There is PDE in $[0,1][0,T]$ where $A$ is complex and $f$ are real: $$iA_t = A_{xx}+f(x,t)A$$ $$A(0,t) = A(1,t) = 0$$
I need to prove that
$$\|A(x,t)\|^2_{C[0,T;H^1_0[0,1])}=\max_t\int_0^1|A_x(x,t)|^2dx = \int_0^1|A_x(x,0)|^2dx = \|A(x,0)\|^2_{H^1_0[0,1]}$$
the first and last equalities are obvious - these are just definitions that I wrote, but I deal with the problem that in the center
So i tried energy method: $$\int_0^liA_tAdx = \int_0^l (A_{xx}A + f(x,t)A^2)dx$$ $$i\frac{d}{dt}\frac{1}{2}\int_0^lA^2dx = -\int_0^lA^2_x dx +\int_0^lfA^2dx$$ $$iE'(t) = -\int_0^lA^2_x dx +\int_0^lfA^2dx$$ Usually I need to prove that $E'(t)$ is equal to $0$ and then $E(t)$, which is similar to the norm in $L_2$, is constant, but this norm is not in $L_2$.
So: What method can be used to show center equality in this Sobolev space? Is it possible to apply the energy method here? Maybe there will be a more complex multiplier function than $A(x,t)$?
I heard that for this we can use the Galerkin and Fourier methods, but at first glance, the Galerkin method is generally about something else. how is it applied?