Nonlinear parabolic problems with a very general quadratic gradient term

Abstract

We study existence and regularity of distributional solutions for a class of nonlinear parabolic problems. The equations we consider have a quasi-linear diffusion operator and a lower-order term, which may grow quadratically in the gradient and may have a very fast growth (for instance, exponential) with respect to the solution. The model problem we refer to is the following \begin{equation} \begin{cases} u_t -\Delta u = \beta(u) |{\nabla} u|^2 +f(x, t), & \ \text{in}\quad \Omega\times (0, T);\\[0.1cm] u(x,t)=0, & \ \text{on}\quad \partial\Omega\times (0, T);\\[0.1cm] u(x, 0)=u_0(x), & \ \text{in}\quad \Omega; \end{cases} \tag*{(0.1)} \label{modello} \end{equation} with $\Omega \subset \mathbb R^N$ a bounded open set, $T>0$, and $\beta(u)\sim e^{|u|}$; as far as the data are concerned, we assume $\exp(\exp(|u_0|))\in L^2({\Omega})$, and $f \in X(0, T; Y({\Omega}))$, where $X, Y$ are Orlicz spaces of logarithmic and exponential type, respectively. We also study a semilinear problem having a superlinear reaction term, a problem that is linked with problem (0.1) by a change of unknown (see (1.3) below). Likewise, we deal with some other related problems, which include a gradient term and a reaction term together.