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B q for parabolic measures

Caroline Sweezy (1998)

Studia Mathematica

If Ω is a Lip(1,1/2) domain, μ a doubling measure on ∂ p Ω , ∂ / ∂ t - L i , i = 0,1, are two parabolic-type operators with coefficients bounded and measurable, 2 ≤ q < ∞, then the associated measures ω 0 , ω 1 have the property that ω 0 ∈ B q ( μ ) implies ω 1 is absolutely continuous with respect to ω 0 whenever a certain Carleson-type condition holds on the difference function of the coefficients of L 1 and L 0 . Also ω 0 ∈ B q ( μ ) implies ω 1 ∈ B q ( μ ) whenever both measures are center-doubling measures. This is B. Dahlberg’s result for elliptic measures extended...

Bifurcations for Turing instability without SO(2) symmetry

Toshiyuki Ogawa, Takashi Okuda (2007)

Kybernetika

In this paper, we consider the Swift–Hohenberg equation with perturbed boundary conditions. We do not a priori know the eigenfunctions for the linearized problem since the SO ( 2 ) symmetry of the problem is broken by perturbation. We show that how the neutral stability curves change and, as a result, how the bifurcation diagrams change by the perturbation of the boundary conditions.

Bifurcations in a modulation equation for alternans in a cardiac fiber

Shu Dai, David G. Schaeffer (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

While alternans in a single cardiac cell appears through a simple period-doubling bifurcation, in extended tissue the exact nature of the bifurcation is unclear. In particular, the phase of alternans can exhibit wave-like spatial dependence, either stationary or travelling, which is known as discordant alternans. We study these phenomena in simple cardiac models through a modulation equation proposed by Echebarria-Karma. As shown in our previous paper, the zero solution of their equation may lose...

Bifurcations of invariant measures in discrete-time parameter dependent cocycles

Anastasia Maltseva, Volker Reitmann (2015)

Mathematica Bohemica

We consider parameter-dependent cocycles generated by nonautonomous difference equations. One of them is a discrete-time cardiac conduction model. For this system with a control variable a cocycle formulation is presented. We state a theorem about upper Hausdorff dimension estimates for cocycle attractors which includes some regulating function. We also consider the existence of invariant measures for cocycle systems using some elements of Perron-Frobenius theory and discuss the bifurcation of parameter-dependent...

Blow-up results for some reaction-diffusion equations with time delay

Hongliang Wang, Yujuan Chen, Haihua Lu (2012)

Annales Polonici Mathematici

We discuss the effect of time delay on blow-up of solutions to initial-boundary value problems for nonlinear reaction-diffusion equations. Firstly, two examples are given, which indicate that the delay can both induce and prevent the blow-up of solutions. Then we show that adding a new term with delay may not change the blow-up character of solutions.

Boundary value problems and duality between Lp Dirichlet and regularity problems for second order parabolic systems in non-cylindrical domains.

Kaj Nyström (2006)

Collectanea Mathematica

In this paper we consider general second order, symmetric and strongly elliptic parabolic systems with real valued and constant coefficients in the setting of a class of time-varying, non-smooth infinite cylindersΩ = {(x0,x,t) ∈ R x Rn-1 x R: x0 &gt; A(x,t)}.We prove solvability of Dirichlet, Neumann as well as regularity type problems with data in Lp and Lp1,1/2 (the parabolic Sobolev space having tangential (spatial) gradients and half a time derivative in Lp) for p ∈ (2 − ε, 2 + ε) assuming...

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