Existence result for nonlinear parabolic problems with $L^1$-data
Volume 37 / 2010
Applicationes Mathematicae 37 (2010), 483-508
MSC: 35K55, 35K61, 35J60, 35Dxx.
DOI: 10.4064/am37-4-6
Abstract
We study the existence of solutions of the nonlinear parabolic problem $$\displaylines{ \frac{\partial u}{\partial t}-\mathop{ \rm div}[|Du-{\mit\Theta}(u)|^{p-2}(Du- {\mit\Theta}(u))] +\alpha(u)=f \quad\ \hbox{in } \mathopen{]}0, T\mathclose{[}\times{\mit\Omega},\cr (|Du-{\mit\Theta}(u)|^{p-2}(Du-{\mit\Theta}(u)))\cdot \eta + \gamma(u)=g \quad\ \hbox{on } \mathopen{]}0, T\mathclose{[}\times\partial{\mit\Omega}, \cr u(0,\cdot )=u_0 \quad\ \hbox{in } {\mit\Omega},\cr} $$ with initial data in $L^1$. We use a time discretization of the continuous problem by the Euler forward scheme.
