課程名稱︰偏微分方程式二
課程性質︰數學系選修
課程教師︰夏俊雄
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2017/06/13
試題 :
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
\[|u(x, t)| \le Ce^{-\gamma t}\ ((x, t) \in U_T).\]