Semi-discretized nonlinear evolution equations as discrete dynamical systems and error analysis.

*(English)*Zbl 0732.35037The paper deals with the study of doubly nonlinear parabolic equations of the type
\[
(P)\quad \frac{d\beta (u)}{dt}-\Delta u+g(t,x,u)=0;\quad \beta (t=0)=\beta (u_ 0)
\]
or of their discrete version with respect to time
\[
(P_ n)\quad \beta (U^ n)-\tau \Delta U^ n+\tau g(n\tau,x,U^ n)=\beta (U^ n);\quad U^ n=u_ 0
\]
with homogeneous Dirichlet boundary condition.

Under suitable assumptions on the data (including \(\beta\) is increasing and onto) existence results and \(L^{\infty}\) estimates are given in the case where \(u_ 0\) belongs to \(L^{\infty}\) or to \(L^ 2\). More restrictive assumptions on g and \(\beta\) leads to uniqueness. A stability analysis of the discrete problem is also performed. A few error estimates for \(u-U^ n\) follow. As far as the semi-discrete dynamical system associated to \((P_ n)\) is concerned the existence of an absorbing set in \(L^{\infty}\) and in \(H^ 1_ 0\) is proved. When g is independent of t this leads to the existence of a global attractor which is shown to be in \(H^ 2\) under the additional assumption that \(\beta '\geq \epsilon >0\).

Under suitable assumptions on the data (including \(\beta\) is increasing and onto) existence results and \(L^{\infty}\) estimates are given in the case where \(u_ 0\) belongs to \(L^{\infty}\) or to \(L^ 2\). More restrictive assumptions on g and \(\beta\) leads to uniqueness. A stability analysis of the discrete problem is also performed. A few error estimates for \(u-U^ n\) follow. As far as the semi-discrete dynamical system associated to \((P_ n)\) is concerned the existence of an absorbing set in \(L^{\infty}\) and in \(H^ 1_ 0\) is proved. When g is independent of t this leads to the existence of a global attractor which is shown to be in \(H^ 2\) under the additional assumption that \(\beta '\geq \epsilon >0\).

Reviewer: D.Blanchard (Mont-Saint-Aignan)

##### MSC:

35K55 | Nonlinear parabolic equations |

37C70 | Attractors and repellers of smooth dynamical systems and their topological structure |

35B40 | Asymptotic behavior of solutions to PDEs |

35B35 | Stability in context of PDEs |