PDE I - August 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: $\mathbb{R}^n$ denotes the standard Euclidean space with $|x|=\sqrt{x_1^2+\ldots+x_n^2}$ for $x=(x_1,\ldots,x_n)\in\mathbb{R}^n$. $\Delta u(x)=\sum_{j=1}^{n}\frac{\partial^2u}{\partial x_j\partial x_j}$ stands for the Laplace operator in $\mathbb{R}^n$. $\Omega$ is used for any open, bounded, and smooth domain in $\mathbb{R}^n$ with $\partial\Omega$ as its boundary, and $\nu(x)$ is the unit out normal at $x\in\partial\Omega$. $\omega_n$ is the surface area for $S^{n-1}=\{x\in\mathbb{R}^n\mid |x|=1\}$.

Problem 1.

Consider the first order equation $u_t(x,t)+2tu_x(x,t)=u$, $x\in\mathbb{R}$, $t\in\mathbb{R}$.

1. Find a solution $u(x,t)$ with initial data $u(x,0)=\sin x$, $x\in\mathbb{R}$.

2. What is the characteristic curve starting from $x_0=0$, $t_0=0$ ? Can you find a $C^1$ solution in a neighborhood of $(0,0)$ such that $u(s^2,s)=\cos s$? Explain your answer.

Problem 2.

Let $\Omega=\{(x,y)\in\mathbb{R}^2\mid\|(x,y)\|=\sqrt{x^2+y^2}<1\}$. Assume that $u$ is a smooth function on $\Omega$ such that

$$\begin{cases}\Delta u=0,&(x,y)\in\Omega,\\u(x,y)=2x^2-y^2+6y,&x^2+y^2=1.\end{cases}$$

1. What is $u(0,0)$? What theorem did you use?

2. What is the maximum and minimum values of $u$ on $\overline\Omega$? What theorem did you use?

3. Find a point $(x_0,y_0)$ on $\partial\Omega$ such that $\frac{\partial u}{\partial\nu}(x_0,y_0)>0$? Briefly explain your answer, where $\nu$ is the unit outnormal at the point $(x_0,y_0)$.

Problem 3.

Let $g(x)$ be a continuous function on $\mathbb{R}^n$, $0<g(x)\le1$ for all $x\in\mathbb{R}^n$, and $\int_{\mathbb{R}^n}g(x)\,dx=1$. Consider the initial value problem

$$\begin{cases}u_t(x,t)=\Delta u(x,t)-u,&x\in\mathbb{R}^n,\ t>0,\\u(x,0)=g(x),&x\in\mathbb{R}^n.\end{cases}$$

1. Prove that there is a solution $u(x,t)$ such that $0<u(x,t)<e^{-t}\min\{1,\frac1{(4\pi t)^{n/2}}\}$ for all $x\in\mathbb{R}^n$ and $t>0$.

2. Can you find another different solution to the initial value problem? Briefly explain your answer.

3. Can you find another different positive solution $u(x,t)$ to the initial value problem? Briefly explain your answer.

4. For any continuous function $f(x,t)$ such that $|f(x,t)|\le5$, $x\in\mathbb{R}^n$, $t\ge0$, prove that there is a bounded solution to the following non-homogeneous equation

$$\begin{cases}u_t=\Delta u-u+f(x,t),&x\in\mathbb{R}^n,\ t>0,\\u(x,0)=g(x),&x\in\mathbb{R}^n.\end{cases}$$

Problem 4.

Let $u(x,y,z,t)$ be a solution of the initial value problem for the wave equation in $\mathbb{R}^3\times\mathbb{R}$

$$\begin{cases}u_{tt}=u_{xx}+u_{yy}+u_{zz},&(x,y,z)\in\mathbb{R}^3,\ t\in\mathbb{R},\\u(x,y,z,0)=0,&(x,y,z)\in\mathbb{R}^3,\\u_t(x,y,z,0)=\frac1{(4+x^2+y^2+z^2)^2},&(x,y,z)\in\mathbb{R}^3.\end{cases}$$

1. Find an explicit formula for $u(0,0,0,t)$. Briefly explain your answer.

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. Is it possible that $u(x,t)$ such that $u(5,0,0,2)-u(0,3,4,2)=2$? Explain your answer.

5. Prove that there is a constant $A>0$ such that

$$F_1(t)=\int_{\mathbb{R}^3}u(x,t)\,dx=At.$$

Problem 5.

Consider the second order differential equation

$$\Delta u(x)=f(x_1,x_2,\ldots,x_n)=\frac{3+\cos x_1+\sin(x_1+x_2+\ldots x_n)}{(1+|x|^2)^n},\quad x\in\mathbb{R}^n.\qquad(1)$$

Note that $\frac1{(1+|x|^2)^n}\le f(x)\le\frac5{(1+|x|^2)^n}$.

1. Prove that there are infinitely many solutions to this equation with $u(0)=1$ when $n\ge3$.

2. Prove that there is one and only one solution $u(x)$ such that $u(x)$ is bounded and $u(0)=1$ when $n\ge3$.

3. Let $g(r)=\int_{|y|=1}u(ry)\,dS_y$. Find $g'(r)$ in terms of the integral of $f(x)$.

4. Prove that $g(r)$ is an increasing and bounded function for any solution $u(x)$ of (1) when $n\ge3$ (even for unbounded solutions of (1) .

5. For $n=2$, prove that there is no bounded solution to (1) by showing that $\lim_{r\to\infty}g(r)=\infty$.

Problem 6.

Consider the nonlinear heat equation

$$\begin{cases}u_t=u_{xx}+u_{yy}-|Du|^2,&x^2+y^2<1,\ t>0,\\u(x,y,0)=(1-x^2-y^2)^2,&x^2+y^2<1,\\u(x,y,t)+xu_x(x,y,t)+yu_y(x,y,t)=u+Du\cdot\nu=0,&x^2+y^2=1,\ t\ge0.\end{cases}$$

1. If $u(x,t)$ is a smooth solution of the equation, prove that $u$ can not attain its positive maximum or negative minimum on the lateral boundary $x^2+y^2=1$.

2. If $u(x,t)$ is a smooth solution of the equation, prove that $0\le u(x,y,t)\le1$ for all $x^2+y^2<1$ and $t>0$.

3. Prove that the equation has at most one solution.