scholarly journals Decay properties for evolution-parabolic coupled systems related to thermoelastic plate equations

2021 ◽  
Author(s):  
Yan Liu ◽  
Zihan Cai ◽  
Shuanghu Zhang
Keyword(s):  
Cauchy Problem ◽  
Phase Space ◽  
Coupled Systems ◽  
Decay Properties ◽  
Representation Of Solutions ◽  
The Cauchy Problem ◽  
Plate Equations ◽  
Thermoelastic Plate ◽  
Local Operators ◽  
Non Local

In this paper, we consider the Cauchy problem for a family of evolution-parabolic coupled systems, which are related to the classical thermoelastic plate equations containing non-local operators. By using diagonalization procedure and WKB analysis, we derive representation of solutions in the phase space. Then, sharp decay properties in a framework of $L^p-L^q$ are investigated via these representations. Particularly, some thresholds for the regularity-loss type decay properties are found.

scholarly journals Decay properties for evolution-parabolic coupled systems related to thermoelastic plate equations

2021 ◽  
Vol 7 (1) ◽  
pp. 260-275
Author(s):  
Zihan Cai ◽  
◽  
Yan Liu ◽  
Baiping Ouyang ◽  
Keyword(s):  
Cauchy Problem ◽  
Phase Space ◽  
Coupled Systems ◽  
Decay Properties ◽  
Representation Of Solutions ◽  
The Cauchy Problem ◽  
Plate Equations ◽  
Thermoelastic Plate ◽  
Local Operators ◽  
Non Local

<abstract><p>In this paper, we consider the Cauchy problem for a family of evolution-parabolic coupled systems, which are related to the classical thermoelastic plate equations containing non-local operators. By using diagonalization procedure and WKB analysis, we derive representation of solutions in the phase space. Then, sharp decay properties in a framework of $ L^p-L^q $ are investigated via these representations. Particularly, some thresholds for the regularity-loss type decay properties are found.</p></abstract>

scholarly journals Dilation Properties for Weighted Modulation Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-29 ◽  
Author(s):  
Elena Cordero ◽  
Kasso A. Okoudjou
Keyword(s):  
Cauchy Problem ◽  
Besov Spaces ◽  
Sharp Estimate ◽  
Modulation Spaces ◽  
Well Posedness ◽  
The Cauchy Problem ◽  
Plate Equations ◽  
Unweighted Case ◽  
Vibrating Plate

We give a sharp estimate on the norm of the scaling operatorUλf(x)=f(λx)acting on the weighted modulation spacesMs,tp,q(ℝd). In particular, we recover and extend recent results by Sugimoto and Tomita in the unweighted case. As an application of our results, we estimate the growth in time of solutions of the wave and vibrating plate equations, which is of interest when considering the well-posedness of the Cauchy problem for these equations. Finally, we provide new embedding results between modulation and Besov spaces.

scholarly journals Global existence and energy decay of solutions to the Cauchy problem for a wave equation with a weakly nonlinear dissipation

2004 ◽  
Vol 2004 (11) ◽  
pp. 935-955 ◽  
Author(s):  
Abbès Benaissa ◽  
Soufiane Mokeddem
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Global Existence ◽  
Stable Set ◽  
Nonlinear Dissipation ◽  
Weakly Nonlinear ◽  
Dissipative Term ◽  
Decay Properties ◽  
Decay Of Solutions ◽  
The Cauchy Problem

We prove the global existence and study decay properties of the solutions to the wave equation with a weak nonlinear dissipative term by constructing a stable set inH1(ℝn).

A finite difference method for the very weak solution to a Cauchy problem for an elliptic equation

2018 ◽  
Vol 26 (6) ◽  
pp. 835-857 ◽  
Author(s):  
Dinh Nho Hào ◽  
Le Thi Thu Giang ◽  
Sergey Kabanikhin ◽  
Maxim Shishlenin
Keyword(s):  
Cauchy Problem ◽  
Weak Solution ◽  
Finite Difference ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Weak Sense ◽  
Very Weak Solution ◽  
Local Boundary ◽  
The Cauchy Problem ◽  
Non Local

Abstract We introduce the concept of very weak solution to a Cauchy problem for elliptic equations. The Cauchy problem is regularized by a well-posed non-local boundary value problem whose solution is also understood in a very weak sense. A stable finite difference scheme is suggested for solving the non-local boundary value problem and then applied to stabilizing the Cauchy problem. Some numerical examples are presented for showing the efficiency of the method.

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