On integral convexity, variational solutions and nonlinear semigroups

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 $\mathbf{F}(u)={\int }_{\mathrm{\Omega }}f(x,Du(x))dx,$ 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 $D_{\xi }f\mathrm{.}$ 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 $f\mathrm{.}$