Consider the heat equation,
$(1)$ $u_t=u_{xx}+f(x,t)$, $0<x<1$, $t>0$
$(2)$ $u(x,0)=\phi(x)$
$(3)$ $u(0,t)=g(t)$, $u(1,t)=h(t)$
When one wants to Show the uniqueness of solution of problem $(1)-(3)$, s/he can use so-called energy method or use maximum principle.
My Question: What is the difference between these method? Is the class of solutions change for each method?
Thanks in advance!...
The methods are different, but they can lead to the same conclusion, the first thing to do is assume that $u_1,u_2$ satisfy the problem, and let $w=u_1-u_2$, then $w$ solves $$w_t-w_{xx}=0$$ $$w(x,0)=0$$ $$w(0,t)=w(1,t)=0$$ To use the energy method we multiply the equation by $w$, integrate and use integration by parts (noting that $w$ is $0$ on the boundary) to obtain: $$\int_0^1ww_t+|w_x|^2\,dx=0$$ and $ww_t=\frac{1}{2}\frac{\partial }{\partial t}(w^2)$, and since $|w_x|^2\ge 0$ we get $$\int_0^1\frac{1}{2}\frac{\partial}{\partial t}(w^2)\,dx\le 0$$ $$\frac{d}{dt}(\frac{1}{2}\int_0^1w^2\,dx)\le 0$$ We let $E(t)=\frac{1}{2}\int_0^1w^2\,dx$, we this is the "energy", and we see that it is decreasing in time (since the derivative is negative) so $E(t)\le E(0)$, i.e. $$\int_0^1w^2(x,t)\,dx\le\int_0^1w^2(x,0)\,dx=0$$ Thus $w=0$, i.e. $u_1=u_2$.
The maximum principle is a different approach, I assume you mean to use the strong maximum principle, where again we take the same $w$, and the strong max principle tells us that if there is a point $(x_0,t_0)\in(0,1)\times(0,T)=:Q(T)$ such that $w(x_0,t_0)=\sup_{\overline{Q}(T)}w$, then $w$ is constant. Now for $w$ by the SMP we see that the maximums of $w$ occur at $x=0,x=1$ or $t=0$, in both cases $w$ is zero here, and we find that $\sup w=0$, thus $w\le0$. i.e. $u_1\le u_2$, now apply this again with $w=u_2-u_1$, and you obtain $u_2\le u_1$, combining these you get $u_1=u_2$.