I want to study how some numerical schemes work on $2$-dimensional reaction-diffusion systems on rectangles with Neumann Boundary conditions and I search for a while for a problem of the form:

$$\begin{cases} \dfrac{\partial u}{\partial t}(t,x)-d_1\Delta u(t,x)=f(u(t,x),v(t,x)),\ (t,x)\in (0,T)\times (a,b)\times (c,d) \\ \dfrac{\partial v}{\partial t}(t,x)-d_2\Delta v(t,x)=g(u(t,x),v(t,x)),\ (t,x)\in (0,T)\times (a,b)\times (c,d) \\ \dfrac{\partial u}{\partial\nu}(t,x)=\dfrac{\partial v}{\partial\nu}(t,x)=0, \text{on the boundary} \\ u(0,x)=u_0(x),\ v(0,x)=v_0(x)\end{cases}$$

that possess an analytical solution $u,v$ with $u,v$ depending both on $x=(x_1,x_2)$ and $t$.

One example that I found was $u=\sin (t)$ and $v=\cos(t)$ but they do not depend on $x$ nor satisfy Neumann bc. You can choose $f,g,d_1,d_2,u_0,v_0$ whatever you want.

Do you know if there is such an example??