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Semi-discretized nonlinear evolution equations as discrete dynamical systems and error analysis. (English) Zbl 0732.35037
The 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 $$\beta '\geq \epsilon >0$$.

##### 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
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