scholarly journals Well-posedness and direct internal stability of coupled non-degenrate Kirchhoff system via heat conduction

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Akram Ben Aissa
Keyword(s):  
Heat Conduction ◽  
Exponential Decay ◽  
Open Subset ◽  
Existence Of Solution ◽  
Integral Inequalities ◽  
Internal Stability ◽  
Well Posedness ◽  
Real Numbers ◽  
Bounded Open Subset

<p style='text-indent:20px;'>In the paper under study, we consider the following coupled non-degenerate Kirchhoff system</p><p style='text-indent:20px;'><disp-formula><label/><tex-math id="FE0"> \begin{document}$\begin{equation} \left \{ \begin{aligned} &amp; y_{tt}-\mathtt{φ}\Big(\int_\Omega | \nabla y |^2\,dx\Big)\Delta y +\mathtt{α} \Delta \mathtt{θ} = 0, &amp;{\rm{ in }}&amp;\; \Omega \times (0, +\infty)\\ &amp; \mathtt{θ}_t-\Delta \mathtt{θ}-\mathtt{β} \Delta y_t = 0, &amp;{\rm{ in }}&amp;\; \Omega \times (0, +\infty)\\ &amp; y = \mathtt{θ} = 0,\; &amp;{\rm{ on }}&amp;\;\partial\Omega\times(0, +\infty)\\ &amp; y(\cdot, 0) = y_0, \; y_t(\cdot, 0) = y_1,\;\mathtt{θ}(\cdot, 0) = \mathtt{θ}_0, \; \; &amp;{\rm{ in }}&amp;\; \Omega\\ \end{aligned} \right. \end{equation} \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$ \end{document}</tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a bounded open subset of <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ \mathtt{α} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ \mathtt{β} $\end{document}</tex-math></inline-formula> be two nonzero real numbers with the same sign and <inline-formula><tex-math id="M5">\begin{document}$ \mathtt{φ} $\end{document}</tex-math></inline-formula> is given by <inline-formula><tex-math id="M6">\begin{document}$ \mathtt{φ}(s) = \mathfrak{m}_0+\mathfrak{m}_1s $\end{document}</tex-math></inline-formula> with some positive constants <inline-formula><tex-math id="M7">\begin{document}$ \mathfrak{m}_0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ \mathfrak{m}_1 $\end{document}</tex-math></inline-formula>. So we prove existence of solution and establish its exponential decay. The method used is based on multiplier technique and some integral inequalities due to Haraux and Komornik[<xref ref-type="bibr" rid="b5">5</xref>,<xref ref-type="bibr" rid="b8">8</xref>].</p>

scholarly journals Thermoelasticity of Moore–Gibson–Thompson type with history dependence in the temperature

2020 ◽  
Vol 120 (1-2) ◽  
pp. 1-21 ◽  
Author(s):  
Monica Conti ◽  
Vittorino Pata ◽  
Ramon Quintanilla
Keyword(s):  
Heat Conduction ◽  
Differential System ◽  
Exponential Decay ◽  
Relaxation Parameter ◽  
Dimensional Case ◽  
Well Posedness ◽  
History Dependence ◽  
One Dimensional ◽  
Thermoelastic Model ◽  
The One

In this paper, we consider a thermoelastic model where heat conduction is described by the history dependent version of the Moore–Gibson–Thompson equation, arising via the introduction of a relaxation parameter in the Green-Naghdi type III theory. The well-posedness of the resulting integro-differential system is discussed. In the one-dimensional case, the exponential decay of the energy is proved.

Some new fractional integral inequalities for generalized relative semi-m-(r; h1, h2)-preinvex mappings via generalized Mittag-Leffler function

2019 ◽  
Vol 26 (1/2) ◽  
pp. 41-55 ◽  
Author(s):  
Artion Kashuri ◽  
Rozana Liko
Keyword(s):  
Fractional Integral ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Auxiliary Result ◽  
Real Numbers ◽  
Positive Real ◽  
Invex Set ◽  
Special Means ◽  
Special Cases ◽  
Differentiable Mappings

The authors discover a new identity concerning differentiable mappings defined on m-invex set via fractional integrals. By using the obtained identity as an auxiliary result, some fractional integral inequalities for generalized relative semi- m-(r;h1,h2)-preinvex mappings by involving generalized Mittag-Leffler function are presented. It is pointed out that some new special cases can be deduced from main results of the paper. Also these inequalities have some connections with known integral inequalities. At the end, some applications to special means for different positive real numbers are provided as well.

Most Infinitely Differentiable Functions are Nowhere Analytic

1973 ◽  
Vol 16 (4) ◽  
pp. 597-598 ◽  
Author(s):  
R. B. Darst
Keyword(s):  
Open Subset ◽  
Real Numbers ◽  
The Real ◽  
Differentiable Functions ◽  
First Category ◽  
Infinitely Differentiable Functions

We define a natural metric, d, on the space, C∞,, of infinitely differentiable real valued functions defined on an open subset U of the real numbers, R, and show that C∞, is complete with respect to this metric. Then we show that the elements of C∞, which are analytic near at least one point of U comprise a first category subset of C∞,.

scholarly journals Non-occurrence of the Lavrentiev phenomenon for a class of convex nonautonomous Lagrangians

2020 ◽  
Vol 18 (1) ◽  
pp. 1-9
Author(s):  
Carlo Mariconda ◽  
Giulia Treu
Keyword(s):  
Convex Function ◽  
Calculus Of Variations ◽  
Open Subset ◽  
Lipschitz Function ◽  
Lavrentiev Phenomenon ◽  
Smooth Functions ◽  
Bounded Open Subset ◽  
Admissible Functions

Abstract We consider the classical functional of the Calculus of Variations of the form $$\begin{array}{} \displaystyle I(u)=\int\limits_{{\it\Omega}}F(x, u(x), \nabla u(x))\,dx, \end{array}$$ where Ω is a bounded open subset of ℝn and F : Ω × ℝ × ℝn → ℝ is a Carathéodory convex function; the admissible functions u coincide with a prescribed Lipschitz function ϕ on ∂Ω. We formulate some conditions under which a given function in ϕ + $\begin{array}{} \displaystyle W^{1,p}_0 \end{array}$(Ω) with I(u) < +∞ can be approximated in the W1,p norm and in energy by a sequence of smooth functions that coincide with ϕ on ∂Ω. As a particular case we obtain that the Lavrentiev phenomenon does not occur when F(x, u, ξ) = f(x, u) + h(x, ξ) is convex and x ↦ F(x, 0, 0) is sufficiently smooth.

scholarly journals Fejér type integral inequalities related with geometrically-arithmetically convex functions with applications

2019 ◽  
Vol 23 (1) ◽  
pp. 51-64
Author(s):  
S. S. Dragomir ◽  
M. A. Latif ◽  
E. Momoniat
Keyword(s):  
Differentiable Function ◽  
Integral Inequality ◽  
Convex Functions ◽  
Symmetric Function ◽  
Integral Inequalities ◽  
Real Numbers ◽  
Positive Real ◽  
Left Part ◽  
Special Means ◽  
Type Integral

A new identity involving a geometrically symmetric function and a differentiable function is established. Some new Fejér type integral inequalities, connected with the left part of Hermite–Hadamard type inequalities for geometrically-arithmetically convex functions, are presented by using the Hölder integral inequality and the notion of geometrically-arithmetically convexity. Applications of our results to special means of positive real numbers are given.

scholarly journals Some Fejér type integral inequalities for geometrically-arithmetically-convex functions with applications

Filomat ◽  
2018 ◽  
Vol 32 (6) ◽  
pp. 2193-2206 ◽  
Author(s):  
Muhammad Latif ◽  
Sever Dragomir ◽  
Ebrahim Momoniat
Keyword(s):  
Integral Inequality ◽  
Convex Functions ◽  
Symmetric Functions ◽  
Integral Inequalities ◽  
Real Numbers ◽  
Positive Real ◽  
Special Means ◽  
Type Integral

In this paper, the notion of geometrically symmetric functions is introduced. A new identity involving geometrically symmetric functions is established, and by using the obtained identity, the H?lder integral inequality and the notion of geometrically-arithmetically convexity, some new Fej?r type integral inequalities are presented. Applications of our results to special means of positive real numbers are given as well.

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