Search Results for « asymptotic behavior »

This paper focuses on the asymptotic behavior of the eigenvalues of the classical Steklov problem in a thin multidomain as it becomes increasingly narrow. The thin multidomain consists of two vertical cylinders, one positioned atop the other. We show that the eigenvalues tend to zero as the domain becomes thinner. In this case, the limiting eigenvalue problem in the upper cylinder corresponds to a one-dimensional eigenvalue problem, while the limiting problem in the lower cylinder is defined in an ( n − 1 ) -dimensional ball. We conclude our investigation by considering the case in which the thinness of the upper and lower cylinders is equal.

In this article, we are devoted to studying the following nonlocal Navier–Stokes equations involving the time–space fractional operators { ∂ t β u + ( − Δ ) α u + u ⋅ ∇ u = ∇ p + λ | u | q − 2 u + g ( x , t ) in Ω × R + , div u = 0 in Ω × R + , u ( x , t ) = 0 in ( R N ∖ Ω ) × R + , u ( x , 0 ) = u 0 ( x ) in Ω , where Ω ⊂ R N is a bounded domain with Lipschitz boundary, ( − Δ ) α is the fractional Laplacian with 0 < α < 1 , ∂ t β is the Riemann–Liouville time fractional derivative with 0 < β < 1 , λ is a parameter, and g ∈ ( L ∞ ( 0 , ∞ ; L 2 ( Ω ) ) ) N . Due to the nonlocal nature of the fractional Laplacian, the classical weak solution frameworks proposed by J. L. Lions and E. Hopf are inapplicable to this problem. To address this challenge, we first construct a regularized problem and establish the existence of its solutions via the Galerkin method combined with fractional calculus techniques. Subsequently, we prove the uniqueness of the global weak solution to the regularized problem. Furthermore, under appropriate assumptions, a decay estimate characterizing the long-time behavior of solutions is derived. Finally, the existence of solutions to the original problem is obtained by analyzing the limiting behavior of the regularized solutions as the regularization parameter vanishes. The main contributions of this work are twofold: (1) Our problem simultaneously incorporates the Riemann–Liouville time fractional derivative and the fractional Laplacian, extending classical Navier–Stokes theory to a nonlocal time–space framework. (2) We develop a new analytical approach to estimate the nonlinear convection term u ⋅ ∇ u in the presence of fractional operators, overcoming challenges posed by the loss of localization.

In this paper the asymptotic behavior of elliptic nonlocal equations in cylindrical domains is studied. These equations are associated with symmetric pure jump Lévy stochastic processes and are generalisation of equations involving the fractional Laplacian operator. First we consider the existence and uniqueness of weak solutions in appropriate Hilbert spaces associated with the Lévy measure and then study the asymptotic behavior in cylinders.

In this note, we elaborate on the asymptotic behavior of the spectral gap of a class of discrete Schrödinger operators defined on a path graph in the limit of infinite volume. We confirm recent results and generalize them to a larger class of potentials using entirely different methods. Notably, we also resolve a conjecture previously proposed in this context. This then yields new insights into the rate at which the spectral gap tends to zero as the volume increases.

We consider a vectorial monotone problem posed on a cylinder ${\varOmega}^\varepsilon=\omega\times (0,\varepsilon L)$ of $\mathbb{R}^N$ and we study its limit behavior as $\varepsilon$ goes to $0$ . We show that the limit solution solves an elliptic system posed on $\omega$ . We give also a corrector result as well as an error estimate under some regularity hypotheses.

We establish pointwise asymptotic estimates at infinity for solutions to the doubly weighted quasi-linear equation: { − div ( w 1 | ∇ u | p − 2 ∇ u ) = w 2 | u | q − 2 u in Ω , u ∈ D 1 , p , w 1 ( Ω ) , where w 1 and w 2 are compatible weights and q > p > 1 is a critical exponent in the sense of Sobolev.

A variational characterization for the shift of eigenvalues caused by a general type of perturbation is derived for second order self-adjoint elliptic differential operators. This result allows the direct extension of asymptotic formulae from simple eigenvalues to repeated ones. Some examples of particular interest are presented theoretically and numerically for the Laplacian operator for the following domain perturbations: excision of a small hole, local change of conductivity, small boundary deformation.