Backward In Time Problem Research Articles

Uniqueness for a high order ill posed problem

In this work, we study a high order derivative in time problem. First, we show that there exists a sequence of elements of the spectrum which tends to infinity and therefore, it is ill posed. Then, we prove the uniqueness of solutions for this problem by adapting the logarithmic arguments to this situation. Finally, the results are applied to the backward in time problem for the generalized linear Burgers’ fluid, a couple of heat conduction problems and a viscoelastic model.

The local well-posed results of Kirchhoff parabolic equation with nonlocal condition

This article is devoted to the Kirchhoff type parabolic problem with a nonlocal condition. Our problem has many applications in physical and biological phenomena. Applying Banach fixed point theory, we first show that the local existence of the mild solution under some prior assumptions on the input data f. Next, we investigate the convergence of the mild solution as ϵ goes to zero, and we have proven the solution of our problem to the solution of backward in time problem when T is enough small.

Terminal value problem for nonlinear parabolic equation with Gaussian white noise

<abstract><p>In this paper, We are interested in studying the backward in time problem for nonlinear parabolic equation with time and space independent coefficients. The main purpose of this paper is to study the problem of determining the initial condition of nonlinear parabolic equations from noisy observations of the final condition. The final data are noisy by the process involving Gaussian white noise. We introduce a regularized method to establish an approximate solution. We establish an upper bound on the rate of convergence of the mean integrated squared error. This article is inspired by the article by Tuan and Nane <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>.</p></abstract>

Stability for the Kelvin–Voigt variable order equations backward in time

Continuous dependence upon the initial data is established for a solution to the equations for a Kelvin–Voigt fluid of variable order, for the backward in time problem. Relatively weak restrictions are required on the base velocity field for a potentially improperly posed problem.

Existence and regularity results for terminal value problem for nonlinear fractional wave equations

We consider the terminal value problem (or called final value problem, initial inverse problem, backward in time problem) of determining the initial value, in a general class of time-fractional wave equations with Caputo derivative, from a given final value. We are concerned with the existence, regularity of solutions upon the terminal value. Under several assumptions on the nonlinearity, we address and show the well-posedness (namely, the existence, uniqueness, and continuous dependence) for the terminal value problem. Some regularity results for the mild solution and its derivatives of first and fractional orders are also derived. The effectiveness of our methods are shown by applying the results to two interesting models: time fractional Ginzburg–Landau equation, and time fractional Burgers equation, where time and spatial regularity estimates are obtained.

On the Partition of Energies for the Backward in Time Problem of Thermoelastic Materials with a Dipolar Structure

We first formulate the mixed backward in time problem in the context of thermoelasticity for dipolar materials. To prove the consistency of this mixed problem, our first main result is regarding the uniqueness of the solution for this problem. This is obtained based on some auxiliary results, namely, four integral identities. The second main result is regarding the temporal behavior of our thermoelastic body with a dipolar structure. This behavior is studied by means of some relations on a partition of various parts of the energy associated to the solution of the problem.

The Backward in Time Problem of Double Porosity Material with Microtemperature

In the present study, the theory of thermoelastodynamics is considered in the case of materials with double porosity structure and microtemperature. The novelty of this study consists in the investigation of a backward in time problem associated with double porous thermoelastic materials with microtemperature. In the first part of the paper, in case of the bounded domains the impossibility of time localization of solutions is obtained. This study is equivalent to the uniqueness of solutions for the backward in time problem. In the second part of the paper, a Phragmen-Lindelof alternative in the case of semi-infinite cylinders is obtained.

Stability of solutions of a class of nonlinear fractional diffusion equations with respect to a pseudo‐differential operator

In this paper, we study the forward and the backward in time problems for a class of nonlinear diffusion equations with respect to the pseudo‐differential operator. Herein, we investigate the stability of the solution of the forward problem in relationship with parameters of the pseudo‐differential operator and initial data. Besides, as known, the backward in time problem is instability. Hence, we give a method to regularize the solution of the backward problem in the case of the parameters are perturbed.

Harmonic vibrations in thermoelastic dynamics with double porosity structure

This paper is a study regarding the harmonic vibrations of materials with a double porosity structure. The amplitude of vibrations corresponding to the backward in time problem is analysed. The aim is to estimate the evolution of the amplitude and to obtain conservation laws.

Spatial Behavior in Thermoelastodynamics with Double Porosity Structure

In the present study, we consider the theory of thermoelastodynamics with double porosity structure. Two situations are studied: for bounded domains, the impossibility of time localization of solutions is obtained, which is equivalent to the uniqueness of solutions for the backward in time problem. For the second study, in the case of semi-infinite cylinders, a Phragmen–Lindelof alternative, as well as an upper bound for the amplitude term when solutions decay, are obtained.

Identification of the initial condition in backward problem with nonlinear diffusion and reaction

In this paper, we study the backward in time problem for nonlinear parabolic equations associated with nonlinear diffusion coefficient. The problem is ill-posed in the sense of Hadamard. Under an a priori assumption on the exact solution in Gevrey space, we propose a new regularization method for stabilising the ill-posed problem. Our results extend some earlier works on backward problems for nonlinear parabolic equations.

A porous thermoelastic diffusion theory of types II and III

In this paper, we derive a nonlinear theory of thermoelasticity with voids and diffusion according to the Green–Naghdi theory of thermomechanics of continua of types II and III. This theory permits propagation of both thermal and diffusion waves at finite speeds. The equations of the linear theory are also obtained. We prove the well-posedness of the linear model and the asymptotic behavior of the solutions by the semigroup theory of linear operators. Finally, we investigate the impossibility of the localization in time of solutions. The main idea to prove this result is to show the uniqueness of solutions for the backward in time problem.

Spatial behavior in dynamical thermoviscoelasticity backward in time for porous media

ABSTRACTIn this article we study the spatial behavior of solutions to the backward in time problem associated with the linear theory of thermoviscoelastic materials with voids. We consider an appropriate time-weighted volume measure associated with the final boundary value problem and then, in terms of such measure, we obtain a first-order partial differential inequality. Some spatial estimates of Saint-Venant type are established.

On a backward parabolic problem with local Lipschitz source

We consider the regularization of the backward in time problem for a nonlinear parabolic equation in the form ut+Au(t)=f(u(t),t), u(1)=φ, where A is a positive self-adjoint unbounded operator and f is a local Lipschitz function. As known, it is ill-posed and occurs in applied mathematics, e.g. in neurophysiological modeling of large nerve cell systems with action potential f in mathematical biology. A new version of quasi-reversibility method is described. We show that the regularized problem (with a regularization parameter β>0) is well-posed and that its solution Uβ(t) converges on [0,1] to the exact solution u(t) as β→0+. These results extend some earlier works on the nonlinear backward problem.

On the backward in time problem for the thermoelasticity with two temperatures

This paper is devoted to the study of the existence, uniqueness, continuous dependence and spatial behaviour of the solutions for the backward in time problem determined by the Type III with two temperatures thermoelastodynamic theory. We first show the existence, uniqueness and continuous dependence of the solutions. Instability of the solutions for the Type II with two temperatures theory is proved later. For the one-dimensional Type III with two temperatures theory, the exponential instability is also pointed-out. We also analyze the spatial behaviour of the solutions. By means of the exponentially weighted Poincare inequality, we are able to obtain a function that defines a measure on the solutions and, therefore, we obtain the usual exponential type alternative for the solutions of the problem defined in a semi-infinite cylinder.

On the Uniqueness in Dynamical Thermoelasticity Backward in Time for Porous Media

The aim of this article is to study some uniqueness criteria for the solutions of boundary-final value problems associated with the linear theory of thermoelastic materials with voids. More precisely, the present study is devoted to the investigation of a backward in time problem associated with porous thermoelastic materials.

Uniqueness and structural stability for the Cattaneo–Christov equations

The recent modification of the Cattaneo equations proposed by Christov is analysed. The solution to both the initial and final value problems is shown to be unique. Further, it is shown that the solution to the backward in time problem depends continuously on the relaxation time.

Some Results in Linear Theory of Thermoelasticity Backward in Time for Microstretch Materials

In this paper the uniqueness of solutions for the backward in time problem of the linear theory of thermo-microstretch elastic materials is shown and the impossibility of the localization in time of the solution of the corresponding forward in time problem is proved. Moreover, the temporal behavior backward in time of thermoelastodynamics processes is studied by establishing the relations describing the asymptotic behavior of the Cesàro means of the different parts of the total energy.

Impossibility of localization in thermo-porous-elasticity with microtemperatures

In this note, we prove the impossibility of the localization in-time for the solutions of the linear porous-thermo-elasticity with microtemperatures theory. This means that the only solution for this problem that vanishes after a finite time is the null solution. From a thermomechanical point of view, this result says that the combination of the porous dissipation, the temperature and the microtemperatures (in the linear theory) is not strong enough to guarantee that the thermomechanical deformations will vanish after a finite time. The main idea to prove this result is to show the uniqueness of solutions for the backward in-time problem.

On the impossibility of localization in linear thermoelasticity

In the middle of the 1990s, Green & Naghdi proposed three theories of thermoelasticity that they labelled as types I, II and II. The type II theory, which is also called thermoelasticity without energy dissipation, is conservative and the solutions cannot decay with respect to time. It is well known that, in general, in the linear theories of thermoelasticity of types I and III, the solutions decay with respect to time. In many situations this decay is at least exponential. In this paper we study whether this decay can be fast enough to guarantee the solutions to be zero in a finite time. We investigate the impossibility of the localization in time of the solutions of linear thermoelasticity for the theories of Green & Naghdi. This means that the only solution that vanishes after a finite time is the null solution. The main idea is to show the uniqueness of solutions for the backward in time problem. To be precise, for type III thermoelasticity we will prove the impossibility of localization of solutions in the case of bounded domains, and for the type I thermoelasticity in the case of exterior domains, even when the solutions can be unbounded, whether the spatial variable goes to infinity.