Zai-yun Zhang

New exact solutions to the perturbed nonlinear Schrödinger’s equation with Kerr law nonlinearity

In this paper, by using the modified mapping method and the extended mapping method, we derive some new exact solutions of the perturbed nonlinear Schrodinger’s equation with Kerr law nonlinearity, which are the linear combination of two different Jacobi elliptic functions and we also consider the solutions in the limit cases.

Global existence and uniform stabilization of a generalized dissipative Klein–Gordon equation type with boundary damping

In this paper, we prove the existence, uniqueness, and uniform stability of strong and weak solutions of the nonlinear generalized Klein-Gordon equation (1.1)_1 (see Sec. I) in bounded domains with nonlinear damped boundary conditions given by (1.1)_3 (see Sec. I) with some restrictions on function f(u), h(∇u), g(ut), and b(x), we prove the existence and uniqueness by means of nonlinear semigroup method and obtain the uniform stabilization by using the multiplier technique.

Global existence and general decay for a nonlinear viscoelastic equation with nonlinear localized damping and velocity-dependent material density

In this article, we investigate a nonlinear viscoelastic equation with nonlinear localized damping and velocity-dependent material density. We prove the global existence of weak solutions and general decay of the energy by using the Faedo–Galerkin method [Z.Y. Zhang and X.J. Miao, Global existence and uniform decay for wave equation with dissipative term and boundary damping, Comput. Math. Appl. 59 (2010), pp. 1003–1018; J.Y. Park and J.R. Kang, Global existence and uniform decay for a nonlinear viscoelastic equation with damping, Acta Appl. Math. 110 (2010), pp. 1393–1406] and the perturbed energy method [Zhang and Miao (2010); X.S. Han, and M.X. Wang, Global existence and uniform decay for a nonlinear viscoelastic equation with damping, Nonlinear Anal. TMA. 70 (2009), pp. 3090–3098], respectively. Furthermore, for certain initial data and suitable conditions on the relaxation function, we show that the energy decays exponentially or polynomially depending the rate of the decay of the relaxation function. This result is an improvement over the earlier ones in the literature.

Well-posedness and unique continuation property for the solutions to the generalized Kawahara equation below the energy space

Abstract In this paper, we investigate the initial value problem (IVP henceforth) associated with the generalized Kawahara equation [Z.Y. Zhang, J.H. Huang, Z.H. Liu and M.B. Sun, On the unique continuation property for the modified Kawahara equation, Adv Math (China).45(2016),pp.80–88] as follows: with initial data in the Sobolev space Benefited from ideas of [Z.Y. Zhang and J.H. Huang, Well-posedness and unique continuation property for the generalized Ostrovsky equation with low regularity, Math Meth Appl Sci. 39(2016),pp.2488–2513; Z.Y. Zhang, J.H. Huang, Z.H. Liu and M.B. Sun, Almost conservation laws and global rough solutions of the defocusing nonlinear wave equation on ; Acta Math Sci.37(2017),pp.385C39], first, we show that the local well-posedness is established for the initial data with ( ) and ( ) respectively. Then,using these results and conservation laws, we also prove that the IVP is globally well-posed for the initial data with ( ). Finally, benefited from ideas of [Z.Y. Zhang and J.H. Huang, Well-posedness and unique continuation property for the generalized Ostrovsky equation with low regularity, Math Meth Appl Sci. 39(2016),pp.2488–2513; Z.Y. Zhang, J.H. Huang, Z.H. Liu and M.B. Sun, On the unique continuation property for the modified Kawahara equation,Adv Math (China).45(2016),pp.80–88], i.e. using complex variables technique and Paley–Wiener theorem, we prove the unique continuation property (UCP henceforth) for the IVP.

Global Attractor for the Generalized Dissipative KDV Equation with Nonlinearity

We discuss global attractor for the generalized dissipative KDV equation with nonlinearity under the initial condition . We prove existence of a global attractor in space , by using decomposition method with cut-off function and Kuratowski-measure in order to overcome the noncompactness of the classical Sobolev embedding.