Renormalized solutions to nonlinear parabolic problems with blowing up coefficients and general measure data (Q2281548)

Let $\Omega$ be a bounded domain in $\mathbb{R}^n,$ $T>0$ and $Q\Omega \times (0,T)$. The authors introduce a new definition of renormalized solutions to a quasilinear parabolic problem associated to a diffusion type equation having continuous coefficients blowing up for a finite value of the known term $\mu \in M_b (Q)$ and an initial data $u_0 \in L^1(\Omega)$ The main result is based on approximate problems whose solutions satisfy the a priori estimates. The authors end the paper with the proof of the main result under more restrictive conditions on test functions. The main argument relies on approximation properties of the measure, with respect to the nonlinear potential of the data and of the truncated potential. The authors use generalized version of this convolution result which uses a mean regularization together with the singular part of $\mu$. The existence of renormalized solution is, then, obtained by passing to the limit in the difference of diffuse and singular terms, and this is where they use the assumption that $\mu$ is equidiffuse.