空間離散型反應擴散方程的馬賽克解

Abstract

本篇論文主要研究空間離散型擴散方程的穩定花樣(stationary patterns)。先經由幾何形式來建構和描繪方程式中的參數條件，所求得方程的解稱為馬賽克解(mosaic solutions)，而利用這些解來描繪的花樣稱為馬賽克花樣(mosaic patterns)。藉由所建構的虛擬合適的基本花樣(pseudo feasible basic patterns)來討論一維和二維網格系統裡的花樣形式(pattern formations)和空間熵(spatial entropy)，也就是說由基本花樣來組成較大的花樣。在有限網格系統裡，討論三種邊界條件對空間熵和花樣的影響。最後，展示一些數值結果來驗證理論。

In this thesis, we study the stationary patterns for spatially discrete reaction diffusion equations. The so-called mosaic patterns and mosaic solutions are characterized and constructed through a geometrical formulation on the parameter conditions. We discuss pattern formations and spatial entropy for one and two dimensional lattices via establishing pseudo basic patterns and feasible basic patterns as well as combining these basic patterns into large patterns. For the systems on finite lattices, we also consider three kinds of boundary conditions and investigate their effects on patterns formations and spatial entropy. Several numerical computations are performed to illustrate our results.