DCDS-B
Global attractors for $p$-Laplacian differential inclusions in unbounded domains
Jacson Simsen José Valero
Discrete & Continuous Dynamical Systems - B 2016, 21(9): 3239-3267 doi: 10.3934/dcdsb.2016096
In this work we consider a differential inclusion governed by a p-Laplacian operator with a diffusion coefficient depending on a parameter in which the space variable belongs to an unbounded domain. We prove the existence of a global attractor and show that the family of attractors behaves upper semicontinuously with respect to the diffusion parameter. Both autonomous and nonautonomous cases are studied.
keywords: upper semicontinuity. differential inclusions Unbounded domains $p$-Laplacian operator attractors
CPAA
Pullback attractors for non-autonomous evolution equations with spatially variable exponents
Peter E. Kloeden Jacson Simsen
Communications on Pure & Applied Analysis 2014, 13(6): 2543-2557 doi: 10.3934/cpaa.2014.13.2543
Dissipative problems in electrorheological fluids, porous media and image processing often involve spatially dependent exponents. They also include time-dependent terms as in equation \begin{eqnarray} \frac{\partial u_\lambda}{\partial t}(t)-\textrm{div}(D_\lambda(t)|\nabla u_\lambda(t)|^{p(x)-2}\nabla u_\lambda(t))+|u_\lambda(t)|^{p(x)-2}u_\lambda(t) = B(t,u_\lambda(t)) \end{eqnarray} on a bounded smooth domain $\Omega$ in $\mathbb{R}^n$, $n\geq 1$, with a homogeneous Neumann boundary condition, where the exponent $p(\cdot)\in C(\bar{\Omega}, \mathbb{R}^+)$ satisfying $p^-$ $:=$ $\min p(x)$ $>$ $2$, and $\lambda$ $\in$ $[0,\infty)$ is a parameter.
The existence and upper semicontinuity of pullback attractors are established for this equation under the assumptions, amongst others, that $B$ is globally Lipschitz in its second variable and $D_\lambda$ $ \in $ $L^\infty([\tau,T] \times \Omega, \mathbb{R}^+)$ is bounded from above and below, is monotonically nonincreasing in time and continuous in the parameter $\lambda$. The global existence and uniqueness of strong solutions is obtained through results of Yotsutani.
keywords: upper semicontinuity. variable exponents pullback attractors Non-autonomous parabolic problems
DCDS-B
On asymptotically autonomous dynamics for multivalued evolution problems
Jacson Simsen Mariza Stefanello Simsen
Discrete & Continuous Dynamical Systems - B 2017, 22(11): 1-11 doi: 10.3934/dcdsb.2018278

In this work we improve the result presented by Kloeden-Simsen-Stefanello Simsen in [8] by reducing uniform conditions. We prove theoretical results in order to establish convergence in the Hausdorff semi-distance of the component subsets of the pullback attractor of a non-autonomous multivalued problem to the global attractor of the corresponding autonomous multivalued problem.

keywords: Multivalued problem pullback attractors asymptotically autonomous dynamics variable exponents
DCDS-B
Non-autonomous reaction-diffusion equations with variable exponents and large diffusion
Antonio Carlos Fernandes Marcela Carvalho Gonçcalves Jacson Simsen
Discrete & Continuous Dynamical Systems - B 2017, 22(11): 1-26 doi: 10.3934/dcdsb.2018217
In this work we prove continuity of solutions with respect to initial conditions and a couple of parameters and we prove upper semicontinuity of a family of pullback attractors for the problem
$\left\{ {\begin{array}{*{20}{l}}{\frac{{\partial {u_s}}}{{\partial t}}(t) - {D_s}{\rm{div}}(|\nabla {u_s}{|^{{p_s}(x) - 2}}\nabla {u_s}) + C(t)|{u_s}{|^{{p_s}(x) - 2}}{u_s} = B({u_s}(t)),\;\;t > \tau ,}\\{{u_s}(\tau ) = {u_{\tau s}},}\end{array}} \right.$
under homogeneous Neumann boundary conditions,
$u_{τ s}∈ H: = L^2(Ω),$
$Ω\subset\mathbb{R}^n$
(
$n≥ 1$
) is a smooth bounded domain,
$B:H \to H$
is a globally Lipschitz map with Lipschitz constant
$L≥ 0$
,
$D_s∈[1,∞)$
,
$C(·)∈ L^{∞}([τ, T];\mathbb{R}^+)$
is bounded from above and below and is monotonically nonincreasing in time,
$p_s(·)∈ C(\bar{Ω})$
,
$p_s^-: = \textrm{min}_{x∈\bar{Ω}}\;p_s(x)≥ p,$
$p_s^+: = \textrm{max}_{x∈\bar{Ω}}\;p_s(x)≤ a,$
for all
$s∈ \mathbb{N},$
when
$p_s(·) \to p$
in
$L^∞(Ω)$
and
$D_s \to ∞$
as
$s \to∞,$
with
$a,p>2$
positive constants.
keywords: Reaction-diffusion equations parabolic problems variable exponents pullback attractors upper semicontinuity
CPAA
Reaction-Diffusion equations with spatially variable exponents and large diffusion
Jacson Simsen Mariza Stefanello Simsen Marcos Roberto Teixeira Primo
Communications on Pure & Applied Analysis 2016, 15(2): 495-506 doi: 10.3934/cpaa.2016.15.495
In this work we prove continuity of solutions with respect to initial conditions and couple parameters and we prove joint upper semicontinuity of a family of global attractors for the problem \begin{eqnarray} &\frac{\partial u_{s}}{\partial t}(t)-\textrm{div}(D_s|\nabla u_{s}|^{p_s(x)-2}\nabla u_{s})+|u_s|^{p_s(x)-2}u_s=B(u_{s}(t)),\;\; t>0,\\ &u_{s}(0)=u_{0s}, \end{eqnarray} under homogeneous Neumann boundary conditions, $u_{0s}\in H:=L^2(\Omega),$ $\Omega\subset\mathbb{R}^n$ ($n\geq 1$) is a smooth bounded domain, $B:H\rightarrow H$ is a globally Lipschitz map with Lipschitz constant $L\geq 0$, $D_s\in[1,\infty)$, $p_s(\cdot)\in C(\bar{\Omega})$, $p_s^-:=\textrm{ess inf}\;p_s\geq p,$ $p_s^+:=\textrm{ess sup}\;p_s\leq a,$ for all $s\in \mathbb{N},$ when $p_s(\cdot)\rightarrow p$ in $L^\infty(\Omega)$ and $D_s\rightarrow\infty$ as $s\rightarrow\infty,$ with $a,p>2$ positive constants.
keywords: parabolic problems variable exponents Reaction-Diffusion equations attractors upper semicontinuity.

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