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This paper is devoted to the study of the nonhomogeneous problem
−div(a(|∇u|)∇u)+a(|u|)u=λV(x)|u|m(x)−2u−μg(x,u) in Ω, u=0 on ∂Ω, where Ω is a bounded smooth domain in RN,λ,μ are positive real numbers, V(x) is a potential, m:¯Ω→(1,∞) is a continuous function, a is mapping such that φ(|t|)t is increasing homeomorphism from ℝ to ℝ and g:¯Ω×ℝ→ℝ is a continuous function. We establish there main results with various assumptions, the first one asserts that any λ0> is an eigenvalue of our problem. The second Theorem states the existence of a constant λ∗ such that every λ∈(0,λ∗) is an eigenvalue of the problem. While the third Theorem claims the existence of a constant λ∗∗ such that every λ∈[λ∗∗,∞) is an eigenvalue of the problem. Our approach relies on adequate variational methods in Orlicz-Sobolev spaces.
In this work we establish a general decay rate for a nonlinear viscoelastic wave equation with boundary dissipation where the relaxation function satisfies g′(t)≤−ξ(t)gp(t),t≥0,1≤p≤32. This work generalizes and improves earlier results in the literature.
In this paper, we consider the following inhomogeneous nonlinear Schrödinger equation (INLS)
i∂tu+Δu+μ |x|−b|u|αu=0,(t,x)∈ℝ×ℝd
with b,α > 0. First, we revisit the local well-posedness in H1(ℝd) for (INLS) of Guzmán [Nonlinear Anal. Real World Appl. 37 (2017), 249-286] and give an improvement of this result in the two and three spatial dimensional cases. Second, we study the decay of global solutions for the defocusing (INLS), i.e. μ=−1 when 0 < α < α⋆ where α⋆=4−2bd−2 for d≥3, and α⋆=∞ for d=1,2
by assuming that the initial data belongs to the weighted L2 space Σ={u∈H1(ℝd): |x|u∈L2(ℝd)}. Finally, we combine the local theory and the decaying property to show the scattering in Σ for the defocusing (INLS) in the case α⋆ < α < α⋆, where α⋆=4−2bd.
This article concerns the discrete-time problem of the damped Schrödinger equation in a bounded domain and the internal stabilization problem is considered. We constructed a well-posed discrete-time scheme and we proved the exponential stability. Numerical simulations are presented to illustrate the theoretical results.
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