台灣大學數學系

九十二學年度第二學期博士班資格考試題

偏微分方程

May 8, 2004

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1.

(20%) Consider the following initial value problem

$$\begin{cases} u_{tt}-\sum_{i,j=1}^3a^{ij}u_{x_ix_j}=0,\qua... ...athbb{R}^3\\ u(x,0)=0,\quad u_t(x,0)=\phi(x), \end{cases}$$

where the matrix $A=(a^{ij})_{i,j=1}^3$ is a positive-definite symmetric matrix with eigenvalues $\lambda_1\leq\lambda_2\leq\lambda_3$.

(a)

Write down the explicit formula of $u(x,t)$ (in integral form).

(b)

Assume supp$\ \phi\subset\{x:\vert x\vert\leq\alpha\}$. For $t>0$, estimate the support of $u(\cdot,t)$.

2.

(20%) Consider the wave equation

$$u_{tt}-\Delta u=0\quad(x,t)\in(0,\infty)\times \mathbb{R}^n.$$

Let $B(x_0,t_0)$ be the ball of radius $t_0>0$ centered at $x_0$ and $C:=\{(x,t):0\leq t\leq t_0,\ \vert x-x_0\vert\leq t_0-t\}$. Using the energy estimate without referring to the explicit formula of the solution to show that if $u(x,t)=u_t(x,t)\equiv 0$ in $B(x_0,t_0)$ then $u\equiv 0$ in $C$. Argue that the speed of propagation is finite. Can the strong maximum principle hold for the wave equation?

3.

(20%) Solve the Cauchy problem: $u_y=u_x^3$ with $u(x,0)=2x^{3/2}$.

4.

(30%) (a) Let $U:=\{u(x)\in C_0^{\infty}(\mathbb{R}^n):$   supp$(u)\subset B(0,r_0)\$   and$\ \lim_{r\to 0}e^{r^{-s}}u=0\ \text{for every}\ s>0\}$, where $r=\vert x\vert$ and $r_0<1$. Derive the following estimate: there exists a constant $C>0$ such that for all $u\in U$ and sufficiently large $s$

$$s^4\int_{\mathbb{R}^n} r^{-2s-2}e^{2r^{-s}}\vert u\vert^2dx\leq c\int_{R^n}r^{s+2}e^{2r^{-s}}\vert\Delta u\vert^2dx.$$

(Hint: set $v=e^{r^{-s}}u$ and use the integration by parts).
(b) Let $u\in H_{loc}^2(\mathbb{R}^n)$ satisfy the differential inequality

$$\vert\Delta u\vert\leq M\vert u\vert$$   for$$\ x\in\Omega,$$

where $M>0$ and $\Omega$ is an open neighborhood of 0. Assume that

$$\lim_{r\to 0}e^{r^{-s}}u=0\$$   for every$$\ s>0.$$

Show that $u$ vanishes near 0. (Hint: use (a) on $\theta u$, where $\theta$ is a suitable cut-off function with $\theta(x)=1$ for $\vert x\vert\leq r_0/2$ and $\theta(x)=0$ for $\vert x\vert\geq r_0$).

5.

(10%) Derive the explicit formula of the solution $u(x,t)$ to

$$\begin{cases} u_{tt}-u_{xx}=p(x,t)\quad x>0,\ t>0\\ u(x... ...g(x)\quad x\geq 0\\ u(0,t)=h(t)\quad t\geq 0, \end{cases}$$

where $f(0)=h(0)$ and $g(0)=h'(0)$. (Hint: divide the first quadrant by the line $x=t$).

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