PDE I - January 2022

Math 847 — Qualifying Examination — January 2022

Instructions: You must show all necessary work to get full or partial credits. You can use a 3 × 5 index card. You can not use your book, cell phone, computer, or other notes. Read all problems through once carefully before beginning work.

Notation: ${ℝ}^n$ denotes the standard Euclidean space with $\left|x\right|=\sqrt{x_1^2+\cdots +x_n^2}$ for $x=(x_1,\dots ,x_n)\in {ℝ}^n$. $\Delta u\left(x\right)={\sum }_{j=1}^n\frac{{\partial }^{2}u}{\partial x_j\partial x_j}$ stands for the Laplace operator in ${ℝ}^n$. $\Omega$ is used for any open, bounded, and smooth domain in ${ℝ}^n$ with $\partial \Omega$ as its boundary, and $\nu \left(x\right)$ is the unit out normal at $x\in \partial \Omega$. ${\omega }_n$ is the surface area for $S^{n-1}=\{x\in {ℝ}^n\mid \left|x\right|=1\}$.

Problem 1

Use the method of characteristics to find a solution $u(x,t)$ to the Burger’s equation

$$\{\begin{array}{cc}{u}_t+u{u}_x=0,\hfill & x\in ℝ,t\ge 0,\hfill \\ u(x,0)=x.\hfill \end{array}$$

near the initial surface $t=0$. Verify your answer by a direct differentiation.

Problem 2

Let $u$ be a non-constant harmonic function in ${ℝ}^n$ (i.e., $\Delta u=0$). Let $B\left(r\right)$ be the solid ball of radius $r$ centered at the origin in ${ℝ}^n$ and $\Omega \left(r\right)$ be the boundary of $B\left(r\right)$. That is,

$$B\left(r\right)=\{x\in {ℝ}^n\mid \left|x\right|\le r\},\Omega \left(r\right)=\{x\in {ℝ}^n\mid \left|x\right|=r\}.$$

1. Prove that $g_1\left(r\right)={\int }_{\Omega \left(1\right)}u\left(ry\right)d{S}_y$ is independent of $r$ for $r\ge 0$.

2. Prove that $g_{\infty }\left(r\right)={sup}_{y\in \Omega \left(1\right)}\left|u\left(ry\right)\right|$ is a strictly increasing function of $r$ for $r\ge 0$, and ${lim}_{r\to \infty }{g}_{\infty }\left(r\right)=\infty$.

3. Prove that $g_2\left(r\right)={\int }_{\Omega \left(1\right)}{\left|u\right|}^2\left(ry\right)d{S}_y$ is a strictly increasing function of $r$ for $r\ge 0$.

4. Can you find a number $r_0>0$ such that $\frac{\partial u\left(x\right)}{\partial \nu }=0$ for all $x\in \Omega \left(r_0\right)$? Briefly explain your answer.

Problem 3

Consider the following initial value problem for the heat equation in ${ℝ}^1\times [0,\infty )$

$$u_t(x,t)=u_{xx}(x,t),u(x,0)=\frac{1+{sin}^{2}x}{1+x^2}.$$

1. Prove that there is one and only one solution such that $0<u<1$ for all $x\in ℝ$ and $t>0$. For this particular solution, prove that $h\left(t\right)={\int }_{ℝ}u(x,t)dx\equiv c_0>0$. Don’t try to find the explicit formula for $u(x,t)$!

2. Is it possible to find another solution $u(x,t)$ to the given initial value problem such that $u(0,2)=-1$? Explain your answer.

3. Prove that the solution $u$ in (a) satisfies the estimate $u(x,t)\ge v(x,t)$ for all $x\in [0,1]$ and $t\ge 0$, where $v(x,t)$ is the solution of the following initial and boundary value problem

$$v_t(x,t)=v_{xx}(x,t),v(x,0)=\frac{4x(1-x)}{1+x^2},v(0,t)=v(1,t)=0.$$

Problem 4

Consider the initial value problem for the wave equation in ${ℝ}^3\times ℝ$

$$\{\begin{array}{cc}{u}_{tt}=u_{xx}+u_{yy}+u_{zz},\hfill & (x,y,z)\in {ℝ}^3,t\in ℝ,\hfill \\ u(x,y,z,0)=0,\hfill & (x,y,z)\in {ℝ}^3,\hfill \\ u_t(x,y,z,0)=\frac{1}{{(1+x^2+y^2+z^2)}^2},\hfill & (x,y,z)\in {ℝ}^3.\hfill \end{array}$$

1. If $u$ is a solution to the initial value problem, find an explicit formula for $u(0,0,0,t)$.

2. Find a solution of the form $u(x,y,z,t)=V(r,t)$ with $r=\sqrt{x^2+y^2+z^2}$.

   [ Hint: Note that $W(r,t)=rV(r,t)$ satisfies $W_{rr}=W_{tt}$. ]

3. Check that $V(0,t)=u(0,0,0,t)$ you obtained from Parts (a) and (b).

4. Can you find a solution $u(x,t)$ such that $u(-1,0,0,2)-u(1,0,0,2)=1$? Explain your answer.

Problem 5

Let $B(0,r)$ be the closed ball in ${ℝ}^n$ ($n\ge 2$) of radius $r$ centered at the origin. Assume that $u\left(x\right)$ is a harmonic function on $B(0,r)$.

1. What is the Poisson’s formula for $u\left(x\right)$?

2. Use this Poisson’s formula for balls to prove

   $$r^{n-2}\frac{r-\left|x\right|}{{(r+\left|x\right|)}^{n-1}}u\left(0\right)\le u\left(x\right)\le r^{n-2}\frac{r+\left|x\right|}{{(r-\left|x\right|)}^{n-1}}u\left(0\right),\left|x\right|<r$$

   if $u$ is nonnegative on $B(0,r)$

3. Show that any harmonic function on whole ${ℝ}^n$ that is bounded above (or bounded below) must be a constant. (Note that it is different form the Liouville’s theorem.)

Problem 6

Consider the following so-called sine-Gordon equation, which appears in differential geometry and relativistic field theory, with initial and boundary data

$$\{\begin{array}{cc}{u}_{tt}=\Delta u-sinu,\hfill & x\in \Omega ,t>0,\hfill \\ u(x,0)=g\left(x\right),\hfill & x\in \Omega ,\hfill \\ u_t(x,0)=h\left(x\right),\hfill & x\in \Omega ,\hfill \\ u(x,t)=0,\hfill & x\in \partial \Omega ,t>0.\hfill \end{array}$$

1. Set $E\left(t\right)={\int }_{\Omega }[u_t^2+{\left|Du\right|}^2+u^2]dx$. Prove that $E\prime \left(t\right)\le 2E\left(t\right)$, $t\ge 0$.

2. Prove that the solution to the original problem is unique.