臺灣大學數學系

八十八學年度第一學期碩博士班資格考試試題

微分方程式

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* 以下七題選四題

1. 
   
   $$\mbox{Solve }\left\{ \begin{array}{l} u_x+(1+y)u_y=u^2\\ u(x,0)=h(x) \end{array}\right.\hspace{15cm}$$
   
   
2. Solve the initial-boundary-value problem
   
   
   $$\begin{array}{lcl} u_{tt}=u_{xx} &for& 0<x<\pi, 0<t,\\ u=0... ...i, t>0,\\ u=1 &for& 0<x<\pi, t=0.\\ \end{array}\hspace{7cm}$$
   
   
3. Suppose $u$ is harmonic on Ω, and the ball $\{x:\vert x-\xi \vert\leq\rho\} \subset\Omega$. Show that
   
   $$\vert Du(\xi )\vert^2\leq\frac1{\omega_n\rho^{n-1}}\int_{\vert x-\xi \vert=\rho} \vert Du(x)\vert^2dS_x,$$
   
   where $\omega_n\rho^{n-1}$ is the surface area of the sphere $\vert x-\xi \vert=\rho$.
4. For $x,y,t\in\mathbb{R}$, $t\ne 0$, define
   
   $$K(x,y,t)=(4\pi\vert t\vert)^{-\frac12}e^{-(x-y)^2/4t}$$
   
   Show that for $s>0,t>0$,
   
   $$K(x,0,s+t)=\int K(x,y,t)K(y,0,s)dy$$
   holds.
5. Assume $f(x)$ is bounded and continuous on $\mathbb{R}^n$ and satisfies $\int_{\mathbb{R}^n}\vert f(y)\vert dy<\infty$. Let $u$ be a bounded solution of
   
   $$\left\{\begin{array}{l} u_t=\triangle u\, for \, x\in\mathbb{R}^n ,t>0\\ u(x,0)=f(x).\\ \end{array}\right.\hspace{15cm}$$
   
   Show that $\lim{t}{\infty}u(x,t)=0$
6. Let $u\in C^2$ and
   
   $$\left\{\begin{array}{l} u_{tt}-\triangle u=0 \, for \, x\in\... ... u(x,0)=f(x), u_t(x,0)=g(x)\\ \end{array}\right.\hspace{15cm}$$
   
   Suppose $f$, $g$ have compact support (that is , $f(x)=0$ and $g(x)=0$ when $\vert x\vert$ is large). Show that there is a constant $C$ such that $\vert u(x,t)\vert\leq\frac{C}{t}$ for $x\in\mathbb{R}^3$, $t>0$ [Hint:
   
   $$u(x,t)=\frac{1}{4\pi t^2}\int_{\vert y-x\vert=t}[tg(y)+f(y)+\sum f_{y_i}(y)(y_i-x_i)]dS_y]$$
   
   
7. 
   
   $$\mbox{Let}\left(\begin{array}{c} w_1(x,y,t)\\ w_2(x,y,t)\\ ... ...u_2(x,y,t)\\ \end{array}\right)\mbox{satisfying}\hspace{15cm}$$
   
   
   $$\left\{\begin{array}{l} \left(\begin{array}{cc} 5 & 1\\ 1 &... ...\quad (x,y)\in \mathbb{R}^3\\ \end{array}\right.\hspace{15cm}$$
   
   Suppose $u$ has compact support in $(x,y)$ for $0\leq t\leq T$. Show that there is a constant $C_T$ such that
   
   $$\int_0^T\int_{\mathbb{R}^2}(u_1^2+u_2^2)dxdy\ dt\leq C_T\int_0^T\int_{\mathbb{R}^2}(w_1^2+w_2^2)dxdy\ dt.$$

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