I'm trying to solve a set of exercises in order to prepare myself for a test. This question verses about energy method on partial differential equations and I would like to ask for help on that, and, if possible, a refference on energy methods and maximum principles. I'm a begginer on partial differential equations.
Show that there is at most one solution to the problem $\begin{cases}u_t=\alpha^2u_{xx}+g,\textrm{ in }(0,L)\times(0,\infty)\\ u(0,t)=u(L,t)=0,t\geqslant 0\\ u(x,0)=u_0(x),\textrm{ in }[0,L]\\ u\in C^2([0,L]\times(0,\infty))\cap C([0,L]\times[0,\infty)) \end{cases}$ if $u$ is a continuous, differentiable by parts, $u_0(0)=u_0(L)=0$, $g\in C((0,L)\times(0,\infty))$, using: (a) the maximum principle; (b) the energy method. Obtain the candidate for a solution, of the form $u(x,t)=\sum_{n=1}^{\infty}c_n(t)\sin\left(\frac{n\pi x}{L}\right)$.
Thanks in advance! P.S.: Oscar Niemeyer lives forever!