Existence of solutions for the following PDE We have that $a:\mathbb{R}^+\times \mathbb{R}\to \mathbb{R}$ be a given function such that $|a(t,x)|+|\partial_x a(t,x)|\leq M$. $\rho_0$ is the initial data where $$\rho_{|t=0} = \rho_0 \in L^1\cap L^{\infty}(\mathbb{R}),\quad \rho_0\geq 0.$$

*

*Show that the iterative scheme
$$ \left\{ \begin{array}{lcc}
             \rho^{(0)}\in L^2((0,T)\times \mathbb{R}),\quad \partial_x \rho^{(0)}\in L^2((0,T)\times \mathbb{R}) \\
             \\ \partial_t \rho^{(n+1)} - \partial_{xx}^2 \rho^{(n+1)} = -\partial_x (a\rho^{(n)}) =   -\rho^{(n)}\partial_x a - a \partial_x \rho^{(n)}\\
             \\ \rho^{(n+1)}_{|t=0} = \rho_0
             \end{array}
   \right.$$
is well defined.

*Show that it is contracting on $C^0 ([0; T]; L^2 (\mathbb{R}))$ endowed with norm
$$N(u)^2 = \sup_{0\leq t\leq T< \infty}\left(e^{-\gamma t}\int_\mathbb{R}|u(t,x)|^2 dx\right)$$
for a well chosen value of $\gamma> 0$.

My fisrt question is, what is the meaning of iterative schema well defined? and how to test it?
For the second item, I have the following idea:
I worked with
$$\small \partial_t (\rho^{(n+1)}-\rho^{(n)}) - \partial_{xx}^2( \rho^{(n+1)} - \rho^{(n)}) = -(\rho^{(n)}-\rho^{(n-1)})\partial_x a - a \partial_x (\rho^{(n)}-\rho^{(n-1)})$$
then, multiplying by $(\rho^{(n+1)}-\rho^{(n)})$ and integrating with respect to $x$ we have that
$$\scriptsize \frac{d}{dt}\int_\mathbb{R}\frac{|\rho^{(n+1)}-\rho^{(n)}|^2}{2}dx + \int_\mathbb{R}|\nabla(\rho^{(n+1)}-\rho^{(n)})|^2 dx = -\int_\mathbb{R} (\rho^{(n+1)}-\rho^{(n)})(\rho^{(n)}-\rho^{(n-1)})\partial_x a dx -\int_\mathbb{R} a(\rho^{(n+1)}-\rho^{(n)})\partial_x(\rho^{(n)}-\rho^{(n-1)}) dx$$
and from this expression, we will find majorities for the expression on the right until we can express a Gronwall lemma inequality.
 A: Concerning (1), I think the question is asking you to check that if $\rho^{(n)} \in L^{2}((0,T) \times \mathbb{R})$ and $\partial_{x} \rho^{(n)} \in L^{2}((0,T) \times \mathbb{R})$, then $\rho^{(n + 1)}$ is well-defined (i.e. there's a solution of the equation) and has the same property.  (I won't give the details, but it looks correct to me.)
Concerning (2), notice that, given $u \in C([0,T]; L^{2}(\mathbb{R}))$, $N_{\gamma}(u)$ can be written as
\begin{equation*}
N_{\gamma}(u)^{2} = \sup \left\{ \int_{-\infty}^{\infty} |u_{\gamma}(t)|^{2} \, dx \, \mid \, t \in [0,T] \right\}.
\end{equation*}
Here $u_{\gamma}$ is defined by $u_{\gamma}(t) = e^{-\gamma t/2} u(t)$.  Hence one way to proceed is to study $\rho_{\gamma}$ instead of $\rho$.
Rather than the scheme, let's show that the solution map $S : \mathcal{S} \to \mathcal{S}$ is a contraction.  Here $\mathcal{S}$ is the space of possible solutions
\begin{equation*}
\mathcal{S} = \{\rho \in C([0,T]; L^{2}(\mathbb{R})) \, \mid \, \rho \restriction_{t = 0} = \rho_{0}\}
\end{equation*}
and, given $\eta \in C([0,T];L^{2}(\mathbb{R}))$, we define $\rho = S\eta$ as the solution of the equation
\begin{equation*}
\left\{ \begin{array}{r l}
\partial_{t} \rho - \partial^{2}_{xx} \rho = - \partial_{x}(a \eta) & \text{in} \, \, (0,T) \times \mathbb{R} \\
\rho \restriction_{t = 0} = \rho_{0} & \text{on} \, \, \{0\} \times \mathbb{R}
\end{array} \right.
\end{equation*}
We want to show that, for an appropriate choice of $\gamma > 0$, there is an $\alpha_{\gamma} \in (0,1)$ such that $N_{\gamma}(S\eta - S\beta) \leq \alpha_{\gamma} N_{\gamma}(\eta - \beta)$.
Fix $\eta, \beta \in \mathcal{S}$.  Let $\rho^{\eta} = S\eta$ and $\rho^{\beta} = S \beta$ and let $\mu = \rho^{\eta} - \rho^{\beta}$ be the error.  If we set $\gamma > 0$, then $\mu_{\gamma}(t) = e^{-\gamma t/2} \mu(t)$ solves the equation
\begin{equation*}
\left\{ \begin{array}{r l}
\partial_{t} \mu_{\gamma} - \partial^{2}_{xx} \mu_{\gamma} + \frac{\gamma}{2} \mu_{\gamma} = - e^{-\gamma t/2} \partial_{x}(a (\eta - \beta)) & \text{in} \, \, (0,T) \times \mathbb{R} \\
\mu \restriction_{t = 0} = 0 & \text{on} \, \, \{0\} \times \mathbb{R}
\end{array} \right.
\end{equation*}
If we multiply the equation by $\mu_{\gamma}$ and integrate-by-parts in the right-hand side, we find
\begin{equation*}
\frac{1}{2} \frac{d}{dt} \left\{ \int_{-\infty}^{\infty} |u_{\gamma}(t)|^{2} \, dx \right\} + \int_{-\infty}^{\infty} |\partial_{x} u_{\gamma}(t)|^{2} \, dx = e^{-\gamma t/2} \int_{-\infty}^{\infty} a (\eta - \beta) \partial_{x} \mu_{\gamma} \, dx - \frac{\gamma}{2} \int_{-\infty}^{\infty} |u_{\gamma}(t)|^{2} \, dx
\end{equation*}
By Young's inequality, given $\nu> 1/2$, we have
\begin{equation*}
e^{-\gamma t/2} \int_{-\infty}^{\infty} a (\eta - \beta) \partial_{x} \mu_{\gamma} \, dx \leq \frac{1}{2 \nu} \int_{-\infty}^{\infty} |\partial_{x} \mu_{\gamma}|^{2} \, dx + \frac{\nu}{2} e^{-\gamma t} \int_{-\infty}^{\infty} a^{2} |\eta - \beta|^{2} \, dx
\end{equation*}
and, thus, the function $F_{\gamma}(t) = \int_{-\infty}^{\infty} |u_{\gamma}(t)|^{2} \, dx$ satisfies
\begin{equation*}
F_{\gamma}'(t) \leq - \gamma F_{\gamma}(t) + \frac{\nu M}{2} e^{-\gamma t} \int_{-\infty}^{\infty} |\eta(t) - \beta(t)|^{2} \, dx.
\end{equation*}
Therefore, we can apply Gronwall's Lemma to find
\begin{equation*}
F_{\gamma}(t) \leq \frac{\nu M}{2} \int_{0}^{t} e^{- \gamma(t - s)} \cdot e^{-\gamma s} \int_{-\infty}^{\infty} |\eta(s) - \beta(s)|^{2} \, dx \, ds \leq \frac{\nu M}{2} \int_{0}^{t} e^{-\gamma (t - s)} \, ds \cdot N_{\gamma}(\eta - \beta)^{2}.
\end{equation*}
which gives
\begin{equation*}
N_{\gamma}(S\eta - S \beta)^{2} = \sup \left\{ F_{\gamma}(t) \, \mid \, t \in [0,T] \right\} \leq \frac{\nu M}{2 \gamma} N_{\gamma}(\eta - \beta)^{2}.
\end{equation*}
Hence we get $\alpha_{\gamma}^{2} = \frac{\nu M}{2 \gamma} < 1$ as soon as $\gamma > \frac{2}{\nu M}$.
Given the above, notice that $N_{\gamma}(\rho^{(n + 1)} - \rho^{(n)}) \leq \alpha_{\gamma} N_{\gamma}(\rho^{(n)} - \rho^{(n - 1)})$ and, thus, $N_{\gamma} (\rho^{(n + 1)} - \rho^{(n)}) \leq \alpha_{\gamma}^{n} N_{\gamma}(\rho^{(1)} - \rho^{(0)})$.  This implies $\{\rho^{(n)}\}_{n \in \mathbb{N}}$ is a Cauchy sequence in $\mathcal{S}$ and it approaches its limit exponentially fast.
