  We study a nonlinear evolutionary partial differential equation that can be
viewed as a generalization of the heat equation where the temperature gradient
is a~priori bounded but the heat flux provides merely \mbox{$L^1$-coercivity}.
Applying higher differentiability techniques in space and time, choosing a
special weighted norm (equivalent to the Euclidean norm in $\mathbb{R}^d$),
incorporating finer properties of integrable functions and using the concept of
renormalized solution, we prove long-time and large-data existence and
uniqueness of weak solution, with an $L^1$-integrable flux, to an initial
spatially-periodic problem for all values of a positive model parameter. If
this parameter is smaller than $2/(d+1)$, where $d$ denotes the spatial
dimension, we obtain higher integrability of the flux. As the developed
approach is not restricted to a scalar equation, we also present an analogous
result for nonlinear parabolic systems in which the nonlinearity, being the
gradient of a strictly convex function, gives an a-priori $L^\infty$-bound on
the gradient of the unknown solution.
