2,691 research outputs found
Multiple positive solutions of parabolic systems with nonlinear, nonlocal initial conditions
In this paper we study the existence, localization and multiplicity of
positive solutions for parabolic systems with nonlocal initial conditions. In
order to do this, we extend an abstract theory that was recently developed by
the authors jointly with Radu Precup, related to the existence of fixed points
of nonlinear operators satisfying some upper and lower bounds. Our main tool is
the Granas fixed point index theory. We also provide a non-existence result and
an example to illustrate our theory.Comment: 28 pages, 1 figure. arXiv admin note: text overlap with
arXiv:1401.135
On an application of Tikhonov's fixed point theorem to a nonlocal Cahn-Hilliard type system modeling phase separation
This paper investigates a nonlocal version of a model for phase separation on
an atomic lattice that was introduced by P. Podio-Guidugli in Ric. Mat. 55
(2006) 105-118. The model consists of an initial-boundary value problem for a
nonlinearly coupled system of two partial differential equations governing the
evolution of an order parameter and the chemical potential. Singular
contributions to the local free energy in the form of logarithmic or
double-obstacle potentials are admitted. In contrast to the local model, which
was studied by P. Podio-Guidugli and the present authors in a series of recent
publications, in the nonlocal case the equation governing the evolution of the
order parameter contains in place of the Laplacian a nonlocal expression that
originates from nonlocal contributions to the free energy and accounts for
possible long-range interactions between the atoms. It is shown that just as in
the local case the model equations are well posed, where the technique of
proving existence is entirely different: it is based on an application of
Tikhonov's fixed point theorem in a rather unusual separable and reflexive
Banach space.Comment: The paper is dedicated to our friend Paolo Podio-Guidugli on the
occasion of his 75th birthday with best wishe
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