Strongly nonlinear parabolic equations with natural growth terms in Orlicz spaces (Q703404)

The paper deals with the parabolic initial-boundary value problem \[ \begin{aligned} {\partial u\over\partial t}+ A(u)+ g(t,x,u,\nabla u)= f&\quad\text{in }Q,\\ u(x,t)= 0 &\quad\text{on }\partial\Omega\times (0,T),\\ u(x,0)= u_0(x)&\quad\text{in }\Omega,\end{aligned}\tag{1} \] where \(\Omega\) is a bounded open set in \(\mathbb{R}^N\) \((N\geq 2)\), \(Q= \Omega\times (0,T)\) with \(T> 0\) and \(A(u)= -\text{div}(a(x, t,u\nabla u))\) with \(a: \Omega\times [0, T]\times\mathbb{R}\times\mathbb{R}^N\to \mathbb{R}^N\) a Leray-Lions type operator satisfying some Orlicz space growth conditions. The authors prove the existence of weak solutions for (1) with the nonlinearity \(g\) having ``natural'' growth with respect to the gradient.