I am difficulty with this assignment:
This project deals with the partial differential equation (PDE) $$ u_t(x, t) + u(x, t)u_x(x, t) = 0, \quad x \in \mathbb{R} \text{ and } t \geq 0 \tag{1} $$ together with the initial condition $$ u(x, 0) = u_0(x), \quad x \in \mathbb{R} \tag{2} $$ where $u_0$ is a differentiable function on $\mathbb{R}$ with continuous derivative $u_0^\prime$. The PDE (1) together with the initial condition (2) is called an initial value problem.
The initial value problem (1–2) is used, among other applications, as a rudimentary model for the motion of a fluid in a thin tube. The unknown function u gives the velocity of the fluid. That is, $u(x, t)$ is the velocity of the fluid at the point $x$ in the tube at time $t$. The initial value $u_0$ prescribes the velocity of the fluid at time $t = 0$.
This project deals with issues related to the solutions of (1–2).
Definition 1. A solution of the initial value problem (1–2) is a function
$$ u \colon \mathbb{R} \times \left[0,\infty \right) \to \mathbb{R} $$
that satisfies the following:
$u$ is continuous on $\mathbb{R} \times \left[0,\infty \right)$.
$u$ is differentiable with continuous first order partial derivatives on $\mathbb{R} \times \left[0,\infty \right)$.
$u$ satisfies the PDE (1) at every point $(x, t) \in \mathbb{R} \times \left[0,\infty \right)$.
$u(x, 0) = u_0(x)$ for every $x \in \mathbb{R}$, that is, $u$ satisfies the initial condition (2).
Concerning the solutions of (1–2), there are two basic questions. (1) Does a solution exist? (2) If a solution exists, what does it look like?
The aim of this project is to address these two questions. The analysis will be based on the method of characteristics.
How do I solve this problem?
There are infinitely many solutions to this problem. But your supplemental conditions, particularly 1. and 2., eliminate many of them. As I understand it, you want $u$ to remain continuous and differentiable for all $t$. (If I have misunderstood, you can stop reading now.) Only very specific initial conditions will produce this behavior. Namely, given any $x_0$, and $x_1$ such that $x_1-x_0>0$ and $u(x_0,t)$, it must be the case that
$u(x_1,0)\ge u(x_0,0)$
I'm not going to be able to give you a formal solution via characteristics, but you can see this in a qualitative way. This equation simply "transports" $u(x,0)$ to some other $x$ at time $t$, and the rate at which it transports $u$ is simply $u$. So what happens if the above condition is violated, that is, what if we have $u(x_0,0)$, but at some $x<x_0$, $x_{-1}$, say, we have $u(x_{-1},0)>u(x_0,0)$? Then, eventually, $u(x_{-1},0)$ will "catch up with" $u(x_0,0)$. When that happens, our function will be multivalued -- both discontinuous and undifferentiable.