Consider a (possibly unbounded) densely defined selfadjoint operator $A$ on $\mathcal{D}(A) \subset C(\Omega)$ for some domain $\Omega \subset \mathbb{R}^d$ with spectrum $\sigma(A) \le \lambda$ for some fixed $\lambda \in \mathbb{R}_{\ge 0}.$ Suppose it generates a semigroup $S(t)$. Suppose I have the bound $$ 0 \le u(t, x) \le f(t,x) + \int_0^t Au(s, x) \ ds, t \in[0, T], $$
for some $f \ge 0.$ Can we find an integral Gronwall-type bound for $u?$ My naive expectation would be something like the following (but I believe it would be wrong): $$ u(t,x) \le f(t,x) +\int_0^tf(s,x) \lambda e^{\lambda(t-s)} \ ds $$ The differential inequality should be simply comparison: $$ \dot{u} \le Au + f , u(0) = u_0 \ \ \Rightarrow u \le v \text{ with } \ \ \dot{v} = Av + f, v(0) = v_0 $$ which follows from $S(\tau)h(s,x) \ge 0, \forall h \ge 0.$
Any "integral" estimate or useful counterexample is welcome, also if you need to add some hypotheis.