addition we study the continuity of attractors with respect to a parameter in a residual dense set. [ABSTRACT FROM AUTHOR]

This paper deals with stability of nonlinear differential equations with parameter with periodic perturbation. We determine values of the parameter under which the solutions of the perturbed systems could be uniformly exponentially stable. Sufficient conditions for global uniform asymptotic stability and/or practical stability in terms of Lyapunov-like functions are obtained in the sense that the trajectories converge to a small ball centered at the origin. Moreover, to illustrate the applicability of our result, we study the stabilization problem for a class of control system. [ABSTRACT FROM AUTHOR]

In this note, we present some recent results in the study of the fractional mean curvature flow, that is a geometric evolution of the boundary of a set whose speed is given by the fractional mean curvature. The flow under consideration is of nonlocal type and presents several interesting difference with respect to the classical mean curvature flow. We will describe the main contributions in this field, with particular emphasis on some tipically nonlocal behaviors which are in contrast with the classical local case.

We consider a class of linear second order differential equations with damping and external force. We investigate the link between a uniform bound on the forcing term and the corresponding ultimate bound on the velocity of solutions, and we study the dependence of that bound on the damping and on the "elastic force". We prove three results. First of all, in a rather general setting we show that different notions of bound are actually equivalent. Then we compute the optimal constants in the scalar case. Finally, we extend the results of the scalar case to abstract dissipative wave-type equations in Hilbert spaces. In that setting we obtain rather sharp estimates that are quite different from the scalar case, in both finite and infinite dimensional frameworks. The abstract theory applies, in particular, to dissipative wave, plate and beam equations. [ABSTRACT FROM AUTHOR]

The chemotaxis‐consumption system with generalized logistic source ut=Δu−∇·(uS(x,u,v)·∇v)+λu−μuα,x∈Ω,t>0,vt=Δv−uv,x∈Ω,t>0,is considered under homogeneous Neumann boundary conditions in a bounded smooth domain Ω⊂ℝn(n≥1) with suitably regular positive initial data. Here λ, μ > 0, α > 1 and S∈C2(Ω¯×[0,∞)2;ℝn×n) is a given matrix‐valued function. We construct globally defined solutions in an appropriately generalized sense and prove that these solutions converge to the spatially homogeneous equilibrium λμ1α−1,0 in the large time limit. [ABSTRACT FROM AUTHOR]

We investigate the solvability of three-dimensional dynamical mixed boundary value problems of electro-magneto-elasticity theory for homogeneous anisotropic bodies with interior cracks. Using the Laplace transform technique, the potential method, and the theory of pseudodifferential equations, we prove the existence and uniqueness theorems and analyze asymptotic properties of solutions near the crack edges and near the lines where the different boundary conditions collide. [ABSTRACT FROM AUTHOR]

We consider in this article a nonlinear vibrating Timoshenko system with thermoelasticity with second sound. We first recall the results obtained in [2] concerning the well-posedness, the regularity of the solutions and the asymptotic behavior of the associated energy. Then, we use a fourth-order finite difference scheme to compute the numerical solutions and we prove its convergence. The energy decay in several cases, depending on the stability number , are numerically and theoretically studied. [ABSTRACT FROM AUTHOR]

We study the asymptotic behavior of solutions of a Fisher equation with free boundaries and the nonlocal term (an integral convolution in space). This problem can model the spreading of a biological or chemical species, where free boundaries represent the spreading fronts of the species. We give a dichotomy result, that is, the solution either converges to $1$ locally uniformly in $\mathbb{R}$, or to $0$ uniformly in the occupying domain. Moreover, we give the sharp threshold when the initial data $u_0=\sigma \phi$, that is, there exists $\sigma^*>0$ such that spreading happens when $\sigma>\sigma^*$, and vanishing happens when $\sigma\leq \sigma^*$.

The paper is devoted to studying the behavior of solutions of the Cauchy problem for large values of time—more precisely, obtaining an asymptotic expansion characterizing the behavior of the solution of the Cauchy problem for a one-dimensional second-order hyperbolic equation with periodic coefficients for large values of the time parameter t. To obtain this asymptotic expansion as t→∞, methods of the spectral theory of differential operators are used, as well as the properties of the spectrum of a non-positive Hill operator with periodic coefficients.

The main goal of this article is to study the behavior of solutions of non-stationary problems at large timescales, namely, to obtain an asymptotic expansion characterizing the behavior of the solution of the Cauchy problem for a one-dimensional second-order hyperbolic equation with periodic coefficients at large values of the time parameter t. To obtain an asymptotic expansion as t→∞, the basic methods of the spectral theory of differential operators are used, as well as the properties of the spectrum of the Hill operator with periodic coefficients in the case when the operator is positive: H0>0.

We consider a weakly coupled system of discounted Hamilton–Jacobi equations set on a closed Riemannian manifold. We prove that the corresponding solutions converge to a specific solution of the limit system as the discount factor goes to 0. The analysis is based on a generalization of the theory of Mather minimizing measures for Hamilton–Jacobi systems and on suitable random representation formulae for the discounted solutions. [ABSTRACT FROM AUTHOR]

We consider a mathematical model describing the production of the components of some living system under the influence of positive and negative feedback. The model is presented in the form of the Cauchy problem for a nonlinear delay integro-differential equation. A theorem of the existence, uniqueness, and nonnegativity of the solutions to the model on the half-axis is proved for nonnegative initial data. The questions of the asymptotic behavior of the solutions and the stability of the equilibria of the model are investigated. Sufficient conditions for the asymptotic stability are obtained for nontrivial equilibria and the boundaries of their attraction domains are estimated. Examples illustrating the application of the obtained theoretical results are given. [ABSTRACT FROM AUTHOR]

We recount and discuss some of the most important methods and blow-up criteria for analyzing solutions of Keller-Segel chemotaxis models. First, we discuss the results concerning the global existence, boundedness and blow-up of solutions to parabolic-elliptic type models. Thereafter we describe the global existence, boundedness and blow-up of solutions to parabolic-parabolic models. The numerical analysis of these models is still at a rather early stage only. We recollect quite a few of the known results on numerical methods also and direct the attention to a number of open problems in this domain. [ABSTRACT FROM AUTHOR]

In this paper, we study several aspects related with solutions of nonlocal problems whose prototype is u t = ∫ ℝ N J (x − y) (u (y , t) − u (x , t)) 𝒢 (u (y , t) − u (x , t)) d y in  Ω × (0 , T) , u (x , 0) = u 0 (x) in  Ω , where we take, as the most important instance, 𝒢 (s) ∼ 1 + μ 2 s 1 + μ 2 s 2 with μ ∈ ℝ as well as u 0 ∈ L 1 (Ω) , J is a smooth symmetric function with compact support and Ω is either a bounded smooth subset of ℝ N , with nonlocal Dirichlet boundary condition, or ℝ N itself. The results deal with existence, uniqueness, comparison principle and asymptotic behavior. Moreover, we prove that if the kernel is rescaled in a suitable way, the unique solution of the above problem converges to a solution of the deterministic Kardar–Parisi–Zhang equation. [ABSTRACT FROM AUTHOR]

The long time behavior of solutions to the autonomous PDE system describing fluid diffusion in a viscoelastic porous medium with capillary hysteresis is studied with homogeneous Dirichlet conditions for the displacement of the solid and homogeneous Neumann boundary conditions for the capillary pressure. Although the set of possible equilibria is very large, a detailed investigation of the hysteresis memory dynamics shows that all global solution trajectories converge to an equilibrium in the state space of all admissible memory configurations as time tends to infinity. [ABSTRACT FROM AUTHOR]

Nonlocal models provide exceptional simulation fidelity for a broad spectrum of scientific and engineering applications. However, wider deployment of nonlocal models is hindered by several modeling and numerical challenges. Among those, we focus on the nontrivial prescription of nonlocal boundary conditions, or volume constraints, that must be provided on a layer surrounding the domain where the nonlocal equations are posed. The challenge arises from the fact that, in general, data are provided on surfaces (as opposed to volumes) in the form of force or pressure data. In this paper we introduce an efficient, flexible, and physically consistent technique for an automatic conversion of surface (local) data into volumetric data that does not have any constraints on the geometry of the domain or on the regularity of the nonlocal solution and that is not tied to any discretization. We show that our formulation is well-posed and that the limit of the nonlocal solution, as the nonlocality vanishes, is the local solution corresponding to the available surface data. Quadratic convergence rates are proved for the strong energy and $L^2$ convergence. We illustrate the theory with one-dimensional numerical tests whose results provide the groundwork for realistic simulations. [ABSTRACT FROM AUTHOR]

In this paper, we study large time behavior of complex-valued solutions to nonlinear Klein-Gordon equation with a gauge invariant quadratic nonlinearity in two spatial dimensions. To find a possible asymptotic behavior, we consider the final value problem. It turns out that one possible behavior is a linear solution with a logarithmic phase correction as in the real-valued case. However, the shape of the logarithmic correction term has one more parameter which is also given by the final data. In the real case the parameter is constant so one cannot see its effect. However, in the complex case it varies in general. The one dimensional case is also discussed. [ABSTRACT FROM AUTHOR]

We explore, by using formal analysis, the existence of mass conserving self-similar solutions for Smoluchowski's coagulation equation when kernel K(x, y) = xλyµ + xµyλ with 0 < λ + µ < 1. [ABSTRACT FROM AUTHOR]

In this paper we study a class of nonlinear quasi-linear diffusion equations involving the fractional p (·) -Laplacian with variable exponents, which is a fractional version of the nonhomogeneous p (·) -Laplace operator. The paper is divided into two parts. In the first part, under suitable conditions on the nonlinearity f, we analyze the problem (P 1) in a bounded domain Ω of R N and we establish the well-posedness of solutions by using techniques of monotone operators. We also study the large-time behaviour and extinction of solutions and we prove that the fractional p (·) -Laplacian operator generates a (nonlinear) submarkovian semigroup on L 2 (Ω). In the second part of the paper we establish the existence of global attractors for problem (P 2) under certain conditions in the potential V. Our results are new in the literature, both for the case of variable exponents and for the fractional p-laplacian case with constant exponent. [ABSTRACT FROM AUTHOR]

In this work, we consider a prey‐predator model with herd behavior under Neumann boundary conditions. For the system without diffusion, we establish a sufficient condition to guarantee the local asymptotic stability of all nontrivial equilibria and prove the existence of limit cycle of our proposed model. For the system with diffusion, we consider the long time behavior of the model including global attractor and local stability, and the Hopf and steady‐state bifurcation analysis from the unique homogeneous positive steady state are carried out in detail. Furthermore, some numerical simulations to illustrate the theoretical analysis are performed to expand our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a predator–prey model with herd behavior and cross-diffusion subject to the zero flux boundary conditions. First, the temporal behavior of the model has been investigated, where Hopf bifurcation has been obtained. Then, by analyzing the characteristic equation it has been proved that the cross-diffusion generate a complex dynamics such as Hopf bifurcation, Turing instability, even Turing–Hopf bifurcation. Further, the impact of the prey herd shape on the spatiotemporal patterns has been discussed. Furthermore, by computing and analyzing the normal form associated with the Turing–Hopf bifurcation point, the spatiotemporal dynamics near the Turing–Hopf bifurcation point has been discussed and also justified by some numerical simulations. [ABSTRACT FROM AUTHOR]

In this work, we consider a one-dimensional porous thermoelastic system with memory effects. We prove a general decay result, for which exponential and polynomial decay results are special cases, depending only on the kernel of the memory effects. Our result is established irrespective of the wave speeds of the system. The result obtained is new and improves previous results in the literature. [ABSTRACT FROM AUTHOR]

After reviewing some of the recent results by S. Chakravarty and Y. Kodama and by Y. Kodama and L. Williams concerning the characterization of the asymptotic behavior of real regular and bounded multi–line soliton solutions to the KP-II equation, we introduce the finite–gap approach to the inverse spectral problem for this family of solutions.

A transmission problem for Kirchhoff‐type wave equations with memory source term on one part of the boundary feedback is considered. By using the Faedo‐Galerkin approximation technique, the method of Lyapunov functional and the energy perturbation technique, we establish well‐posedness of global solution and derive a general decay estimate of the energy. [ABSTRACT FROM AUTHOR]

In this paper, we study the multiplicity of solutions for a p–Kirchhoff type problem driven by a nonlocal integro-differential operator. As a particular case, we consider the following problem: where is the fractional p–Laplacian, with sp

Functional-differential equations ẋ = f (t, ϕ(·)) with piecewise continuous right-hand sides are studied. It is assumed that the set M of discontinuity points of the right-hand side has empty interior in contrast to being a measure zero sets, as in the case of differential equations without delay. This assumption is made largely because the domain of the function f is infinite-dimensional. Solutions to the equations under consideration are understood in A. F. Filippov's sense. The main results are theorems on the asymptotic behavior of solutions formulated with the use of invariantly differentiable Lyapunov functionals with constant-sign derivatives. Nonautonomous systems are difficult to deal with because ω-limit sets of their solutions do not possess invariance-type properties, whereas sets of zeros of derivatives of Lyapunov functionals may depend on the variable t and extend beyond the space of variables ϕ(·). For discontinuous nonautonomous systems, there arises the issue of constructing the limiting differential equations with the use of shifts fτ(t + τ, ϕ(·)) of the function f. We introduce the notion of limiting differential inclusion without employing limit passages on sequences of shifts of discontinuous or multivalued mappings. The properties of such inclusions are studied. Invariance-type properties of ω-limit sets of solutions and analogs of LaSalle's invariance principle are established. [ABSTRACT FROM AUTHOR]

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In this work we prove, by using the Faedo-Galerkin method, the existence of at least one solution for a class of p-Laplacian nonautonomous problems with dynamic boundary conditions. We also extend to the multivalued context results on the existence of 𝓓-pullback attractor - the pullback attractor for a determined basin of attraction 𝓓 - and we prove the existence of such attractors for the class of nonautonomous problems we are concerned about. [ABSTRACT FROM AUTHOR]