We consider the Bessel's parabolic operator of exponent $\gamma$ and a rhs of the form F(r,u). The boundary conditions in $r=0$ and $r=1$ are linear in $u$ and $ u_{r}$. We use the Galerkin and compactness method in appropriate Sobolev spaces with weight to prove the existence of a unique weak solution of the problem on $(0,T)$, for every $T>0$. We also prove that if the initial condition is bounded, then so is the solution. Finally we study asymptotic behavior of the solution and give numerical results.