Existence of renormalized solutions for parabolic equations without the sign condition and with three unbounded nonlinearities

Abstract

We study the problem \begin{gather*} \frac{\partial b(x, u)}{\partial t}-{\mathop{\rm div}}(a(x,t,u,D u))+H(x,t,u,Du) = \mu\quad \text{in } Q=\varOmega\times (0,T),\\ b(x,u)|_{t=0}=b(x,u_0)\quad\text{in } \varOmega,\\ u=0\quad \text{in } \partial\varOme