課程名稱︰偏微分方程式一
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）︰2017/01/03
試題 :
1. Suppose Ω = (-π, π) and $\Omega_T = \Omega \times (0, T]$. The parabolic
boundary of $\Omega_T$ is defined as $\partial\Omega_T = \overline{\Omega_T}
\backslash \Omega_T$. Now, we consider the following equation
\begin{equation}\label{eq1}
u_t = u_{xx}
\end{equation}
supplemented with the initial-boundary condition
\begin{equation}\label{eq2}
\begin{cases}
u(x, 0) = x(\pi-x),\text{if }x \ge 0; x(\pi+x),\text{if }x < 0,\\
u(\pm\pi, t) = 0.
\end{cases}
\end{equation}
(A) (20%) Now, you solve the equations (\ref{eq1})-(\ref{eq2}) by the follo-
wing Fourier series scheme: Set
\begin{equation}\label{eq3}
u(x, t) = \sum_{n=1}^\infty a_n(t) \sin nx,
\end{equation}
and evaluate each $a_n(t)$ and show the convergence of (\ref{eq3}) with
the $a_n(t)$ you obtain.
(B) (20%) Show that the solution you obtained in (A) satisfies the initial
condition.
2. (20%) For fixed $x \in \mathbb R^N, T \in \mathbb R, r > 0$, we define the
heat ball
\[E(x, t; r) := \left\{
(y, s) \in \mathbb R^{N+1}:
s \le t, [4\pi(t-s)]^{\frac N2} \exp\frac{|x-y|^2}{4(t-s)} \le r^N
\right\}.\]
Suppose the differentiable function u(x, t) satisfies the heat equation on
some $\Omega_T \subset \mathbb R^{N+1}$ and $E(x, t; \rho) \subset \Omega_T$.
We define
\[f(r) = \frac1{4r^N}\iint_{E(x, t; r)} u(y, s)\frac{|x-y|^2}{(t-s)^2}dyds.\]
Show that f'(r) = 0 for 0 < r < ρ.
3. (A) (10%) State the maximum principle for the Cauchy problem
\[\begin{cases}
u_t(x, t) = \Delta u(x, t)\text{ in }\mathbb R^N \times (0, T),\\
u(x, 0) = g(x),
\end{cases}\]
where $g(x) \in L^\infty(\mathbb R^N) \cap C(\mathbb R^N)$.
(B) (20%) Prove the maximum principle for the Cauchy problem by using the
maximum principle for the bounded domain.
4. (80%) Solve the following differential equations:
\[\begin{cases}
uu_x + 2u_y = 1,\\
u(x, x) = \frac12 x.
\end{cases}\]
\[\begin{cases}
u_{tt} - 4u_{xx} = 0,\\
u_t(x, 0) = x,\\
u(x, 0) = e^x.
\end{cases}\]
\[\begin{cases}
x_1 u_{x_2} - x_2 u_{x_1} = u \text{ in }\Omega,\\
u = g \text{ on }\Gamma,
\end{cases}\]
where Ω is the quadrant ${x_1 > 0, x_2 > 0}$ and $\Gamma = \{x_1 > 0, x_2
= 0\}$.
\[\begin{cases}
uu_{x_1} + u_{x_2} = 1,\\
u(x_1, x_1) = \frac12 x_1.
\end{cases}\]