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 $L_p$ 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

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

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

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 $$u" + q(t)g(u) = 0, \quad t \in [0, T],$$ where $g: [0, +\infty[\, \to \mathbb{R}$ is positive on $\,]0, +\infty[\,$ and $q(t)$ is an indefinite weight. Complementary to previous investigations in the case $\int_0^T q(t) < 0$, we provide existence results for a suitable class of weights having (small) positive mean, when $g'(x) < 0$ at infinity. Our proof relies on a shooting argument for a suitable equivalent planar system of the type $$x' = y, \qquad y' = h(x)y^2 + q(t),$$ with $h(x)$ 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 $\tau=0$ 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 $u''+a(x)g(u)=0$. The weight $a(x)$ is allowed to change its sign. We assume that the function $g\colon\mathopen{[}0,+\infty\mathclose{[}\to\mathbb{R}$ is continuous, $g(0)=0$ and satisfies suitable growth conditions, so as the case $g(s)=s^{p}$, with $p>1$, is covered. In particular we suppose that $g(s)/s$ is large near infinity, but we do not require that $g(s)$ 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