An inequality involving norms.

Question

I have to know how we can show the following inequality:

$\|u\|_{2}\leq\|u_{0}\|_{2}+\int_{0}^{t}\|u_{t}(t,x)\|_{2}dt$

where

$\|u\|_{2}=\Big(\int_{\Omega}u^{2}dx\Big)^{1/2}$, $u_{0}=u(x,0)$, $u_{t}=\frac{\partial u}{\partial t}$, $\Omega$ is an open bounded domain in $R^{n}$ and $u=u(x,t)$.

Answer 1

Assuming that $u \in H^1(0, T; L^2(\Omega))$, you have $$u(T) = u(0) + \int_0^T u_t(t) \, dt.$$ By the triangle inequality, we find $$\| u(T) \|_{L^2(\Omega)} \le \| u(0)\|_{L^2(\Omega)} + \left\| \int_0^T u_t(t) \, dt\right\|_{L^2(\Omega)} \le \| u(0)\|_{L^2(\Omega)} + \int_0^T \| u_t(t) \|_{L^2(\Omega)} \, dt. $$