作者xavier13540 (柊 四千)
看板NTU-Exam
标题[试题] 105-2 夏俊雄 偏微分方程式二 期末考
时间Wed Apr 16 18:21:12 2025
课程名称︰偏微分方程式二
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）︰2017/06/13
试题 :
1. (20 points) Prove that there exists a constant C such that for any function
$f(t) \in C^1([0, 1])$, we have the following inequality
\[\sup_{0\le t\le1}|f(t)| \le C(\|f\|_{L^2(0, 1)}+\|f'\|_{L^2(0, 1)}).\]
2. (60 points) Let U be a smooth, bounded and connected domain in $\mathbb R^N$.
We denote $U_T = U \times (0, T]$. Let $f(x, t) \in L^2(0, T; L^2(U))$ and
$g(x) \in H^1_0(U)$. Prove that
(ⅰ) There exists a weak solution $u \in L^2(0, T; H^1_0(U))$ with $u' \in
L^2(0, T; H^{-1}(U))$ of
\[\begin{cases}
u_t - \Delta u = f(x, t) & \text{in }U_T\\
u = 0 & \text{on }\partial U \times [0, T]\\
u(x, 0) = g(x) & \text{on }U \times \{t = 0\}.
\end{cases}\]
(ⅱ) The solution in fact satisfies
\[
u \in L^2(0, T; H^2(U)) \cap L^\infty(0, T; H^1_0(U)),
u' \in L^2(0, T; L^2(U)).
\]
(ⅲ) If, in addition,
\[g \in H^2(U), f' \in L^2(0, T; L^2(U)),\]
then
\[
u \in L^\infty(0, T; H^2(U)),
u' \in L^\infty(0, T; L^2(U)) \cap L^2(0, T; H^1_0(U)),
u'' \in L^2(0, T; H^{-1}(U)).
\]
Remark: Do not forget to show that the weak solution satisfies the initial
condition.
3. (20 points) Suppose that u is a smooth solution of
\[\begin{cases}
u_t - \Delta u + cu = 0 & \text{in }U \times (0, \infty)\\
u = 0 & \text{on }\partial U \times [0, \infty)\\
u = g & \text{on }U \times \{t = 0\}
\end{cases}\]
and the function c satisfies
c ≧ γ > 0.
Here γ is a positive constant and U is a smooth, bounded and connected do-
main in $\mathbb R^N$. Prove that
\[|u(x, t)| \le Ce^{-\gamma t}\ ((x, t) \in U_T).\]
--
第01话 似乎在课堂上听过的样子 第02话 那真是太令人绝望了
第03话 已经没什麽好期望了 第04话 被当、21都是存在的
第05话 怎麽可能会all pass 第06话 这考卷绝对有问题啊
第07话 你能面对真正的分数吗 第08话 我，真是个笨蛋
第09话 这样成绩，教授绝不会让我过的 第10话 再也不依靠考古题
第11话 最後留下的补考 第12话 我最爱的学分
--
※ 发信站: 批踢踢实业坊(ptt.cc), 来自: 36.230.44.40 (台湾)
※ 文章网址: https://webptt.com/cn.aspx?n=bbs/NTU-Exam/M.1744798875.A.A47.html