Asymptotic Analysis

We first study the existence, uniqueness and well-posedness of a general class of integro-differential diffusion equation on L p ( G ) ( 1 < p < + ∞ , G is a graded Lie group). We show the explicit solution of the considered equation. The equation involves a nonlocal in time operator (with a general kernel) and a positive Rockland operator acting on G . Also, we provide L p ( G ) − L q ( G ) ( 1 < p ⩽ 2 ⩽ q < + ∞ ) norm estimates and time decay rate for the solutions. In fact, by using some contemporary results, one can translate the latter regularity problem to the study of boundedness of its propagator which strongly depends on the traces of the spectral projections of the Rockland operator. Moreover, in many cases, we can obtain time asymptotic decay for the solutions which depends intrinsically on the considered kernel. As a complement, we give some norm estimates for the solutions in terms of a homogeneous Sobolev space in L 2 ( G ) that involves the Rockland operator. We also give a counterpart of our results in the setting of compact Lie groups. Illustrative examples are also given.

In this article we study the long-time dynamics of the dynamical system generated by the Ericksen–Leslie model. More precisely, we discretize the Ericksen–Leslie equations in time using the implicit Euler scheme, and with the aid of the discrete Gronwall lemmas we prove that the scheme is uniformly bounded. Moreover, using the theory of the multi-valued attractors we prove in a particular case the convergence of the global attractors generated by the numerical scheme to the global attractor of the continuous system as the time-step approaches zero.

In this article we continue our research in (Yang and Estrada in Asymptot. Anal. 95(1–2) (2015) 1–19), about the asymptotic expansion of thick distributions. We compute more examples of asymptotic expansion of integral transforms using the techniques developed in (Yang and Estrada in Asymptot. Anal. 95(1–2) (2015) 1–19). Besides, we derive a new “Laplace Formula” for the situation in which a point singularity is allowed.

This article focuses on the existence and non-existence of solutions for the following system of local and nonlocal type { − ∂ x x u + ( − Δ ) y s 1 u + u − u 2 s 1 − 1 = κ α h ( x , y ) u α − 1 v β in R 2 , − ∂ x x v + ( − Δ ) y s 2 v + v − v 2 s 2 − 1 = κ β h ( x , y ) u α v β − 1 in R 2 , u , v ⩾ 0 in R 2 , where s 1 , s 2 ∈ ( 0 , 1 ) , α , β > 1 , α + β ⩽ min { 2 s 1 , 2 s 2 } , and 2 s i = 2 ( 1 + s i ) 1 − s i , i = 1 , 2 . The existence of a ground state solution entirely depends on the behaviour of the parameter κ > 0 and on the function h. In this article, we prove that a ground state solution exists in the subcritical case if κ is large enough and h satisfies (H). Further, if κ becomes very small, then there is no solution to our system. The study of the critical case, i.e., s 1 = s 2 = s , α + β = 2 s , is more complex, and the solution exists only for large κ and radial h satisfying (H1). Finally, we establish a Pohozaev identity which enables us to prove the non-existence results under some smooth assumptions on h.

The wave equation with stochastic rapidly oscillating coefficients can be classically homogenized on bounded time intervals; solutions converge in the homogenization limit to solutions of a wave equation with constant coefficients. This is no longer true on large time scales: Even in the periodic case with periodicity ε , classical homogenization fails for times of the order ε − 2 . We consider the one-dimensional wave equation with random rapidly oscillation coefficients on scale ε and are interested in the critical time scale ε − β from where on classical homogenization fails. In the general setting, we derive upper and lower bounds for β in terms of the growth rate of correctors. In the specific setting of i.i.d. coefficients with matched impedance, we show that the critical time scale is ε − 1 .