PDE I - January 2026

PDE I MTH 847 QUALIFYING EXAM January 9, 2026

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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 7 and 8 are optional for extra credit.

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

Problem 1.

[10 points] (a) 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, $f(x,t)=e^t$, $g(x)=|x|^2$.

2. Let $f\in C(\mathbb{R}^n\times(0,\infty))$ and $g\in C^1(\mathbb{R}^n)\cap L^\infty(\mathbb{R}^n)$. Assume that $\|f(\cdot,t)\|_\infty=\sup\{f(x,t):x\in\mathbb{R}^n\}<\infty$ for all $t>0$. Prove that

$$|u(x,t)|\le e^{-ct}\|g\|_\infty+\int_0^t e^{c(s-t)}\|f(\cdot,s)\|_\infty\,ds$$

for all $(x,t)\in\mathbb{R}^n\times(0,\infty)$.

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

$$\begin{cases}\Delta u=0&\text{in }B(0,1),\\u(x)=8x_1^2-8x_1+4x_2^2&\text{on }\partial B(0,1).\end{cases}$$

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] (a) Let $a$ be a number, $0<a<1$. Consider the set

$$A=\{(x,t)\in\mathbb{R}^n\times(0,\infty):|x|\ge1,\ 0<t<a\}.$$

Prove that the wave equation

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

has a solution $u\in C^2(\mathbb{R}^n\times(0,\infty))$ that is not identically zero, but vanishes on the set $A$.

2. 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 $(0,0,0,\frac12)$, $(0,0,0,3)$, $(10,0,0,8)$, and the values of $u_t(x,t)$, $u_x(x,t)$ at $(0,0,0,\frac12)$.

Problem 4.

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

$$\begin{cases}u_{tt}=u_{xx},&x>0,\ t>0,\\u(x,0)=g(x),\ u_t(x,0)=0,&x>0,\\u_t(0,t)-8u_x(0,t)=0,&t>0,\end{cases}$$

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 $\Gamma_T=\overline{U_T}-U_T$. Let $q\in C(\overline U)$ be a nonnegative function and $f\in C(\overline{U_T})$. Consider the boundary value problem

$$\begin{cases}u_t-\Delta u+(1+q(x))u=f&\text{in }U_T,\\\frac{\partial u}{\partial\nu}=0&\text{on }\partial U\times(0,T),\\u=0&\text{on }U\times\{t=0\}.\end{cases}$$

Prove that a solution $u\in C^{2,1}(\overline{U_T})$ of this boundary value problem satisfies the energy inequality

$$\int_U|u(x,t)|^2\,dx\le\int_0^t\int_U e^{s-t}|f(x,s)|^2\,dx\,ds$$

for all $0<t<T$.

Problem 6.

[10 points] Use the method of characteristics to solve the first order equation

$$(u_{x_1})^2+2x_2^2u_{x_2}=1,\quad(x_1,x_2)\in\mathbb{R}^2$$

with the condition

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

Problem 7.

[10 points] Problem 7 is optional for extra credit.

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

1. Prove that there is a solution of the nonhomogeneous heat equation

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

satisfying the estimate

$$|u(x,t)|\le\sup\{|g(y)|:y\in\mathbb{R}^n\}+t\sup\{|f(y,s)|:y\in\mathbb{R}^n,\ 0\le s\le t\}$$

using Duhamel’s principle.

2. Prove that there is a bounded solution $u$ if $|f(x,t)|\le\frac1{1+t^2}$ for all $(x,t)\in\mathbb{R}^n\times[0,\infty)$.

Problem 8.

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

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

$$a\Delta u+b\cdot Du-cu=0$$

in $U$ with positive constants $a>0$, $c>0$, and $b\in\mathbb{R}^n$.

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

$$a\Delta u+b\cdot Du-cu=0\text{ in }U,\quad u=0\text{ on }\partial U,$$

then $u=0$.