Let $u$ be a solution of the heat equation $$u_t - u_{xx} = 0, \quad t>0, x \in \mathbb{R}$$ with initial data $u(0,\cdot) = u_0$. Fix $\alpha >0$. How can I estimate (without using explicitly the heat kernel) $$\sup_{t>0}\int_{\mathbb{R}} t^\alpha |u_x|^2 \ dx,$$ in terms of the initial data? Could you point out a reference where such an estimate is obtained?
Is it fair to call what we obtain a decay estimate?
The initial condition has been badly stated since the answer would then be$$u(x,t)=u_0\quad,\quad \forall x,t$$this makes sense since when all the bar has the same temperature, there is no cause to change it anywhere and anytime, but if a only a limited part of bar has a different temperature and the rest have different ones, heat then can be propagated (from high temperature parts to low temperature parts).