Finite dimensional global attractor for a class of doubly nonlinear parabolic equations

2006 ◽  
Vol 4 (1) ◽  
pp. 163-182 ◽  
Author(s):  
Alain Miranville
Keyword(s):  
Global Attractor ◽  
Parabolic Equations ◽  
Nonlinear Parabolic Equations ◽  
Time Behavior ◽  
Finite Dimensional ◽  
Nonlinear Parabolic ◽  
Long Time ◽  
Two Space Dimensions ◽  
Doubly Nonlinear ◽  
Finite Fractal Dimension

AbstractOur aim in this paper is to study the long time behavior of a class of doubly nonlinear parabolic equations. In particular, we prove the existence of the global attractor which has, in one and two space dimensions, finite fractal dimension.

scholarly journals Global Attractor for Doubly Nonlinear Parabolic Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-16
Author(s):  
Yongjun Li ◽  
Suyun Wang ◽  
Yanhong Zhang
Keyword(s):  
Parabolic Equation ◽  
Global Attractor ◽  
Parabolic Equations ◽  
Nonlinear Parabolic Equations ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Nonlinear Parabolic ◽  
Long Time ◽  
Doubly Nonlinear ◽  
Doubly Nonlinear Parabolic Equation

Our aim in this paper is to study the long-time behavior for a class of doubly nonlinear parabolic equations. First we show that the problem has a unique solution. Then we prove that the semigroup corresponding to the problem is norm-to-weak continuous in Lq and H01. Finally we establish the existence of global attractor of the problem in Lq and H01.

scholarly journals Harnack’s inequality for doubly nonlinear equations of slow diffusion type

2021 ◽  
Vol 60 (6) ◽  
Author(s):  
Verena Bögelein ◽  
Andreas Heran ◽  
Leah Schätzler ◽  
Thomas Singer
Keyword(s):  
Vector Field ◽  
Harnack Inequality ◽  
Parabolic Equations ◽  
Full Range ◽  
Nonlinear Parabolic Equations ◽  
Growth Conditions ◽  
Diffusion Type ◽  
Slow Diffusion ◽  
Nonlinear Parabolic ◽  
Doubly Nonlinear

AbstractIn this article we prove a Harnack inequality for non-negative weak solutions to doubly nonlinear parabolic equations of the form $$\begin{aligned} \partial _t u - {{\,\mathrm{div}\,}}{\mathbf {A}}(x,t,u,Du^m) = {{\,\mathrm{div}\,}}F, \end{aligned}$$ ∂ t u - div A ( x , t , u , D u m ) = div F , where the vector field $${\mathbf {A}}$$ A fulfills p-ellipticity and growth conditions. We treat the slow diffusion case in its full range, i.e. all exponents $$m > 0$$ m > 0 and $$p>1$$ p > 1 with $$m(p-1) > 1$$ m ( p - 1 ) > 1 are included in our considerations.

The coupled Cahn–Hilliard/Allen–Cahn system with dynamic boundary conditions

2021 ◽  
pp. 1-27
Author(s):  
Ahmad Makki ◽  
Alain Miranville ◽  
Madalina Petcu
Keyword(s):  
Fractal Dimension ◽  
Boundary Conditions ◽  
Well Posedness ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Dynamic Boundary Conditions ◽  
Finite Dimensional ◽  
Dynamic Boundary ◽  
Long Time ◽  
Finite Fractal Dimension

In this article, we are interested in the study of the well-posedness as well as of the long time behavior, in terms of finite-dimensional attractors, of a coupled Allen–Cahn/Cahn–Hilliard system associated with dynamic boundary conditions. In particular, we prove the existence of the global attractor with finite fractal dimension.

Intrinsic scaling method for doubly nonlinear parabolic equations and its application

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Masashi Misawa ◽  
Kenta Nakamura
Keyword(s):  
Parabolic Equation ◽  
Parabolic Equations ◽  
Large Data ◽  
Nonlinear Parabolic Equations ◽  
Sobolev Type ◽  
Scaling Method ◽  
Nonlinear Parabolic ◽  
Doubly Nonlinear ◽  
Doubly Nonlinear Equation ◽  
Doubly Nonlinear Parabolic Equation

Abstract In this article, we consider a fast diffusive type doubly nonlinear parabolic equation, called 𝑝-Sobolev type flows, and devise a new intrinsic scaling method to transform the prototype doubly nonlinear equation to the 𝑝-Sobolev type flows. As an application, we show the global existence and regularity for the 𝑝-Sobolev type flows with large data.

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