Asymptotic Analysis

The article is about an elliptic problem defined on a stratified Lie group. Both sub and superlinear cases are considered whose solutions are guaranteed to exist in light of the interplay between the nonlinearities and the weak L 1 datum. The existence of infinitely many solutions is proved for suitable values of λ , p , q by using the Symmetric Mountain Pass Theorem.

In the present paper, we study the existence and concentration of multiple normalized solutions to the following nonlinear biharmonic Schrödinger equation: { ε 4 Δ 2 u + V ( x ) u = λ u + h ( u ) , x ∈ R N , ∫ R N | u | 2 d x = c 2 ε N , x ∈ R N , where ε > 0 is a positive parameter, λ ∈ R is unknown and appears as a Lagrange multiplier, and V is a positive potential such that inf Λ V < inf ∂ Λ V for some open bounded subset Λ ⊂ R N ( N ⩾ 5 ) . Applying the penalization techniques and Ljusternik–Schnirelmann theory, we obtain multiple mormalized solutions u ε . When ε → 0 , these solutions concentrates around a local minimum of V.

This paper extends the results of Alves and Thin (2023), which considered the nonlinear Schrödinger equations with general nonlinearities, to the biharmonic Schrödinger equations. We develop a truncated skill to obtain the minimum via careful analysis. Moreover, we also obtain orbital stability of the solutions.

In this paper, we consider the following mixed local and nonlocal hyperbolic equation: { u t t − Δ u + μ ( − Δ ) s u = | u | p − 2 u , in Ω × R + , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , in Ω , u ( x , t ) = 0 , in ( R N ∖ Ω ) × R 0 + , where s ∈ ( 0 , 1 ) , N > 2 , p ∈ ( 2 , 2 s * ] , μ is a nonnegative real parameter, Ω ⊂ R N is a bounded domain with Lipschitz boundary ∂ Ω , Δ is the Laplace operator, ( − Δ ) s is the fractional Laplace operator. By combining the Galerkin approach with the modified potential well method, we obtain the global existence, vacuum isolating, and blow-up of solutions for the aforementioned problem, provided certain assumptions are fulfilled. Specifically, we study the existence of global solutions for the above problem in the cases of subcritical and critical initial energy levels, as well as the finite time blow-up of solutions. Then, we investigate the blow-up of solutions for the above problem in the case of supercritical initial energy level, as well as upper and lower bounds of blow-up time of solutions.

This paper is concerned with the well-posedness and long-time dynamics of a class of beam/plate equations with rotational inertia and nonlinear energy damping. The model is derived from nonlocal dissipative energy models for flight structures, as proposed by Balakrishnan-Taylor (Proceedings Damping 89, Flight Dynamics Lab and Air Force Wright Aeronautical Labs, WPAFB, 1989). Our main results address the existence of compact global attractors. The work complements the degenerate coefficient case left open by Sun and Yang (J. Math. Anal. Appl., Volume 512, Issue 2, 2022).

In this paper, we embark on a captivating exploration of the stabilization of locally transmitted problems within the realm of two interconnected wave systems. To begin, we wield the formidable Arendt-Batty criteria (Trans. Am. Math. Soc. 306(2) (1988) 837–852) to affirm the resolute stability of our system. Then, with an artful fusion of a frequency domain approach and the multiplier method, we unveil the exquisite phenomenon of exponential stability, a phenomenon that manifests when the waves of the second system synchronize their propagation speeds. In cases where these speeds diverge, our investigation reveals a graceful decay of our system’s energy, elegantly characterized by a polynomial decline at a rate of t − 1 .