Asymptotic Analysis

In this paper, we study the following generalized quasilinear Schrödinger equation − div ( ε 2 g 2 ( u ) ∇ u ) + ε 2 g ( u ) g ′ ( u ) | ∇ u | 2 + V ( x ) u = K ( x ) | u | p − 2 u + | u | 22 ∗ − 2 u , x ∈ R N , where N ⩾ 3 , ε > 0 , 4 < p < 22 ∗ , g ∈ C 1 ( R , R + ) , V ∈ C ( R N ) ∩ L ∞ ( R N ) has a positive global minimum, and K ∈ C ( R N ) ∩ L ∞ ( R N ) has a positive global maximum. By using a change of variable, we obtain the existence and concentration behavior of ground state solutions for this problem with critical growth, and establish a phenomenon of exponential decay. Moreover, by Ljusternik–Schnirelmann theory, we also prove the existence of multiple solutions.

This work focuses on the topological sensitivity analysis of a three-dimensional parabolic type problem. The considered application model is described by the heat equation. We derive a new topological asymptotic expansion valid for various shape functions and geometric perturbations of arbitrary form. The used approach is based on a rigorous mathematical framework describing and analyzing the asymptotic behavior of the perturbed temperature field.

In this paper, we first study the cone Moser–Trudinger inequalities and their best exponents on both bounded and unbounded domains R + 2 . Then, using the cone Moser–Trudinger inequalities, we study the asymptotic behavior of Cerami sequences and the existence of weak solutions to the nonlinear equation − Δ B u = f ( x , u ) , in  x ∈ int ( B ) , u = 0 , on  ∂ B , where Δ B is an elliptic operator with conical degeneration on the boundary x 1 = 0 , and the nonlinear term f has the subcritical exponential growth or the critical exponential growth.

This paper is concerned with smooth solutions of the non-isentropic Euler–Poisson system for ion dynamics. The system arises in the modeling of semi-conductor, in which appear one small parameter, the momentum relaxation time. When the initial data are near constant equilibrium states, with the help of uniform energy estimates and compactness arguments, we rigorously prove the convergence of the system for all time, as the relaxation time goes to zero. The limit system is the drift-diffusion system.

This paper studies the wave-breaking mechanism and dynamical behavior of solutions near the explicit self-similar singularity for the two component Camassa–Holm equations, which can be regarded as a model for shallow water dynamics and arising from the approximation of the Hamiltonian for Euler’s equation in the shallow water regime.

We consider the Landau Hamiltonian H 0 , self-adjoint in L 2 ( R 2 ) , whose spectrum consists of an arithmetic progression of infinitely degenerate positive eigenvalues Λ q , q ∈ Z + . We perturb H 0 by a non-local potential written as a bounded pseudo-differential operator Op w ( V ) with real-valued Weyl symbol V , such that Op w ( V ) H 0 − 1 is compact. We study the spectral properties of the perturbed operator H V = H 0 + Op w ( V ) . First, we construct symbols V , possessing a suitable symmetry, such that the operator H V admits an explicit eigenbasis in L 2 ( R 2 ) , and calculate the corresponding eigenvalues. Moreover, for V which are not supposed to have this symmetry, we study the asymptotic distribution of the eigenvalues of H V adjoining any given Λ q . We find that the effective Hamiltonian in this context is the Toeplitz operator T q ( V ) = p q Op w ( V ) p q , where p q is the orthogonal projection onto Ker ( H 0 − Λ q I ) , and investigate its spectral asymptotics.

In this paper, we prove the global well-posedness of the 2-D magnetic Prandtl model in the mixed Prandtl/Hartmann regime when the initial data is a small perturbation of the Hartmann layer in Sobolev space.