课程名称︰偏微分方程式一
课程性质︰数学系选修
课程教师︰夏俊雄
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2016/12/01
考试时限（分钟）：50
试题 :
1. (50 points) Suppose $f(x, t) \in C^2_1(\mathbb R^N \times (0, T])$ has a com-
pact support in $\mathbb R^N \times (0, T]$. Show that the function
\[u(x, t) := \int_0^t\int_{\mathbb R^N} \phi(x-y, t-s) f(y, s) dyds\]
solves the nonhomogeneous heat equation
\[u_t = \Delta u + f(x, t),\]
where φ is the heat kernel.
2. (50 points) Consider the following one dimensional wave equation
\[\begin{cases}
u_{tt}-u_{xx} = 0, & (x, t) \in (0, 2) \times (0, 100),\\
u_t(x, 0) = x^3(x-2)^3, & x \in [0, 2],\\
u(x, 0) = x^5(x-2)^3, & x \in [0, 2],\\
u(0, t) = u(2, t) = 0, & t \in [0, 100].
\end{cases}\]
Use the idea of extension to compute u(1/3, 1) and u(5/3, 2).
3. (20 points) What is the most exciting common property that the boundaries of
one-dimensional, two-dimensional and three-dimensional manifolds enjoy?
(Don't think it in a mathematical way. Not even try to answer this question
by English.)