Solutions of a PDE in two dimensions

Question

I should solve the following PDE: $$\dfrac{\partial u(x,y,t)}{\partial t}=a\nabla^2u(x,y,t)+b\nabla u(x,y,t)+g(x,y,t)-c u(x,y,t)$$ where $(a,b,c)$ are constants, $$u(x,y,0)=u_0$$ $$u(x,y,t)=0$$ for $0\le x\le \rho_0$, $y=\sqrt{\rho_0^2-x^2}$, $$u(x,y,t)=0$$ for $0\le x\le\rho_1$, $y=\sqrt{\rho_1^2-x^2}$. Are there suggestions or hints? Thanks.