# On the Existence of Regular Global Attractor for $$p$$ p -Laplacian Evolution Equation

On the Existence of Regular Global Attractor for $$p$$ p -Laplacian Evolution Equation In this study, we consider the nonlinear evolution equation of parabolic type \begin{aligned} u_t-\mathrm{div}\big (|\nabla u|^{p-2}\nabla u\big )+f(u)=g. \end{aligned} u t - div ( | ∇ u | p - 2 ∇ u ) + f ( u ) = g . We analyze the long time dynamics (in the sense of global attractors) under very general conditions on the nonlinearity $$f$$ f . Since we do not assume any polynomial growth condition on it, the main difficulty arises at first in the proof of well-posedness. In fact, the very first contribution to this problem is a pioneering paper (Efendiev and Ôtano, Differ Int Equ 20:1201–1209, 2007 ) where the well-posedness result has been shown by exploiting the technique from the theory of maximal monotone operators. However, from some physical aspects, to obtain the solution in variational sense might be demanding which requires limiting procedure on the approximate solutions. In this work, we are interested in variational (weak) solution. The critical issue in the proof of well-posedness is to deal with the limiting procedure on $$f$$ f which is overcome utilizing the weak convergence tecniques in Orlicz spaces (Geredeli and Khanmamedov, Commun Pure Appl Anal 12:735–754, 2013 ; Krasnosel’skiĭ and Rutickiĭ, Convex functions and Orlicz spaces, 1961 ). Then, proving the existence of the global attractors in $$L^2(\Omega )$$ L 2 ( Ω ) and in more regular space $$W_0^{1,p}(\Omega )$$ W 0 1 , p ( Ω ) , we show that they coincide. In addition, if $$f$$ f is monotone and $$g=0$$ g = 0 , we give an explicit estimate of the decay rate to zero of the solution. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

# On the Existence of Regular Global Attractor for $$p$$ p -Laplacian Evolution Equation

, Volume 71 (3) – Jun 1, 2015
16 pages

/lp/springer_journal/on-the-existence-of-regular-global-attractor-for-p-p-laplacian-TTF7dLo15D
Publisher
Springer Journals
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-014-9268-y
Publisher site
See Article on Publisher Site

### Abstract

In this study, we consider the nonlinear evolution equation of parabolic type \begin{aligned} u_t-\mathrm{div}\big (|\nabla u|^{p-2}\nabla u\big )+f(u)=g. \end{aligned} u t - div ( | ∇ u | p - 2 ∇ u ) + f ( u ) = g . We analyze the long time dynamics (in the sense of global attractors) under very general conditions on the nonlinearity $$f$$ f . Since we do not assume any polynomial growth condition on it, the main difficulty arises at first in the proof of well-posedness. In fact, the very first contribution to this problem is a pioneering paper (Efendiev and Ôtano, Differ Int Equ 20:1201–1209, 2007 ) where the well-posedness result has been shown by exploiting the technique from the theory of maximal monotone operators. However, from some physical aspects, to obtain the solution in variational sense might be demanding which requires limiting procedure on the approximate solutions. In this work, we are interested in variational (weak) solution. The critical issue in the proof of well-posedness is to deal with the limiting procedure on $$f$$ f which is overcome utilizing the weak convergence tecniques in Orlicz spaces (Geredeli and Khanmamedov, Commun Pure Appl Anal 12:735–754, 2013 ; Krasnosel’skiĭ and Rutickiĭ, Convex functions and Orlicz spaces, 1961 ). Then, proving the existence of the global attractors in $$L^2(\Omega )$$ L 2 ( Ω ) and in more regular space $$W_0^{1,p}(\Omega )$$ W 0 1 , p ( Ω ) , we show that they coincide. In addition, if $$f$$ f is monotone and $$g=0$$ g = 0 , we give an explicit estimate of the decay rate to zero of the solution.

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Jun 1, 2015

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly ### Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. ### Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve ### Freelancer DeepDyve ### Pro Price FREE$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations