I am trying to do the following exercise
Consider the heat equation in one space dimension with homogeneous Neumann Boundary conditions : $u_t=cu_{xx}$ in $(0,L)\times (0,\infty), u(x,0)=u_0(x)$ for $x\in [0,L], u_{x}(0,t)=u_x(L,t)=0$ for $t\geq 0$ and let $u$ be a classical solution. Show that in the $L^2-$norm $u(x,t)\rightarrow U_0$ as $t\rightarrow \infty$, where $U_0=\frac{1}{L}\int_{0}^L u(x,t)dx$.
It's easy to see that the average temperature is conserved ,i.e., $\frac{1}{L}\int_{0}^L u(x,t)dx = \frac{1}{L}\int_{0}^L u_0(x)dx$ for every $t>0$. And so if we consider $w(x,t):=u(x,t)-U_0$, we will have that for each $t$ there exists $x(t)$ such that $w(x(t),t)=0$.
Now I am trying to show that $E'(t)\leq -CE(t)$ for $E(t):=\int_{0}^L w^2(x,t)dx $, and then by Gronwall's inequality we will get the desired result.
All I have gotten is that $E'(t)= -2c\int_{0}^L w_x^2(x,t)dx$, and I don't know how to proceed from here.
Any help is appreciated, thanks in advance.