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’) u_1(s,r’) u_0(s,r)\ F_1(x(t,r),y(t,r))&=\rho_1(r)+\int_0^t ds\int_0^1dr’J(r-r’) u_1(s,r’) u_0(s,r)-u_1(s,r) \end{align}

and I can prove that, when $ T$ is small enough, the map $ (F_1, F_2)$ 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?