Existence results for a class of nonlinear parabolic equations with two lower order terms
Volume 41 / 2014
Applicationes Mathematicae 41 (2014), 207-219
MSC: Primary 47A15; Secondary 46A32, 47D20.
DOI: 10.4064/am41-2-8
Abstract
We investigate the existence of renormalized solutions for some nonlinear parabolic problems associated to equations of the form $$\begin{cases} \frac{\partial (e^{\beta u}-1)}{\partial t}-{\rm div}(|\nabla u|^{p-2}\nabla u)+ {\rm div}(c(x,t)|u|^{s-1}u)+b(x,t)|\nabla u|^{r}=f &{\rm in}\ Q=\varOmega\times (0,T),\\ u(x,t)=0 &{\rm on}\ \partial\varOmega \times(0,T),\\ (e^{\beta u}-1)(x,0)=(e^{\beta u_{0}}-1)(x) &{\rm in}\ \varOmega. \end{cases} $$ with $ s =\frac{N+2}{N+p}(p-1)$, $\displaystyle c(x,t)\in (L^{\tau}(Q_{T}))^{N}$, $ \tau =\frac{N+p}{p-1}$, $ r =\frac{N(p-1)+p}{N+2}$, $ b(x,t)\in L^{N+2,1}(Q_{T})$ and $ f\in L^{1}(Q).$
