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\begin{center}
{\sc Fall 2017 \\ [1ex] Partial Differential Equation Comprehensive Exam} 
\end{center}


\vspace{0.1in}
 {\sl Do any six problems. Clearly indicate in the table below which problems you want to be graded. \indent If you do not select any problems we will grade the first 6 problems. Good luck!}
\vspace{.1in}

\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
Problems& 1 & 2& 3& 4 & 5& 6& 7& 8\\
\hline
 & & & & & & & &  \\
 \hline
\end{tabular}

\vspace{0.2in}
\begin{enumerate}

\item Let $\Omega \subset \R^{2}$ denote the positive quadrant, i.e.
$$
\Omega := \left\{ \x = (x,y) \in \R^{2} : x>0, \, y>0 \right\}.
$$
Use the method of reflection to find the Green's function $G(\x,\y)$ for $\Omega,$ and use it to state a solution to the Dirichlet problem
$$
\Delta u(\x) = f(\x) \;\; \x \in \Omega \qquad \text{and} \qquad u(\x) = g(\x) \;\; \x \in \partial \Omega
$$
for Poisson's equation.

\item Let $\Omega \subset \mathbb{R}^{n}$ denote a bounded, open domain, and assume that $u \in C^{2}(\Omega)$ is twice differentiable. Show that $u(\mathbf{x})$ is harmonic in $\Omega$ if and only if $u(\x)$ satisfies the mean value property. You may use the identities
$$
\int_{B_{r}(\x)} u(\z) \, \rd \z = \int^{r}_{0} \left( \int_{\partial B_{s}(\x) } u(\z) \, \rd \sigma_{\z} \right) \; \rd s, \quad |\partial B_{r}(\x)| = r^{n-1} |\partial B_{1}(\0)|, \quad  |B_{r}(\x)| = \frac{r^n}{n} |\partial B_{1}(\0)|
$$
$$
\int_{\partial B_{r}(\x)} u(\z) \, \rd \sigma_{\z} = r^{n-1} \int_{\partial B_{1}(\0)} u(\x + r \y) \, \rd \sigma_{\y}
$$
without proof in your argument.

\item Let $u(x,t)$ denote the solution to the initial value problem
\begin{align*}
u_{tt} &= u_{xx} \\
u(x,0) &= f(x) \\
u_{t}(x,0) &= g(x),
\end{align*}
where the initial conditions $f \in C^{2}(\mathbb{R})$ and $g \in C^{1}(\mathbb{R})$ have compact support. That is, there exists an $R > 0$ so that $f(x) = 0$ and $g(x) = 0$ whenever $|x| \geq R$. Define $k(t)$ and $p(t)$ as
$$
k(t) := \int_{\mathbb{R}} u^{2}_{t} (x,t) \, \rd x \qquad p(t) := \int_{\mathbb{R}} u^{2}_{x}(x,t) \, \rd x.
$$
\begin{enumerate}
\item
Show the total energy $E(t) = k(t) + p(t)$ is constant in time.
\item
Show that $k(t) = p(t)$ whenever $t \geq 2R$.
\end{enumerate}

\item Let $\Omega \subset \R^{n}$ denotes a bounded, connected domain with smooth boundary. Use Green's identity and the energy method to show that $u(\x,t) = 0$ is the unique solution to the following parabolic PDE with bi-harmonic diffusion:
\begin{align*}
u_{t} &= -\Delta( \Delta u ) \qquad \, \x \in \Omega, \; t > 0, \\
\Delta u(\x,t) &= 0 \quad \qquad \qquad \;\, \x \in \partial \Omega, \; t > 0, \\
u(\x,t)  &= 0 \quad \qquad \qquad \;\, \x \in \partial \Omega, \; t > 0, \\
u(\x,0) &= 0 \quad \qquad \qquad \;\, \x \in \Omega, \; t = 0.
\end{align*}

%\item Let
%$$
%K(\x,\y) := \frac{ r^2 - |\x|^2}{ r|\partial B_{1}(\0)||\x-\y|^{n}}
%$$
%denote the Poisson kernel $K(\x,\y)$ for the ball $B_{r}(\0) \subset \R^{n}$ of radius $r$ centered at the origin. 
%\begin{enumerate}
%\item Use the Poisson kernel to prove Harnack's inequality --- if $u(\x) \geq 0$ and $u(\x)$ is harmonic in the ball $B_{r}(\0) $ then
%$$
%\frac{ r^{n-2}(r - |\x|)}{ (r + |\x|)^{n-1} } u(\0) \leq u(\x) \leq \frac{ r^{n-2}(r + |\x|) }{ (r - |\x|)^{n-1}}u(\0)
%$$
%for any $\x \in B_{r}(\0)$.
%\item Use Harnack's inequality to prove Liouville's theorem --- if $\Delta u(\x) = 0$ for all $\x \in \R^{n}$ and $u(\x) \leq M$ for all $\x \in \R^{n}$ then $u(\x) = u(\0)$ for all $\x \in \R^{n}$.
%\end{enumerate}

\item
Suppose that $u \in C^{2,1}(\Omega_T) \cap C( \overline{\Omega}_T )$ solves the heat equation
\begin{align}\label{eq:par}
u_{t}(\x,t) &= \Delta u(\x,t) \quad \, \text{in} \quad \Omega_{T} := \{ (\x,t) : \x \in \Omega , \; 0 < t < T\}\\
u(\x,t) &= h(\x,t) \quad \quad \x \in \partial \Omega, \; t > 0 \nonumber\\
u(\x,0) &= g(\x) \quad \quad \;\;\;\x \in \Omega, \; t=0 \nonumber
\end{align}
in $\Omega \subset \R^n$ a bounded domain with smooth boundary. Let
$$
\partial_{ {\mathrm{p}} } \Omega_{T} := \{ (\x,t) \in \overline{\Omega}_T : \x \in \partial \Omega \;\; \text{or} \;\; t = 0 \}
$$
denote the parabolic boundary.
\begin{enumerate}[(a)]
\item State and prove the weak maximum principle for the heat equation.
\item Let $\Omega = B_{1}(\0)$ denote the unit ball in $\R^{2}$ centered at the origin. Show the solution to \eqref{eq:par} satisfies the inequality
$$
\mathrm{e}^{-8t}\left(1 - |\x|^2\right)^{2} \leq u(\x,t) \leq \mathrm{e}^{-4t}(1-|\x|^2)  
$$
if $ g(\x) = 1-|\x|^2$ and $h(\x,t) = 0$. You may use the identities $\Delta |\x|^2 = 4$ and $\Delta |\x|^4 = 16|\x|^2$ (valid in two dimensions) without proof.
\end{enumerate}

%\item 
%Let $u(x,t)$ be the smooth solution to the heat equation 
%$$
%u_{t}(x,t) = u_{xx}(x,t), x\in(0,1), t>0, \quad u(x,0)= x(1-x), \quad u(0,t)=u(1,t)=0.
%$$
%Show that the solution $u(x,t)$ satisfies the inequalities
%$$
%0 < u(x,t) < \mathrm{e}^{-t}x(1-x) \qquad \text{if} \qquad 0<x<1,\;\,t>0.
%$$

%\item Consider the wave equation in the first quadrant $x>0,t>0$
%\begin{align*}
%u_{tt}  &= u_{xx}, \quad \;\,  0<x<\infty, \ t >0,\\
%u(x,0) &=  f(x), \quad 0<x<\infty,\\
%u_t(x,0)  &=g(x), \quad 0<x<\infty,\\
%u(0,t)  &= 0,\qquad\;\, t>0,
%\end{align*}
%where $f \in C^2([0,\infty))$ and $g \in C^1([0,\infty))$ satisfy $f(0)=f'(0)=g(0)=0$.
%\begin{enumerate}
%\item
%Solve the problem using the odd extensions of $f$ and $g$.
%\item
%Sketch the domain of dependence of a point $(x_0,t_0)$ where $0<x_0<\infty$ and $t_0>0$.
%\item
%Sketch the region of influence of a point $x_0$ where $0 < x_0 < \infty$.
%\end{enumerate}

\item
Consider the pure initial value problem for the damped wave equation,
\begin{equation}\label{eq:damp}
u_{tt} + \alpha u_{t} = u_{xx} \qquad u(x,0) = g(x) \;\; u_{t}(x,0) = h(x),
\end{equation}
for $\alpha > 0$ a drag coefficient.
\begin{enumerate}
\item Fix a point $x_0 \in \R$ and a time $t_0 > 0$. For $0 \leq t \leq t_0$ let 
$$
E(t) := \frac1{2} \int^{x_0 + (t_0-t)}_{x_0 - (t_0-t)} u^{2}_{t}(x,t) + u^{2}_{x}(x,t) \, \rd x
$$
denote the total energy (kinetic plus potential) in the interval $(x_0-(t_0-t),x_0+t_0-t)$. Use Leibniz rule and the PDE \eqref{eq:damp} to show that
$$
E^{\prime}(t) \leq 0
$$
for $0 < t < t_0,$ and so $E(t)$ is non-increasing.
\item Suppose $g(x) = h(x) = 0$ in the interval $(x_0 - t_0, x_0 + t_0)$. Show that 
$$
u(x,t) = 0
$$
in the entire triangular region $T := \{ (x,t) : x_0 - (t_0-t) \leq x \leq x_0 + (t_0-t), \; 0 \leq t \leq t_0\}$.
\item Show that solutions of the initial value problem \eqref{eq:damp} are unique.
\end{enumerate}

\item
Use the method of characteristics to find a solution to the quasi-linear PDE
$$
u_x u_y = u \qquad \text{with} \quad u(x,y)=y^2 \quad \text{on} \quad \Gamma := \{ (x,y) : x=0 \}.
$$
Is the solution uniquely defined near $(x, y) = (0, 0)$? If so, state the solution. If not, find another solution.

\item
Find the entropy solution to Burger's equation
\begin{align*}
u_{t} + uu_{x} = 0 \quad x \in \R, \; t > 0 \\
u(x,0) = \begin{cases}
0 & \text{if} \quad x < 0\\
1 & \text{if} \quad 0 < x < 1\\
0 & \text{if} \quad x > 1.
\end{cases}
\end{align*}
Sketch the characteristics in the $(x,t)$ plane, and clearly indicate any shocks that occur.

\end{enumerate}
\end{document}