4,214 research outputs found
PDEs in Moving Time Dependent Domains
In this work we study partial differential equations defined in a domain that
moves in time according to the flow of a given ordinary differential equation,
starting out of a given initial domain. We first derive a formulation for a
particular case of partial differential equations known as balance equations.
For this kind of equations we find the equivalent partial differential
equations in the initial domain and later we study some particular cases with
and without diffusion. We also analyze general second order differential
equations, not necessarily of balance type. The equations without diffusion are
solved using the characteristics method. We also prove that the diffusion
equations, endowed with Dirichlet boundary conditions and initial data, are
well posed in the moving domain. For this we show that the principal part of
the equivalent equation in the initial domain is uniformly elliptic. We then
prove a version of the weak maximum principle for an equation in a moving
domain. Finally we perform suitable energy estimates in the moving domain and
give sufficient conditions for the solution to converge to zero as time goes to
infinity.Comment: pp 559-577. Without Bounds: A Scientific Canvas of Nonlinearity and
Complex Dynamics (2013) p. 36
Non-local quasilinear parabolic equations
This is a survey of the most common approaches to quasi-linear parabolic evolution equations, a discussion of their advantages and drawbacks, and a presentation of an entirely new approach based on maximal regularity. The general results here apply, above all, to parabolic initial-boundary value problems that are non-local in time. This is illustrated by indicating their relevance for quasi-linear parabolic equations with memory and, in particular, for time-regularized versions of the Perona–Malik equation of image processing
A Class of Free Boundary Problems with Onset of a new Phase
A class of diffusion driven Free Boundary Problems is considered which is
characterized by the initial onset of a phase and by an explicit kinematic
condition for the evolution of the free boundary. By a domain fixing change of
variables it naturally leads to coupled systems comprised of a singular
parabolic initial boundary value problem and a Hamilton-Jacobi equation. Even
though the one dimensional case has been thoroughly investigated, results as
basic as well-posedness and regularity have so far not been obtained for its
higher dimensional counterpart. In this paper a recently developed regularity
theory for abstract singular parabolic Cauchy problems is utilized to obtain
the first well-posedness results for the Free Boundary Problems under
consideration. The derivation of elliptic regularity results for the underlying
static singular problems will play an important role
On the Maxwell-Stefan approach to multicomponent diffusion
We consider the system of Maxwell-Stefan equations which describe
multicomponent diffusive fluxes in non-dilute solutions or gas mixtures. We
apply the Perron-Frobenius theorem to the irreducible and quasi-positive matrix
which governs the flux-force relations and are able to show normal ellipticity
of the associated multicomponent diffusion operator. This provides
local-in-time wellposedness of the Maxwell-Stefan multicomponent diffusion
system in the isobaric, isothermal case.Comment: Based on a talk given at the Conference on Nonlinear Parabolic
Problems in Bedlewo, Mai 200
Positive solutions to indefinite Neumann problems when the weight has positive average
We deal with positive solutions for the Neumann boundary value problem
associated with the scalar second order ODE where is positive on and is an indefinite weight. Complementary to previous
investigations in the case , we provide existence results
for a suitable class of weights having (small) positive mean, when
at infinity. Our proof relies on a shooting argument for a suitable equivalent
planar system of the type with
a continuous function defined on the whole real line.Comment: 17 pages, 3 figure
Global Continua of Positive Equilibria for some Quasilinear Parabolic Equation with a Nonlocal Initial Condition
This paper is concerned with a quaslinear parabolic equation including a
nonlinear nonlocal initial condition. The problem arises as equilibrium
equation in population dynamics with nonlinear diffusion. We make use of global
bifurcation theory to prove existence of an unbounded continuum of positive
solutions
Uniqueness of a Negative Mode About a Bounce Solution
We consider the uniqueness problem of a negative eigenvalue in the spectrum
of small fluctuations about a bounce solution in a multidimensional case. Our
approach is based on the concept of conjugate points from Morse theory and is a
natural generalization of the nodal theorem approach usually used in one
dimensional case. We show that bounce solution has exactly one conjugate point
at with multiplicity one.Comment: 4 pages,LaTe
Existence of positive solutions of a superlinear boundary value problem with indefinite weight
We deal with the existence of positive solutions for a two-point boundary
value problem associated with the nonlinear second order equation
. The weight is allowed to change its sign. We assume
that the function is
continuous, and satisfies suitable growth conditions, so as the case
, with , is covered. In particular we suppose that is
large near infinity, but we do not require that is non-negative in a
neighborhood of zero. Using a topological approach based on the Leray-Schauder
degree we obtain a result of existence of at least a positive solution that
improves previous existence theorems.Comment: 12 pages, 4 PNG figure
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