I am trying to solve the following heat equation problem on the square [0,1]x[0,1]. \begin{equation*} \begin{gathered} u_t = u_{xx} + u_{yy} + f(x,y,t), \qquad u(x,y,0) = 0, \qquad u=0 \text{ on boundary} \\ f(x,y,t) = 10(\sin\pi x)^{10}(\sin\pi y)^{10}(\sin\pi t)^{10}. \end{gathered}\end{equation*}
I am trying to solve it with Mathematica using NDSolve but I am stuck and unsure of what I have done. I have
NDSolve[{Derivative[0, 0, 1][u][x, y, t] ==
Derivative[0, 2, 0][u][x, y, t] +
Derivative[2, 0, 0][u][x, y, t] +
10*((Sin[Pi*x])^10)*((Sin[Pi*y])^10)*((Sin[Pi*t])^10),
u[x, y, 0] == 0}, u, {x, 0, 1}, {y, 0, 1}, {t, 0, 1}]
I am not sure that this is actually correct, and I do not know how to enforce the zero boundary condition on my square. Also, I think this equation has an exact solution. In the event that it does, can I also solve it analytically using Mathematica? I would appreciate any help on this as I am still a complete novice in Mathematica. I thank all helpers.