台灣大學數學系

九十一學年度第二學期博士班資格考試題

偏微分方程

 

May 10, 2003

 

[回上頁]

Choose 4 problems from below.

1. Solve the equations:
(a) $u_x+yu_y-u_z=-u, u(x,y,0)=x+y.$
(b) $u_xu_y=u, u(x,0)=x^2.$

2. Let $u\in C^2(R\times [0,\infty))$ solve

$$u_{tt}-u_{xx}=0, u(x,0)=f(x), u_t(x,0)=g(x).$$

Suppose $f,g$ have compact support. Show that
(a) $\int_{-\infty}^{\infty}\{u_t^2+u_x^2\}\,dx$ is a constant in $t$,
(b) $\int_{-\infty}^{\infty}\{u_t^2-u_x^2\}\,dx=0$ for large $t$.

3. Let $u$ solve $u_t+6uu_x+u_{xxx}=0$    for $x\in R,t>0.$ Suppose $u$ has the form $u(x,t)=v(x-ct)$ for some constant $c$ with $v(s),v'(s),$$v''(s)$ $\rightarrow 0$ as $s\rightarrow \infty$ or $s\rightarrow -\infty$. Find an explicit formula for $u$.

4. Let $u$ be a $C^2$ solution of $\triangle u=0$ in $\Omega$ and $\{x:\vert x-x_o\vert\leq\rho\}\subset \Omega$. Show that
(a) $u(x_o)=\frac1{\omega_n\rho^{n-1}}\int_{\vert x-x_o\vert=\rho} u(x)\,dS_x,$ where $\omega_n\rho^{n-1}$ is the surface area of the sphere $\vert x-x_o\vert=\rho$.
(b) $\vert Du(x_o)\vert^2\leq\frac1{\omega_n\rho^{n-1}}\int_{\vert x-x_o\vert=\rho} \vert Du(x)\vert^2\,dS_x.$

5. Suppose $f(x)$ is bounded and continuous in $R^n$ which satisfies $\int_{R^n}\vert f(x)\vert\,dx$$<\infty$. Let $u$ be a bounded solution of

$$u_t=\triangle u$$    for $$x\in R^n,t>0; u(x,0)=f(x).$$

Show that $lim_{t\to\infty}u(x,t)=0$.

6. Let $u$ be a $C^1$ solution of $u_t+uu_x=0$ in each of two regions separated by a smooth curve $x=\xi(t)$. Suppose $u$ is continuous, but $u_x$ has a jump discontinuity on the curve. Prove that

$$\frac{d\xi}{dt}=u(\xi(t),t).$$

[回上頁]