Continuous dependence on the geometry and on the initial time for a class of parabolic problems II

Extending the investigations initiated in an earlier paper, the authors deal in this paper with the solutions of another class of initial‐boundary value problems for which continuous dependence inequalities on the geometry and the initial time are established. Copyright © 2007 John Wiley & Sons, Ltd.     

Blow up,decay bounds and continuous dependence inequalities for a class of quasilinear parabolic problems

This paper deals with a class of semilinear parabolic problems. In particular, we establish conditions on the data sufficient to guarantee blow up of solution at some finite time, as well as conditions which will insure that the solution exists for all time with exponential decay of the solution and its spatial derivatives. In the case of global existence, we also investigate the continuous dependence of the solution with respect to some data of the problem. Copyright © 2005 John Wiley & Sons, Ltd.     

Continuous dependence results for a class of nonlinear heat equations in a cylindrical region

In this paper, we derive bounds for the solutions of a quasilinear heat equation in a finite cylindrical region if the far end and the lateral surface are held at zero temperature, and a nonzero temperature is applied at the near end. Some continuous dependence inequalities are also obtained. We also investigate the case in which a given heat flux is prescribed at the near end, instead of a given temperature. Copyright © 2008 John Wiley & Sons, Ltd.     

Some remarks on the asymptotic behaviour of the solutions of a class of parabolic problems

This paper deals with a class of semilinear parabolic problems. We establish sufficient conditions on the data forcing the solution to blow up at finite time τ and derive an upper bound for τ. Moreover, we show that if the problem is modified in some way, the solution decays exponentially in time and depends continuously on the data. Copyright © 2005 John Wiley & Sons, Ltd.     

Spatial decay bounds and continuous dependence on the data for a class of parabolic initial-boundary value problems

In this paper we study a semilinear heat equation in a long cylindrical region if the far end and the lateral surface are held at zero temperature and a nonzero temperature is applied at the near end. Our aim is to derive some explicit spatial decay bounds for the solution and its derivatives and to show that the solution depends continuously on the data at the near end of the cylinder.     

Continuous dependence on spatial geometry for the generalized Maxwell–Cattaneo system

In this paper the authors establish continuous dependence of the temperature on the spatial geometry in an initial‐boundary value problem for the generalized Maxwell–Cattaneo system of equations. Copyright © 2001 John Wiley & Sons, Ltd.     

Continuous dependence on the time and spatial geometry for the navier-stokes equations

In this paper we derive explicit a priori inequalities which imply stability of the solution of the initial-boundary value problem for the Navier-Stokes equations under perturbations of the initial time geometry and of the spatial geometry. These inequalities bound the solution perturbation In L₂ in terms of some well defined measure of the perturbation in geometry. We establish continuous dependence on spatial geometry in both two and three dimensions and continuous dependence on initial geometry in two dimensions. In the latter problem the three dimensional case will be somewhat more complicated due to the different form of the Sobolev inequality.     

Continuous dependence inequalities for a class of quasilinear parabolic problems

In this paper, we establish continuous dependence inequalities for the solutions u(x, t) of a class of nonlinear parabolic initial-boundary value problems and their gradients when the data are subject to variations. Dedicated to Joseph Hersch on the occasion of his 80th birthday (Received: February 24, 2005; revised: March 14, 2005)     

Continuous dependence inequalities for a class of quasilinear parabolic problems

In this paper, we establish continuous dependence inequalities for the solutions u(x, t) of a class of nonlinear parabolic initial-boundary value problems and their gradients when the data are subject to variations.     

Continuous dependence on initial geometry for a class of abstract equations in Hilbert space

Conditions are given under which members of a class of uniformly bounded solutions to the Cauchy problem associated with equations of the form Mu_tt ? Nu = F, in Hilbert space, depend continuously on perturbations of the initial geometry; our results generalize a similar theorem of Knops and Payne for classical solutions to initial-boundary value problems in linear elastodynamics.     

Smooth solutions to a class of free boundary parabolic problems

We establish existence, uniqueness, and regularity results for solutions to a class of free boundary parabolic problems, including the free boundary heat equation which arises in the so-called ``focusing problem' in the mathematical theory of combustion. Such solutions are proved to be smooth with respect to time for positive , if the data are smooth.

     

A continuous dependence result for a nonstandard system of phase field equations

The present note deals with a nonstandard system of differential equations describing a two‐species phase segregation. This system naturally arises in the asymptotic analysis carried out recently by the same authors, as the diffusion coefficient in the equation governing the evolution of the order parameter tends to zero. In particular, an existence result has been proved for the limit system in a very general framework. On the contrary, uniqueness was shown by assuming a constant mobility coefficient. Here, we generalize this result and prove a continuous dependence property in the case that the mobility coefficient suitably depends on the chemical potential. Copyright © 2013 John Wiley & Sons, Ltd.     

Continuous dependence on modelling for a complex Ginzburg–Landau equation with complex coefficients

Continuous dependence on a modelling parameter is established for solutions of a problem for a complex Ginzburg–Landau equation. A homogenizing boundary condition is also used to discuss the continuous dependence results. We derive a priori estimates that indicate that solutions depend continuously on a parameter in the governing differential equation. Copyright © 2004 John Wiley & Sons, Ltd.     

A note on continuous dependence of solutions of dynamic equations on time scales

We give a precise formulation and a proof as constructive as possible of the widely accepted claim that solutions of a dynamic equation depend continuously on the base time scale. Our approach to this problem is via Euler polygons which opens possibilities for development of numerical analysis of dynamic equations on time scales.     

Continuous dependence on the initial and flux functions for solutions of conservation laws

We prove the continuous dependence on the initial and flux functions for the entropy solutions to the Cauchy problem for conservation laws. Accordingly, we can show that the continuous dependence on the flux function for the entropy solutions depends only on the sup norm, not on the Lipschitz norm.     

一类非线性演化方程初值问题的幂级数解

研究了一类非线性演化方程初值问题.通过不变子空间方法,这类初值问题被约化为常微分方程组的初值问题.这类初值问题是适定的.本文给出了这类初值问题关于时间变量t的幂级数解.