  In this paper we provide a different approach for existence of the
variational solutions of the gradient flows associated to functionals on
Sobolev spaces studied in \cite{BDDMS20}. The crucial condition is the
convexity of the functional under which we show that the variational solutions
coincide with the solutions generated by the nonlinear semigroup associated to
the functional. For integral functionals of the form $\mathbf F(u)=\int_\Omega
f(x,Du(x)) dx,$ where $f(x,\xi)$ is $C^1$ in $\xi$, we also make some remarks
on the connections between convexity of $\mathbf F$ (called the integral
convexity of $f$) and certain monotonicity conditions of the gradient map
$D_\xi f.$ In particular, we provide an example to show that even for functions
of the simple form $f=f(\xi)$, the usual quasimonotonicity of $D_\xi f$ is not
sufficient for the integral convexity of $f.$
