在這篇論文裡，我們主要討論的是半線性拋物型方程及其相關離散化問題的特殊解。

在第一個部分，我們討論的是一個微分方程系統，其中的非線性項具有雙穩定性質，且其介質具有離散性和週期性。我們研究的是其對應旅行波解的存在性、唯一性和穩定性。而這樣的一個問題是從一個具有週期性質的半線性拋物型方程問題，將其空間變數離散化而來。我們主要是利用比較原理、算子分析和造出上、下解來得到旅行波解的唯一性和穩定性；並將微分方程轉換成積分方程來得到存在性。

在第二個部分，我們研究的是一個具有初值及邊界條件，且具有強吸收之非線性項的熱方程問題。我們已經知道在某個初始條件之下，這個解在有限時間會產生殆核解，而這個產生殆核解的速度會比一般所謂的自我相似解的速度還快。在一個符合動態原理的假設條件下，利用結合內部與外部展開式的方法，我們在形式上可以得到殆核解的速度。另外，在所對應的柯西問題上，我們造了特殊解使其具有特定的速度且滿足我們動態原理的假設。

In this thesis, we study special solutions of some
semilinear parabolic equations and their discrete analogue.

In the first part, we study the existence, uniqueness, and stability of traveling waves for a system of ordinary differential equations with bistable nonlinearity in discrete periodic media. This system arises from a spatial discrete version of some semilinear parabolic equations with periodic nonlinearity. The main tools to derive the uniqueness and asymptotic stability are comparison principle, spectrum analysis of the linearized operator
around a steady state, and the construction of suitable
super/subsolutions. To derive the existence of traveling wave, we first convert the system to an integral equation.
Then we establish the existence traveling wave for this system of ordinary differential equations.

In the second part, we study the solution of initial
boundary value problem for the heat equation with a strong absorption term. It is well-known that the solution develops a dead-core in finite time for a large class of initial data. It is also known that the exact dead-core rate is faster than the corresponding self-similar rate. By using the idea of matching, we formally derive the exact dead-core rates under a dynamical theory assumption. Moreover, we also construct some special solutions for
the corresponding Cauchy problem satisfying this dynamical theory assumption. These solutions provide some examples with certain given polynomial rates.

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