Let $u(x,t) = \sum_{j=1}^\infty u_j^0 e^{-j^2t}\sin(jx)$ be the solution for the heat equation, where $u_j^0 = \sqrt{\frac{2}{\pi}}\int_0^\pi u^0(x)\sin(jx)\;dx$ are the Fourier coefficients of the initial data $u^0$.
Suppose that $u_j^0 = C/j$. Prove that $\left\lVert\frac{d}{dt}u(\cdot,t)\right\rVert\leq \frac{C}{t^{3/4}}$ as $t \rightarrow 0$.
This is a problem from "Numerical Solution of Partial Differential Equations by the Finite Element Method", by Claes Johnson. I am having trouble to relate the decay rate of the Fourier coefficient with the boundness on the time derivative.
The book also uses as an example that $\left\lVert \frac{d}{dt}u(\cdot,t)\right\rVert\leq \frac{C}{t^{1/4}}$ when $u_j^0 = C/j^2$ and $t \rightarrow 0$.
While I can see that $\left\lVert\frac{d}{dt}u(\cdot,t)\right\rVert\leq \frac{C}{t}$ is true because:
$$\frac{d}{dt}u(x,t) = \sum_{j=1}^\infty u_j^0 (-j^2)\frac{t}{t} e^{-j^2t}\sin(jx) \leq \sum_{j=1}^\infty u_j^0 \frac{C}{t}\sin(jx)$$
I don't see how I can use that $u_j^0 = C/j$ to change the value from $1$ to $\frac{3}{4}$.