此篇論文主要探討定義在整個R^n空間上的熱方程解的基本性質，尤其我們著重於探討一維熱方程解的基本性質。因為一維的熱方程比n維的熱方程較容易處理，且具備一些特有的性質。對於n維的熱方程(n > 1)，除了探討解的基本性質外，我們也探討在R^n 有界區域中的散度定理(divergence theorem) 和格林表現公式(Green representation formula)。最後，我們探討二階線性拋物方程在R^n有界區域中的弱極大值原理。

This thesis explores several important fundamental properties of solutions to the heat equation defined on the entire space R^n, with or without initial data. In particular, we focus on the one-dimensional heat equation, which is easier to manage and has some special interesting properties not shared by the case for general n > 1.
For general n > 1, in addition to some basic fundamental properties, we also discuss the divergence theorem and Green's representation formula for the heat equation on bounded domains of R^n.
Finally, we discuss the weak maximum principle for general linear second-order parabolic equations on bounded domains of R^n.

1 The physical motivation for parabolic equation.................................. 1
2 The heat equation on R^n; no initial data........................................2
3 The heat equation on R^n with initial data (solution is not unique !!!)........ 23
4 The heat equation on R with initial data (n = 1)................................42
5 The divergence theorem for heat equation; Green's representation formula....... 82
6 Weak maximum principle for general second-order parabolic linear equations..... 87

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