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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: \overline{\Omega}\times ℝ \to ℝ is a continuous function. We establish there main results with various assumptions, the first one asserts that any \lambda0> is an eigenvalue of our problem. The second Theorem states the existence of a constant \lambda^{*} such that every \lambda \in (0,\lambda^{*}) is an eigenvalue of the problem. While the third Theorem claims the existence of a constant \lambda^{**} such that every \lambda \in [\lambda^{**},\infty) 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^{\prime }\left( t\right) \leq -\xi \left( t\right) g^{p} % \left( t\right) , t\geq 0, 1\leq p\leq \frac{3}{2}. This work generalizes and improves earlier results in the literature.
In this paper, we consider the following inhomogeneous nonlinear Schrödinger equation (INLS)
i\partial_t u + \Delta u + \mu |x|^{-b}|u|^\alpha u = 0, \quad (t,x)\in ℝ \times ℝ^d
with b, \alpha > 0. First, we revisit the local well-posedness in H^1(ℝ^d) 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. \mu=-1 when 0 < \alpha < \alpha^\star where \alpha^\star = \frac{4-2b}{d-2} for d\geq 3, and \alpha^\star = \infty for d=1, 2
by assuming that the initial data belongs to the weighted L^2 space \Sigma =\{u \in H^1(ℝ^d) : |x| u \in L^2(ℝ^d) \}. Finally, we combine the local theory and the decaying property to show the scattering in \Sigma for the defocusing (INLS) in the case \alpha_\star < \alpha < \alpha^\star, where \alpha_\star = \frac{4-2b}{d}.
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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