Let $\rho_1:[0,1]\to [0,1]$ and $J:\mathbb R\to \mathbb R^+$ both continuous and bounded.

I have the following system of PDE's

\begin{align} \begin{cases} \frac{\partial}{\partial t} u_0(t,r)=- J* u_1(t,r) u_0(t,r)\\ \frac{\partial}{\partial t}u_1(t,r)=J*u_1(t,r)u_0(t,r)-u_1(t,r)\\ u_0(0,r)=1-\rho_1(r), u_1(0,r)=\rho_1(r) \end{cases} \end{align} where * denotes the convolution operator.

I would like to prove that there exists unique a local solution of the previous system. I would like to prove that this solution is also continuous.

Is it correct the following argument?

I consider the maps $F_0$ and $F_1$ defined in $L_c^\infty([0, T]\times [0,1])^2$ which contains all the functions bounded by a constant $c$

\begin{align} F_0(x(t,r),y(t,r))&=1-\rho_1(r)+\int_0^t ds\int_0^1dr'J(r-r') y(s,r') x(s,r)\\ F_1(x(t,r),y(t,r))&=\rho_1(r)+\int_0^t ds\int_0^1dr'J(r-r') y(s,r') x(s,r)-y(s,r) \end{align}

and I can prove that, when $T$ is small enough, the map $(F_0, F_1)$ is a contraction in $L_c^\infty([0, T]\times [0,1])^2$,.

Then by the contraction mapping theorem I can conclude that there exists a unique fixed point of $(F_0, F_1)$ which is a local solution of the previous PDE's system.

Is that correct? There is any chance to prove the continuity of my local solution?

My idea is to apply the Contraction mapping theorem in $C_c^\infty([0, T]\times [0,1])^2$ the set of all continuous function bounded by a constant $c$... Is that possible?