There is a standard method to prove the existence and uniqueness in $[0, +\infty)\times (0,1)$ of a PDE as the following:
$\frac{\partial}{\partial t}u(t,r)=\frac{1}{2}r-\frac{1}{4}+\frac{1}{2}\frac{\partial}{\partial r}u(t,r)$
$u(0,r)=f(r)$,
where $f\in C^\infty(0, 1)$? Thanks for your help
Existence - if you set $v = u + r^2/2 - r/2$ then the equation becomes the homogeneous transport equation $$ \partial_t v = \frac12\partial_r v$$ thus $v(t, r)$ is given by a function $v(t,r) = V(\frac12 t+r)$ that depends only on $\frac12 t+r$. So since $$v(0,r) = V(r) = f(r) + \frac{r^2}2 - \frac r2 $$
then $$ u(t, r) = v(t,r) - \frac{r^2}2 + \frac r2 = f\left(\frac12t+r\right) + \frac{(\frac12t+r)^2}2 - \frac{\frac12t+r}2 - \frac{r^2}2 + \frac r2.$$
If $u_1,u_2$ were two solutions then $w = u_1 - u_2$ is a function that solves the homogeneous transport equation $$ \partial_t w = \frac12 \partial_r w$$ with zero initial condition at $t=0$, so it is $0$ for all time.