asymptotic expansion of solutions
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2019 ◽  
Vol 149 (5) ◽  
pp. 1371-1404
Author(s):  
Soohyun Bae

AbstractWe consider the infinite multiplicity of entire solutions for the elliptic equation Δu + K(x)eu + μf(x) = 0 in ℝn, n ⩾ 3. Under suitable conditions on K and f, the equation with small μ ⩾ 0 possesses a continuum of entire solutions with a specific asymptotic behaviour. Typically, K behaves like |x|ℓ at ∞ for some ℓ > −2 and the entire solutions behave asymptotically like − (2 + ℓ)log |x| near ∞. Main tools of the analysis are comparison principle for separation structure, asymptotic expansion of solutions near ∞, barrier method and strong maximum principle. The linearized operator for the equation has two characteristic behaviours related with the stability and the weak asymptotic stability of the solutions as steady states for the corresponding parabolic equation.





2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Weishi Yin ◽  
Fei Xu ◽  
Weipeng Zhang ◽  
Yixian Gao

This paper is devoted to finding the asymptotic expansion of solutions to fractional partial differential equations with initial conditions. A new method, the residual power series method, is proposed for time-space fractional partial differential equations, where the fractional integral and derivative are described in the sense of Riemann-Liouville integral and Caputo derivative. We apply the method to the linear and nonlinear time-space fractional Kuramoto-Sivashinsky equation with initial value and obtain asymptotic expansion of the solutions, which demonstrates the accuracy and efficiency of the method.



2016 ◽  
Vol 2016 ◽  
pp. 1-18
Author(s):  
Le Thi Phuong Ngoc ◽  
Le Huu Ky Son ◽  
Tran Minh Thuyet ◽  
Nguyen Thanh Long

This paper is devoted to the study of a nonlinear Carrier wave equation in an annular membrane associated with Robin-Dirichlet conditions. Existence and uniqueness of a weak solution are proved by using the linearization method for nonlinear terms combined with the Faedo-Galerkin method and the weak compact method. Furthermore, an asymptotic expansion of a weak solution of high order in a small parameter is established.



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