Prove that energy function $Ε(t)=\int\limits_{0}^{L}u^2(x,t)\,\mathrm{d}x$ is strictly decreasing for the problem ($k,\mu>0$): $$ \left \{\begin {array}{lll} u_t=ku_{xx}-\mu u &,~ 0<x<L,~t>0\\ u(x,0)= \phi(x)>0 &, 0<x<L \end{array} \right., $$ where $\phi(0)=\phi(L)=0$ and $u(0,t)=u(L,t)=0$ for all $t>0$.
Attempt. Using integration py parts we get for all $t>0$
$$E'(t) =-2k\int\limits_{0}^{L}u_{x}^2\,\mathrm{d}x- 2\mu \int\limits_{0}^{L}u^2\,\mathrm{d}x\leqslant 0.$$ If for some $t_0>0$ we have $E'(t_0)=0$, then using continuity of $u$ we derive $u(x,t_0)=0$ for all $x\in (0,L)$. My question is:
How does this contradict hypothesis $u(x,0)>0$ for $0<x<L$?
(note that $u$ does not satisfy the heat equation, in order to attain its minimum of the parabolic boundary).