  We provide a simple method for obtaining new Liouville theorems for scaling
invariant superlinear parabolic problems with gradient structure. To illustrate
the method we prove Liouville theorems (guaranteeing nonexistence of positive
classical solutions) for the following model problems: the scalar nonlinear
heat equation $$ u_t-\Delta u=u^p \qquad\hbox{in}\ {\mathbb R}^n\times{\mathbb
R}, $$ its vector-valued generalization with a $p$-homogeneous nonlinearity and
the linear heat equation in ${\mathbb R}^n_+\times{\mathbb R}$ complemented by
nonlinear boundary conditions of the form $\partial u/\partial\nu=u^q$. Here
$\nu$ denotes the outer unit normal on the boundary of the halfspace ${\mathbb
R}^n_+$ and the exponents $p,q>1$ satisfy $p<n/(n-2)$ and $q<(n-1)/(n-2)$ if
$n>2$ (or $p<(n+2)/(n-2)$ and $q<n/(n-2)$ if $n>2$ and some symmetry of the
solutions is assumed). As a typical application of our nonexistence results we
provide optimal universal estimates for positive solutions of related problems
in bounded and unbounded domains.
