Attractors for the semiflow associated with a class of doubly nonlinear parabolic equations

2006/11/15 by Giulio Schimperna, Schimperna, Giulio, Antonio Segatti +1
Computer Science · Engineering · Mathematics · #35B40 #35B41 #35K55 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.math/0611464

openalex publication_date 2006/11/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A doubly nonlinear parabolic equation of the form α(u_t)-Δu+W'(u)= f, complemented with initial and either Dirichlet or Neumann homogeneous boundary conditions, is addressed. The two nonlinearities are given by the maximal monotone function α and by the derivative W' of a smooth but possibly nonconvex potential W; f is a known source. After defining a proper notion of solution and recalling a related existence result, we show that from any initial datum emanates at least one solution which gains further regularity for t>0. Such "regularizing solutions" constitute a semiflow S for which uniqueness is satisfied for strictly positive times and we can study long time behavior properties. In particular, we can prove existence of both global and exponential attractors and investigate the structure of ω-limits of single trajectories.

Citations

• Long-time behavior of second order evolution equations with nonlinear damping

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