Structural stability for the Forchheimer equations interfacing with a Darcy fluid in a bounded region in $\mathbb{R}^{3}$

The structural stability for the Forchheimer fluid interfacing with a Darcy fluid in a bounded region in $\mathbb{R}^{3}$ R 3 was studied. We assumed that the nonlinear fluid was governed by the Forchheimer equations in $\Omega _{1}$ Ω 1 , while in $\Omega _{2}$ Ω 2 , we supposed that the flow satisfies the Darcy equations. With the aid of some useful a priori bounds, we were able to demonstrate the continuous dependence results for the Forchheimer coefficient λ.

Set-valued problems under bounded variation assumptions involving the Hausdorff excess

<p style='text-indent:20px;'>In the very general framework of a (possibly infinite dimensional) Banach space <inline-formula><tex-math id="M1">\begin{document}$ X $\end{document}</tex-math></inline-formula>, we are concerned with the existence of bounded variation solutions for measure differential inclusions</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE100"> \begin{document}$ \begin{equation} \begin{split} &amp;dx(t) \in G(t, x(t)) dg(t),\\ &amp;x(0) = x_0, \end{split} \end{equation}\;\;\;\;\;\;(1) $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M2">\begin{document}$ dg $\end{document}</tex-math></inline-formula> is the Stieltjes measure generated by a nondecreasing left-continuous function.</p><p style='text-indent:20px;'>This class of differential problems covers a wide variety of problems occuring when studying the behaviour of dynamical systems, such as: differential and difference inclusions, dynamic inclusions on time scales and impulsive differential problems. The connection between the solution set associated to a given measure <inline-formula><tex-math id="M3">\begin{document}$ dg $\end{document}</tex-math></inline-formula> and the solution sets associated to some sequence of measures <inline-formula><tex-math id="M4">\begin{document}$ dg_n $\end{document}</tex-math></inline-formula> strongly convergent to <inline-formula><tex-math id="M5">\begin{document}$ dg $\end{document}</tex-math></inline-formula> is also investigated.</p><p style='text-indent:20px;'>The multifunction <inline-formula><tex-math id="M6">\begin{document}$ G : [0,1] \times X \to \mathcal{P}(X) $\end{document}</tex-math></inline-formula> with compact values is assumed to satisfy excess bounded variation conditions, which are less restrictive comparing to bounded variation with respect to the Hausdorff-Pompeiu metric, thus the presented theory generalizes already known existence and continuous dependence results. The generalization is two-fold, since this is the first study in the setting of infinite dimensional spaces.</p><p style='text-indent:20px;'>Next, by using a set-valued selection principle under excess bounded variation hypotheses, we obtain solutions for a functional inclusion</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE102"> \begin{document}$ \begin{equation} \begin{split} &amp;Y(t)\subset F(t,Y(t)),\\ &amp;Y(0) = Y_0. \end{split} \end{equation}\;\;\;\;(2) $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>It is shown that a recent parametrized version of Banach's Contraction Theorem given by V.V. Chistyakov follows from our result.</p>

Continuous Dependence of Solutions of Integer and Fractional Order Non-Instantaneous Impulsive Equations with Random Impulsive and Junction Points

This paper gives continuous dependence results for solutions of integer and fractional order, non-instantaneous impulsive differential equations with random impulse and junction points. The notion of the continuous dependence of solutions of these equations on the initial point is introduced. We prove some sufficient conditions that ensure the solutions to perturbed problems have a continuous dependence. Finally, we use numerical examples to demonstrate the obtained theoretical results.

Continuous dependence and uniqueness for lateral Cauchy problems for linear integro-differential parabolic equations

Via Carleman estimates we prove uniqueness and continuous dependence results for lateral Cauchy problems for linear integro-differential parabolic equations without initial conditions. The additional information supplied prescribes the conormal derivative of the temperature on a relatively open subset of the lateral boundary of the space-time domain.

Qualitative properties in strain gradient thermoelasticity with microtemperatures

This paper is devoted to the strain gradient theory of thermoelastic materials whose microelements possess microtemperatures. The work is motivated by an increasing use of materials which possess thermal variation at a microstructure level. In the first part of this paper we deduce the system of basic equations of the linear theory and formulate the boundary-initial-value problem. We establish existence, uniqueness, and continuous dependence results by the means of semigroup theory. Then, we study the one-dimensional problem and establish the analyticity of solutions. Exponential stability and impossibility of localization are consequences of this result. In the case of the anti-plane problem we derive uniqueness and instability results without assuming the positivity of the mechanical energy. Finally, we study equilibrium theory and investigate the effects of a concentrated heat source in an unbounded body.

Existence and Continuous Dependence for Fractional Partial Hyperbolic Differential Equations

This paper is concerned with a class of fractional hyperbolic partial differential equations with the Caputo derivative. Existence and continuous dependence results of solutions are obtained under the hypothesis of the Lipschitz condition without any restriction on the Lipschitz constant. Examples are discussed to illustrate the results.