80 research outputs found
Convergence of Rothe scheme for hemivariational inequalities of parabolic type
This article presents the convergence analysis of a sequence of piecewise
constant and piecewise linear functions obtained by the Rothe method to the
solution of the first order evolution partial differential inclusion
, where the multivalued term
is given by the Clarke subdifferential of a locally Lipschitz functional. The
method provides the proof of existence of solutions alternative to the ones
known in literature and together with any method for underlying elliptic
problem, can serve as the effective tool to approximate the solution
numerically. Presented approach puts into the unified framework known results
for multivalued nonmonotone source term and boundary conditions, and
generalizes them to the case where the multivalued term is defined on the
arbitrary reflexive Banach space as long as appropriate conditions are
satisfied. In addition the results on improved convergence as well as the
numerical examples are presented.Comment: to appear in: International Journal of Numerical Analysis and
Modelin
On non-autonomously forced Burgers equation with periodic and Dirichlet boundary conditions
We study the non-autonomously forced Burgers equation
on the space interval with two sets of the boundary conditions:
the Dirichlet and periodic ones. For both situations we prove that there exists
the unique bounded trajectory of this equation defined for all . Moreover we demonstrate that this trajectory attracts all
trajectories both in pullback and forward sense. We also prove that for the
Dirichlet case this attraction is exponential
Minimality properties of set-valued processes and their pullback attractors
We discuss the existence of pullback attractors for multivalued dynamical
systems on metric spaces. Such attractors are shown to exist without any
assumptions in terms of continuity of the solution maps, based only on
minimality properties with respect to the notion of pullback attraction. When
invariance is required, a very weak closed graph condition on the solving
operators is assumed. The presentation is complemented with examples and
counterexamples to test the sharpness of the hypotheses involved, including a
reaction-diffusion equation, a discontinuous ordinary differential equation and
an irregular form of the heat equation.Comment: 33 pages. A few typos correcte
On renormalized solutions to elliptic inclusions with nonstandard growth
We study the elliptic inclusion given in the following divergence form
\begin{align*}
& -\mathrm{div}\, A(x,\nabla u) \ni f\quad \mathrm{in}\quad \Omega,
& u=0\quad \mathrm{on}\quad \partial \Omega.
\end{align*}
As we assume that , the solutions to the above problem are
understood in the renormalized sense. We also assume nonstandard, possibly
nonpolynomial, heterogeneous and anisotropic growth and coercivity conditions
on the maximally monotone multifunction which necessitates the use of the
nonseparable and nonreflexive Musielak--Orlicz spaces. We prove the existence
and uniqueness of the renormalized solution as well as, under additional
assumptions on the problem data, its relation to the weak solution. The key
difficulty, the lack of a Carath\'{e}odory selection of the maximally monotone
multifunction is overcome with the use of the Minty transform
Informational structures and informational fields as a prototype for the description of postulates of the integrated information theory
Informational Structures (IS) and Informational Fields (IF) have been recently introduced
to deal with a continuous dynamical systems-based approach to Integrated Information Theory (IIT). IS and IF contain all the geometrical and topological constraints in the phase space. This allows one to characterize all the past and future dynamical scenarios for a system in any particular state. In this paper, we develop further steps in this direction, describing a proper continuous framework
for an abstract formulation, which could serve as a prototype of the IIT postulates.National Science Center of PolandUMO-2016/22/A/ST1/00077Junta de AndalucÃaMinisterio de Economia, Industria y Competitividad (MINECO). Españ
Convergence of non-autonomous attractors for subquintic weakly damped wave equation
We study the non-autonomous weakly damped wave equation with subquintic
growth condition on the nonlinearity. Our main focus is the class of
Shatah--Struwe solutions, which satisfy the Strichartz estimates and are
coincide with the class of solutions obtained by the Galerkin method. For this
class we show the existence and smoothness of pullback, uniform, and cocycle
attractors and the relations between them. We also prove that these
non-autonomous attractors converge upper-semicontinuously to the global
attractor for the limit autonomous problem if the time-dependent nonlinearity
tends to time independent function in an appropriate way
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