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

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Publisher
Springer US
Copyright
Copyright © 2015 by Springer Science+Business Media New York
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

References

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