Let $u:\mathbb{R}\to\mathbb{R}^3$ where $u(t)=(u_1(t),u_2(t), u_3(t))$ be a function that satisfies $$\frac{d}{dt}|u(t)|^2+|u|^2\le 1,\tag{1}$$where $|\cdot|$ is the Euclidean norm. According to Temam's book paragraph 2.2 on page 32 number (2.10), inequality (1) implies $$|u(t)|^2\le|u(0)|^2\exp(-t)+1-\exp(-t),\tag{2}$$but I do not understand why (1) implies (2).
inequality in a differential equation
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analysis
ordinary-differential-equations
dynamical-systems