# Why is this true? (standard a-priori estimate on linear parabolic PDE)

I read this:

We see that $$\frac{1}{2}\lVert u(T) \rVert^2_{L^2} \leq \frac{1}{2}\lVert u(0) \rVert^2_{L^2} + \lVert f \rVert_{L^2(0,T;H^{-1})}\lVert u \rVert_{L^2(0,T;H^1_0)}$$ and since $T$ is arbitrary, we have $$\lVert u \rVert^2_{L^\infty(0,T;L^2)} \leq \lVert u(0) \rVert^2_{L^2} + \lVert f \rVert^2_{L^2(0,T;H^{-1})}$$

How is that true? I see he used Young's inequality on the RHS but what happened to the other term of Young's inequality?

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Can you give a reference to the source of this statement? –  user53153 Feb 2 '13 at 4:35
@5PM See math.ucdavis.edu/~hunter/notes/nonlinev.pdf, page 66. –  george.s Feb 2 '13 at 9:41