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）︰2017/01/03 试题 : 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 $\Gamma = \{x_1 > 0, x_2 = 0\}$. \[\begin{cases} uu_{x_1} + u_{x_2} = 1,\\ u(x_1, x_1) = \frac12 x_1. \end{cases}\] -- 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 --

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