PDE I - August 2024

PDE I MTH 847 QUALIFYING EXAM August 23, 2024

Name: ________ Signature: ________

Write clearly and coherently! Show all your work! This is a closed book exam, notes, electronic devices, cell phones, etc., can not be used. You can use a 4 × 6 index card. Read all problems through once before beginning your work. Problem 8 is optional for extra credit.

Problem Points: 1., 2., 3., 4., 5., 6., 7., 8.; Total:

Problem 1.

[10 points] Find the solution of the following initial value problem for the transport equation

$$u_t+b\cdot Du+cu=f\text{ in }\mathbb{R}^n\times(0,\infty),\quad u=g\text{ on }\mathbb{R}^n\times\{t=0\}.$$

Here $c\in\mathbb{R}$, $b\in\mathbb{R}^n$ are constants, $f(x,t)=e^{-2t}$, $g(x)=|x|^{2}$.

Problem 2.

[10 points] Let $B(0,1)=\{x\in\mathbb{R}^2:|x|<1\}$ be the open unit ball.

1. Find the solution of the boundary value problem

$$\begin{cases}\Delta u=0&\text{in }B(0,1),\\u(x)=2x_1^2+2x_1+x_2^2&\text{on }\partial B(0,1).\end{cases}$$

2. Find the value $u(0)$.

3. Find $\max\{u(x),x\in B(0,1)\}$ and $\min\{u(x),x\in B(0,1)\}$.

4. Is there a point $x\in B(0,1)$ such that $u(x)=0$?

Problem 3.

[10 points] Consider the initial value problem for the $n=3$ dimensional wave equation

$$u_{tt}-\Delta u=0,\text{ in }\mathbb{R}^3\times(0,\infty),\quad u=0,\ u_t=h\text{ on }\mathbb{R}^3\times\{t=0\}.$$

The function $h\in C_c^\infty(\mathbb{R}^3)$ satisfies the conditions that $0\le h(x)\le1$ for all $x\in\mathbb{R}^3$, $h(x)=1$ if $|x|\le1$ and $h(x)=0$ if $|x|\ge2$. Calculate the values of $u(x,t)$ at the points $(0,0,0,\frac14)$, $(0,0,0,4)$, $(0,7,0,5)$, and the value of $u_t(x,t)$ at $(0,0,0,1/4)$.

Problem 4.

[10 points] Solve the mixed initial-boundary value problem for the wave equation in one dimension

$$\begin{cases}u_{tt}=u_{xx},&x>0,\ t>0,\\u(x,0)=g(x),\ u_t(x,0)=0,&x>0,\\u_t(0,t)=2u_x(0,t),&t>0,\end{cases}$$

where the function $g\in C^2(\mathbb{R}_+)$ vanishes near $x=0$.

Problem 5.

[10 points] Let $U$ be a bounded open set in $\mathbb{R}^n$ with $C^1$ boundary. The time $T>0$ is given Let $U_T=U\times(0,T]$, and $\Gamma_T=\overline{U_T}-U_T$. Consider the boundary value problem

$$\begin{cases}u_t-\Delta u+u^{5}=0&\text{in }U_T,\\u=0&\text{on }\Gamma_T.\end{cases}$$

Let $u\in C^{2,1}(U_T)$ be a solution of this boundary value problem. Prove that $u\equiv0$ in $U_T$.

Problem 6.

[10 points] Let $U$ be a bounded open set in $\mathbb{R}^2$. Suppose $u\in C^2(U)$ satisfies

$$au_{xx}+bu_{yy}+cu_x+du_y-(1+e^2)u=0$$

in $U$ with positive constants $a>0$, $b>0$.

1. Show that $u$ cannot have a positive maximum or a negative minimum in the interior of $U$.

2. Use this to show that if $u\in C^2(U)\cap C(\overline U)$ is a solution of the boundary value problem

$$au_{xx}+bu_{yy}+cu_x+du_y-(1+e^2)u=0\text{ in }U,\quad u=0\text{ on }\partial U,$$

then $u=0$.

Problem 7.

[10 points] Let $f\in C(\mathbb{R}^n\times[0,\infty))$ and $g\in C(\mathbb{R}^n)$ be bounded functions.

Prove that there is a bounded solution $u$ of the nonhomogeneous heat equation

$$u_t-\Delta u=f\text{ in }\mathbb{R}^n\times(0,\infty),\quad u(x,0)=g(x)\text{ on }\mathbb{R}^n$$

if $|f(x,t)|\le\frac1{1+t^4}$ for all $(x,t)\in\mathbb{R}^n\times[0,\infty)$. Use Duhamel’s principle.

Problem 8.

[10 points] Problem 8 is optional for extra credit.

Use the method of characteristics to solve the first order equation

$$(u_{x_1})^2+2x_2^2u_{x_2}=2,\quad(x_1,x_2)\in\mathbb{R}^2$$

with the condition

$$u(x_1,1)=x_1^2,\quad x_1\in\mathbb{R}.$$