Nonlinear parabolic equations with a lower order term and $L^1$ data
Rosaria Di Nardo · Communications on Pure & Applied Analysis · 2010
In this paper we prove the existence of a renormalized solution for a class ofnonlinear parabolic problems whose prototype is$\frac{\partial u}{\partial t}-\Delta_p u+$ div $(c(x,t)|u|^{\gamma-1}u) =f $ in $Q_T$$u(x,t)=0$ on $\partial\Omega\times(0,T) $$u(x,0)=u_0 (x)$ in $\Omega,$where $Q_T=\Omega\times(0,T),$ $\Omega$ is an open andbounded subset of $\mathcal{R}^N$, $N\geq2,$ $T>0,$ $\Delta_p$ is the so called $p$-Laplace operator,$\gamma=\frac{(N+2)(p-1)}{N+p},$ $c(x,t)\in(L^{\tau}(Q_{T}))^N,$ $\tau=\frac{N+p}{p-1},$ $\ f\in L^1 (Q_T), $$u_{0}\in L^1(\Omega).$