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 $L^1$-coercivity. We use the concept of renormalized solutions and higher differentiability techniques to prove existence and uniqueness of weak solution with $L^1$-integrable flux for all values of a positive model parameter $a$. If this parameter is smaller than $2/(d+1)$, where $d$ denotes the spatial dimension, we obtain higher integrability of the flux. We also relate the studied problem to problems in fluid mechanics.