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A posteriori error estimation for semilinear parabolic optimal control problems with application to model reduction by POD

Eileen Kammann, Fredi Tröltzsch, Stefan Volkwein (2013)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We consider the following problem of error estimation for the optimal control of nonlinear parabolic partial differential equations: let an arbitrary admissible control function be given. How far is it from the next locally optimal control? Under natural assumptions including a second-order sufficient optimality condition for the (unknown) locally optimal control, we estimate the distance between the two controls. To do this, we need some information on the lowest eigenvalue of the reduced Hessian....

A strong maximum principle for the Paneitz operator and a non-local flow for the Q -curvature

Matthew J. Gursky, Andrea Malchiodi (2015)

Journal of the European Mathematical Society

In this paper we consider Riemannian manifolds ( M n , g ) of dimension n ≥ 5 , with semi-positive Q -curvature and non-negative scalar curvature. Under these assumptions we prove (i) the Paneitz operator satisfies a strong maximum principle; (ii) the Paneitz operator is a positive operator; and (iii) its Green’s function is strictly positive. We then introduce a non-local flow whose stationary points are metrics of constant positive Q -curvature. Modifying the test function construction of Esposito-Robert, we show...

Abstract parabolic problems with non-Lipschitz critical nonlinearities

Konrad J. Wąs (2010)

Colloquium Mathematicae

The Cauchy problem for a semilinear abstract parabolic equation is considered in a fractional power scale associated with a sectorial operator appearing in the linear main part of the equation. Existence of local solutions is proved for non-Lipschitz nonlinearities satisfying a certain critical growth condition.

Application of Rothe's method to a parabolic inverse problem with nonlocal boundary condition

Yong-Hyok Jo, Myong-Hwan Ri (2022)

Applications of Mathematics

We consider an inverse problem for the determination of a purely time-dependent source in a semilinear parabolic equation with a nonlocal boundary condition. An approximation scheme for the solution together with the well-posedness of the problem with the initial value u 0 ∈ H 1 ( Ω ) is presented by means of the Rothe time-discretization method. Further approximation scheme via Rothe’s method is constructed for the problem when u 0 ∈ L 2 ( Ω ) and the integral kernel in the nonlocal boundary condition is symmetric.

Asymptotic behavior of a sixth-order Cahn-Hilliard system

Alain Miranville (2014)

Open Mathematics

Our aim in this paper is to study the asymptotic behavior, in terms of finite-dimensional attractors, of a sixth-order Cahn-Hilliard system. This system is based on a modification of the Ginzburg-Landau free energy proposed in [Torabi S., Lowengrub J., Voigt A., Wise S., A new phase-field model for strongly anisotropic systems, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 2009, 465(2105), 1337–1359], assuming isotropy.

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...

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.

Certified reduced-basis solutions of viscous Burgers equation parametrized by initial and boundary values

Alexandre Janon, Maëlle Nodet, Clémentine Prieur (2013)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We present a reduced basis offline/online procedure for viscous Burgers initial boundary value problem, enabling efficient approximate computation of the solutions of this equation for parametrized viscosity and initial and boundary value data. This procedure comes with a fast-evaluated rigorous error bound certifying the approximation procedure. Our numerical experiments show significant computational savings, as well as efficiency of the error bound.

Continuity of the quenching time in a semilinear parabolic equation

Théodore Boni, Firmin N'Gohisse (2008)

Annales UMCS, Mathematica

In this paper, we consider the following initial-boundary value problem [...] where Ω is a bounded domain in RN with smooth boundary ∂Ω, p > 0, Δ is the Laplacian, v is the exterior normal unit vector on ∂Ω. Under some assumptions, we show that the solution of the above problem quenches in a finite time and estimate its quenching time. We also prove the continuity of the quenching time as a function of the initial data u0. Finally, we give some numerical results to illustrate our analysis.

Control of Traveling Solutions in a Loop-Reactor

Y. Smagina, M. Sheintuch (2010)

Mathematical Modelling of Natural Phenomena

We consider the stabilization of a rotating temperature pulse traveling in a continuous asymptotic model of many connected chemical reactors organized in a loop with continuously switching the feed point synchronously with the motion of the pulse solution. We use the switch velocity as control parameter and design it to follow the pulse: the switch velocity is updated at every step on-line using the discrepancy between the temperature at the front...

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