915 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
Multivalued anisotropic problem with Fourier boundary condition involving diffuse Radon measure data and variable exponents
We study a nonlinear anisotropic elliptic problem under Fourier type boundary condition governed by a general anisotropic operator with variable exponents and diffuse Radon measure data which does not charge the sets of zero p(·)-capacity. We prove an existence and uniqueness result of entropy or renormalized solution
Analysis of a Cahn--Hilliard system with non-zero Dirichlet conditions modeling tumor growth with chemotaxis
We consider a diffuse interface model for tumor growth consisting of a
Cahn--Hilliard equation with source terms coupled to a reaction-diffusion
equation, which models a tumor growing in the presence of a nutrient species
and surrounded by healthy tissue. The well-posedness of the system equipped
with Neumann boundary conditions was found to require regular potentials with
quadratic growth. In this work, Dirichlet boundary conditions are considered,
and we establish the well-posedness of the system for regular potentials with
higher polynomial growth and also for singular potentials. New difficulties are
encountered due to the higher polynomial growth, but for regular potentials, we
retain the continuous dependence on initial and boundary data for the chemical
potential and for the order parameter in strong norms as established in the
previous work. Furthermore, we deduce the well-posedness of a variant of the
model with quasi-static nutrient by rigorously passing to the limit where the
ratio of the nutrient diffusion time-scale to the tumor doubling time-scale is
small.Comment: 33 pages, minor typos corrected, accepted versio
Analytic and Asymptotic Methods for Nonlinear Singularity Analysis: a Review and Extensions of Tests for the Painlev\'e Property
The integrability (solvability via an associated single-valued linear
problem) of a differential equation is closely related to the singularity
structure of its solutions. In particular, there is strong evidence that all
integrable equations have the Painlev\'e property, that is, all solutions are
single-valued around all movable singularities. In this expository article, we
review methods for analysing such singularity structure. In particular, we
describe well known techniques of nonlinear regular-singular-type analysis,
i.e. the Painlev\'e tests for ordinary and partial differential equations. Then
we discuss methods of obtaining sufficiency conditions for the Painlev\'e
property. Recently, extensions of \textit{irregular} singularity analysis to
nonlinear equations have been achieved. Also, new asymptotic limits of
differential equations preserving the Painlev\'e property have been found. We
discuss these also.Comment: 40 pages in LaTeX2e. To appear in the Proceedings of the CIMPA Summer
School on "Nonlinear Systems," Pondicherry, India, January 1996, (eds) B.
Grammaticos and K. Tamizhman
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