An efficient spectral-collocation difference method for two-dimensional Schrödinger equation with Neumann boundary conditions

2020 ◽  
Vol 79 (8) ◽  
pp. 2322-2335
Author(s):  
Xin Li ◽  
Luming Zhang
Keyword(s):  
Schrödinger Equation ◽  
Boundary Conditions ◽  
Difference Method ◽  
Schrodinger Equation ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Two Dimensional ◽  
Spectral Collocation

Immediate smoothing and global solutions for initial data in L1 × W1,2 in a Keller–Segel system with logistic terms in 2D

2020 ◽  
pp. 1-21
Author(s):  
Johannes Lankeit
Keyword(s):  
Initial Data ◽  
Boundary Conditions ◽  
Global Solutions ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Two Dimensional ◽  
Image Position

This paper deals with the logistic Keller–Segel model \[ \begin{cases} u_t = \Delta u - \chi \nabla\cdot(u\nabla v) + \kappa u - \mu u^2, \\ v_t = \Delta v - v + u \end{cases} \] in bounded two-dimensional domains (with homogeneous Neumann boundary conditions and for parameters χ, κ ∈ ℝ and μ > 0), and shows that any nonnegative initial data (u0, v0) ∈ L1 × W1,2 lead to global solutions that are smooth in $\bar {\Omega }\times (0,\infty )$ .

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