1. **Problem Statement:** We need to solve the homogeneous spherical partial differential equation (PDE): $$\frac{\partial u}{\partial t} = \frac{1}{r^2} \frac{\partial}{\partial r} \left(r^2 \frac{\partial u}{\partial r}\right) + \frac{1}{r^2 \sin \phi} \frac{\partial}{\partial \phi} \left(\sin \phi \frac{\partial u}{\partial \phi}\right) + \frac{1}{r^2 \sin^2 \phi} \frac{\partial^2 u}{\partial \theta^2} + k$$ with boundary conditions (BC): $$u(R, \theta, \phi, t) = 0$$ $$\frac{\partial u}{\partial \theta}(r, 0, \phi, t) = \frac{\partial u}{\partial \theta}(r, \frac{\pi}{2}, \phi, t) = 0$$ $$\frac{\partial u}{\partial \phi}(r, \theta, 0, t) = \frac{\partial u}{\partial \phi}(r, \theta, \pi, t) = 0$$ and initial condition (IC): $$u(r, \theta, \phi, 0) = \beta(r, \theta, \phi)$$ 2. **Method and Formula:** This is a spherical heat/diffusion equation with source term $k$. We use separation of variables and spherical harmonics expansion. The general approach: - Assume solution $u(r, \theta, \phi, t) = R(r) \Theta(\theta) \Phi(\phi) T(t)$. - Use eigenfunction expansions for angular parts satisfying BCs. - Radial part satisfies Sturm-Liouville problem with BC $u(R, \theta, \phi, t)=0$. - Time dependence typically exponential decay or growth. 3. **Boundary Conditions Explanation:** - $u(R, \theta, \phi, t) = 0$ means the solution vanishes on the spherical surface of radius $R$. - Neumann BCs on $\theta$ and $\phi$ imply zero flux or symmetry at boundaries. 4. **Solution Sketch:** - Expand $u$ in spherical harmonics $Y_l^m(\theta, \phi)$ which satisfy angular BCs. - Solve radial ODE for each mode: $$\frac{1}{r^2} \frac{d}{dr} \left(r^2 \frac{dR}{dr}\right) - \frac{l(l+1)}{r^2} R + \lambda R = 0$$ with $R(R) = 0$. - Time evolution: $$\frac{dT}{dt} = -\lambda T + k$$ - Use initial condition to find coefficients. 5. **Summary:** The problem reduces to solving eigenvalue problems in $r$ and angular variables, then combining with time evolution and source term $k$. The solution is a series: $$u(r, \theta, \phi, t) = \sum_{l,m} R_{l}(r) Y_l^m(\theta, \phi) T_{l,m}(t)$$ where $R_l$ satisfy radial BC, $Y_l^m$ satisfy angular BCs, and $T_{l,m}$ solve ODEs with source $k$. This is a classical spherical PDE problem solved by separation of variables and spherical harmonics expansion.