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{ \large \textbf{\textsc{Math 5587 -- Homework 4 (Due Thursday Sept 29)}}}
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\begin{enumerate}
\item For a solution $u(x,t)$ of the wave equation
\[u_{tt} - u_{xx} = 0,\]
the energy density is defined as $e(x,t)=(u_t^2 + u_x^2)/2$ and the momentum density is $p(x,t) = u_tu_x$. 
\begin{enumerate}
\item Show that $e_t = p_x$ and $p_t = e_x$. 
\item Show that both $e$ and $p$ also satisfy the wave equation.
\end{enumerate}
\item Let $u(x,t)$ and $v(x,t)$ be functions such that
\[u_t - ku_{xx} \leq v_t-kv_{xx}\]
on the rectangular strip $U_T=(a,b)\times (0,T]$. Prove the following \textbf{comparison principle}: 
\begin{center} If $u \leq v$ on $\Gamma$ then $u \leq v$ everywhere in $U_T$. \end{center}
 Recall $\Gamma$ is the parabolic boundary of $U_T$, i.e., the sides $x=a$ and $x=b$, and base $t=0$.  [Hint: Apply the maximum principle to $w:=u-v$.] 
\item Consider the nonlinear heat equation 
\[u_t - ku_{xx} + bu_x^2 = 0 \ \ \ \text{ for } \ -\infty < x < \infty \text{ and } t>0,\]
subject to an initial condition $u(x,0) = f(x)$. This type of PDE arises in stochastic optimal control theory. In this question you will derive a representation formula for the solution $u(x,t)$.
\begin{enumerate}
\item Define the Cole-Hopf transformation $w(x,t) = e^{-\frac{b}{k}u(x,t)}$. Show that $w$ is a solution of the linear heat equation
\[w_t - kw_{xx} = 0.\]
\item Use the fundamental solution of the heat equation to solve for $w(x,t)$.
\item Invert the Cole-Hopf transformation to find a formula for $u$.
\end{enumerate}
\item Consider the heat equation
\[u_t - ku_{xx} = 0 \ \ \ \text{ for } 0 < x < \ell \text{ and } t>0,\]
subject to homogeneous Dirichlet boundary conditions 
\[u(0,t) = 0 = u(\ell,t) \ \ \ \text{ for } t>0,\]
and initial condition 
\[u(x,0) = f(x) \ \ \ \text{ for } 0 < x < \ell.\]
We assume that $f$ is \emph{nonnegative}, that is, $f(x) \geq 0$ for all $x$.
\begin{enumerate}
\item Use the comparison principle (Problem 2), or the maximum principle, to show that $u(x,t) \geq 0$ for all $x$ and $t$.
\item Show that $u_x(0,t) \geq 0$ and $u_x(\ell,t) \leq 0$ for all $t$. [Hint: Use the definition of these partial derivatives and (a).]
\item Show that the total heat
\[H(t) = \int_0^\ell u(x,t) \, dx\]
is decreasing in $t$. That is, show that $H'(t) \leq 0$. Give a short explanation of why heat is decreasing and not conserved.
\item Show that for each $0 < x < \ell$
\[\lim_{t\to \infty} u(x,t) = 0.\]
That is, all of the heat in the rod eventually dissipates.
[Hint: Define
\[v(x,t) := \Phi(x,t+1) = \frac{1}{\sqrt{4\pi k(t+1)}}\exp\left(-\frac{x^2}{4k(t+1)}\right),\]
where $\Phi$ is the fundamental solution of the heat equation. Recall that $v$ satisfies the heat equation $v_t-kv_{xx}=0$ for $t>-1$. Explain how to use the comparison principle from problem 2 to show that $u \leq Cv$ for an appropriate constant $C$ depending on $f$ and $k$. Complete the proof from here.]
\end{enumerate}
\item \textbf{Maximum principle:} Consider the heat equation 
\[\text{(H)} \ \left\{ \begin{aligned}
u_t - u_{xx} &= 0,&&   -\infty <  x < \infty, \ t>0\\
u(x,0) &=\phi(x),&&  -\infty < x < \infty.
\end{aligned}\right.\]
As it turns out, there are infinitely many solutions $u$ of the above heat equation. All but one solution are ``non-physical'' and grow exponentially fast as $x\to \pm \infty$. In this question, you will show that if $\phi$ is bounded, then there is a unique bounded solution $u(x,t)$. The proof involves the maximum principle that we dicussed in class for bounded domains. 

Throughout the question let $u$ be a bounded solution of (H); this means there exists $C>0$ such that $|u(x,t)|\leq C$ for all $(x,t)$.
\begin{enumerate}
\item Show that $w(x,t) = x^2 + 2t$ solves the heat equation $w_t = w_{xx}$.
\item For every $\eps>0$ show that 
\[u(x,t)  \leq \eps w(x,t)  + M  \ \ \text{ for all } x \in \R \ \text{ and } \ t>0,\]
where $M>0$ is any number satisfying $\phi(x) \leq M$ for all $x \in \R$. [Hint: For $N>0$ let $R_N$ denote the rectangle
\[R_N = [-N,N]\times[0,N] = \{(x,t) \, : \, -N \leq x \leq N \text{ and } 0 \leq t \leq N\}.\]
Show that there exists $\bar{N}>0$ such that for all $N>\bar{N}$, $u \leq \eps w + M$ on the sides $x=-N$ and $x=N$, and base $t=0$ of $R_N$. Then apply the comparison principle from Problem 2.]
\item Let $M>0$ such that $\phi(x) \leq M$ for all $x \in \R$. Show that $u \leq M$. 
\item Show that there is at most one bounded solution $u$ of (H) when $\phi$ is bounded. [Hint: Take two bounded solutions $u,v$ and consider $w:=u-v$. Then choose $M=0$ to show that $u\leq v$. Complete the proof from here.]
\end{enumerate}
It is possible to prove a stronger result; namely that there is at most one solution $u$ of (H) satisfying the exponential growth estimate
\[u(x,t) \leq A e^{ax^2}\]
for constants $A>0$ and $a>0$. The proof is similar to this exercise, except that $w$ has a different form (similar to HW3 \#5). This means that the ``non-physical'' solutions all grow faster than $Ae^{ax^2}$ as $x \to \pm\infty$.
\item Consider the heat equation on the half line 
\[u_t - ku_{xx} = 0 \ \ \ \text{ for } \ \ \ x>0 \text{ and } t>0,\]
with homogeneous Neumann boundary conditions $u_x(0,t)=0$ for all $t>0$, and intial condition $u(x,0)=f(x)$ for $x>0$. Use the method of even extensions, as outlined in the notes, to show that the solution $u(x,t)$ is given by
\[u(x,t) = \frac{1}{\sqrt{4\pi kt}} \int_0^\infty \left(e^{-(x-y)^2/4kt} + e^{-(x+y)^2/4kt} \right) f(y) \, dy.\]
\end{enumerate}


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