1. I start with simple differential inequality: find $u\in C^1[0,1]$ such that $u(0) = 0$ and $$ u'(t)\leq -u(t) $$ for all $t\in [0,1]$. Using Gronwall's lemma one can see that $u\leq 0$. On the other hand it seems to be the only solution, since this inequality keep $u$ non-decreasing for $u<0$ and non-increasing for $u>0$. Is it right that only $u=0$ satisfies this inequality?
2. With Gronwall's lemma one can see that any solution of $$ u'(t)\leq\beta(t)u(t) $$ is bounded from above by the solution of $$ u'(t) = \beta(t)u(t). $$
So there are two main results:
(i) there is a solution of $u'\leq \beta u$ which dominates any other solution.
(ii) this solution is attained on the correspondent equation.
Are there any similar results on PD inequalities of the type $$ u_t(t,x)\leq L_x u(t,x) $$ where $L_x$ is a differential operator in $x$ variable (of first or second order).
The main question for me if (i) is valid for such inequalities.