Nonlinear differential inequalities

Question

Is there any classic method to solve nonlinear inequalities such as $$ (f(t))^{p}\leq C_{0} f'(t)+C_{1}(f'(t))^{p},\qquad p\geq1, \ C_{0}, C_{1}>0, $$ where $f, f'\geq 0$?

Answer 1

For $s > 0$, let $G(s)$ be the positive solution of $C_0 x + C_1 x^p = s^p$. The separable differential equation

$$ f(t)^p = C_0 f'(t) + C_1 f'(t)^p $$

has implicit solution

$$ \int_1^{f(t)} \dfrac{ds}{G(s)} = t + C$$

Note that $G(s) \sim s/C_1^{1/p}$ as $s \to +\infty$ while $G(s) \sim s^p/C_0$ as $s \to 0+$, so the improper integral of $1/G(s)$ diverges at $0$ and $\infty$. Thus these solutions exist for $t \in \mathbb R$ with $f(t) \to 0$ as $t \to -\infty$ and $f(t) \to +\infty$ as $t \to +\infty$.

Let $H(s)$ be an increasing function on $[0,\infty)$ with $H(s) \ge G(s)$, so that $$C_0 H(s) + C_1 H(s)^p \ge C_0 G(s) + C_1 G(s)^p = s^p$$ Then the solution of $f'(t) = H(f(t))$ satisfies your inequality, and can similarly be written as $$ \int_1^{f(t)} \dfrac{ds}{H(s)} = t + C $$ and this will exist for $t \in \mathbb R$ with $f(t) \to 0$ as $t \to -\infty$ and $f(t) \to +\infty$ as $t \to +\infty$ if the improper integral of $1/H(s)$ diverges at $0$ and $\infty$.