\documentclass[]{article}
\headheight=8pt     \topmargin=0pt
\textheight=624pt   \textwidth=432pt
\oddsidemargin=18pt \evensidemargin=18pt

\usepackage{amssymb,amsmath,amsthm}
\usepackage{epsfig}
\usepackage{subfigure}
\usepackage{enumerate}
\usepackage{color}
\usepackage{graphicx}
\usepackage{enumerate}

\DeclareMathOperator{\Tr}{Tr}
\pagestyle{plain}

\theoremstyle{plain}

\begin{document}

\title{MTH 352/652:
Homework \#2}
\vspace{-1in}
\date{Due Date: February 02, 2024}
\maketitle

\section{Problems for Everyone}
\begin{enumerate}
\item Let $c\neq 0$ and suppose $u(t,x)$ solves the following initial value problem
\begin{equation*}
u_t+cu_x=0 \text{ and } u(0,x)=f(x).
\end{equation*}
Suppose $f$ is continuous and satisfies $\lim_{|x|\rightarrow \infty}f(x)=0$. Prove that $\lim_{t\rightarrow \infty}u(t,x)=0$.
\item Solve the following initial value problems in the region $x\in \mathbb{R}$, $t>0$
\begin{enumerate}
\item $u_t+xtu_x=0$ \text{ and } $u(0,x)=f(x)$
\item $u_t+xu_x=e^t$ \text{ and } $u(0,x)=f(x)$
\end{enumerate}

\item Solve the following initial value problems in the region $x\in \mathbb{R}$, $t>0$
\begin{enumerate}
\item $u_t+xu_x=-tu$ \text{ and } $u(0,x)=f(x)$
\item $tu_t+xu_x=-2u$ \text{ and } $u(0,x)=f(x)$
\item $u_t+u_x=-tu$ \text{ and } $u(0,x)=f(x)$
\end{enumerate}

\item Consider the following initial value problems in the region $x\in \mathbb{R}$, $t>0$:
\begin{equation*}
u_t+u_x+u^2=0 \text{ and } u(0,x)=f(x).
\end{equation*}
\begin{enumerate}
\item Find the general solution to this initial value problem.
\item Show that if $f(x)$ is bounded and positive, i.e., $0\leq f(x)\leq M$, then the solution exists for all $t>0$ and
\begin{equation*}
\lim_{t\rightarrow \infty}u(t,x)=0.
\end{equation*}
\item Show that if $f(x)$ is negative, so $f(x)<0$ at some $x\in \mathbb{R}$, then the solution blows up in finite time:
\begin{equation*}
\lim_{t\rightarrow \tau^{-}}u(t,y)=-\infty
\end{equation*}
for some $\tau>0$ and some $y\in \mathbb{R}$. 
\end{enumerate}

\item Consider the equation
\begin{equation*}
u_t+xu_x=0
\end{equation*}
with the boundary condition $u(t,0)=\phi(t)$.
\begin{enumerate}
\item For $\phi(t)=t$, show that no solution exists.
\item For $\phi(t)=1$, show that there are infinitely many solutions.
\end{enumerate}
\end{enumerate}
\end{document}