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!
The existence is guaranteed by Theorem (Theorem 2, page 50, Evans, L.C. Partial Differential Equations). Let $ T>0$ arbitrary and suppose $w=u_{1}-u_{2}$. Then $$\begin{cases}w_{t}=\alpha^2w_{xx}, \textrm{ in }(0,L)\times(0,T]\\ w(x,0)=0, x\in(0,L)\\ w(x,t)=0, \{0,L\}\times[0,T] \end{cases}$$ By the principle of maximum
max $w$ in$[0,L]\times[0,T]=$max $w$ in $\{0,L\}\cup\{0,T\}=0$. Then, $w=0$ in $[0,T], \forall T>0\Longrightarrow u_{1}=u_{2}$.
Item b), I do not know :(