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Existence result for nonlinear parabolic problems with \(L^1\)-data. (English) Zbl 1207.35195

The authors study the existence of solutions of the nonlinear parabolic problem
\[ \begin{aligned} \frac{\partial u}{\partial t}-\text{div}(|Du-\Theta(u)|^{p-2}(Du-\Theta(u))) +\alpha(u)=&f \quad\;\text{in } ]0, T[\times\Omega,\\ (|Du-\Theta(u)|^{p-2}(Du-\Theta(u)))\cdot \eta + \gamma(u)=&g \quad\;\text{on } ]0, T[\times\partial\Omega, \\ u(0,\cdot )=&u_0 \quad \text{in } \Omega,\end{aligned} \]
with initial data in \(L^1\). Such kind of equations describe several physical phenomena such as the filtration of a fluid in a partially satured porous medium and the flow through a porous medium in a turbulent regime The authors use a time discretization of the continuous problem by the Euler forward scheme.

MSC:

35K61 Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations
35K59 Quasilinear parabolic equations
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