Consider the boundary value problem (BVP) $u_t$ = $u_{xx}$ for $t>0$ and $x \in (0,1)$ with initial condition
$$u(x,0) = \sin^4(2\pi x),\quad x \in(0,1) $$
and boundary conditions $u(0,t) = u(1,t) = 0$ for all $t>0$.
a) Show that if $u(x,t)$ is a solution of the BVP, then for all $t>0$ and $x \in (0,1)$ we have $0 \leq u(x,t) \leq 1$. (supposed to use separation of variables by the way of the boundary problems chapter from the pde strauss text) $$$$ b) Show that if u(x,t) solves the BVP, then it is unique. (Is this where I am supposed to follow the uniqueness method?) $$$$ c) Use part b, to show that if u(x,t) is the regular solution of the BVP, then $\forall$t>0, x $\in$(0,1) = u(1-x,t). (This one I do not know how or where to begin.) $$$$ d) For each t>0 define M(t) = max{u(x,t):0 $\leq$ x $\leq$ 1}, and E(t) = $\int$$_0$$^1$ u$^2$(x,t) dx. Show that M(t) and E(t) are non-increasing functions of t. (Do I just take the integral of this E function?) I really need a hint on parts b and c, I still do not know what to do or how to start it...
HINT: the property you are looking for is known as the maximum principle of (parabolic) partial differential equations.
The main idea is that if $u$ a solution a BVP for parabolic (or elliptic) partial differential equation, then the minimum and maximum values are reached by $u$ on the boundary of the domain.
A good reference to look up the proof is Evans, although you can find pretty much everything you need in the internet.
Maximum principle is used in the proof of uniqueness of solution of heat equation.