scholarly journals Existence, Nonexistence, and Stability of Solutions for a Delayed Plate Equation with the Logarithmic Source

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Hazal Yüksekkaya ◽  
Erhan Pișkin ◽  
Salah Mahmoud Boulaaras ◽  
Bahri Belkacem Cherif ◽  
Sulima Ahmed Zubair
Keyword(s):  
Time Delay ◽  
Source Term ◽  
Existence Of Solutions ◽  
Logarithmic Sobolev Inequality ◽  
Potential Well ◽  
Stability Of Solutions ◽  
Plate Equation ◽  
Global Existence Of Solutions ◽  
Frictional Damping ◽  
The Stability

In this work, we study a plate equation with time delay in the velocity, frictional damping, and logarithmic source term. Firstly, we obtain the local and global existence of solutions by the logarithmic Sobolev inequality and the Faedo-Galerkin method. Moreover, we prove the stability and nonexistence results by the perturbed energy and potential well methods.

scholarly journals Global existence, energy decay and blow-up of solutions for wave equations with time delay and logarithmic source

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Sun-Hye Park
Keyword(s):  
Time Delay ◽  
Global Existence ◽  
Wave Equations ◽  
Blow Up ◽  
Logarithmic Sobolev Inequality ◽  
Decay Rates ◽  
Infinite Time ◽  
Global Existence Of Solutions ◽  
Frictional Damping ◽  
T Tau

AbstractIn this paper, we study the wave equation with frictional damping, time delay in the velocity, and logarithmic source of the form $$ u_{tt}(x,t) - \Delta u (x,t) + \alpha u_{t} (x,t) + \beta u_{t} (x, t- \tau ) = u(x,t) \ln \bigl\vert u(x,t) \bigr\vert ^{\gamma } . $$ u t t ( x , t ) − Δ u ( x , t ) + α u t ( x , t ) + β u t ( x , t − τ ) = u ( x , t ) ln | u ( x , t ) | γ . There is much literature on wave equations with a polynomial nonlinear source, but not much on the equations with logarithmic source. We show the local and global existence of solutions using Faedo–Galerkin’s method and the logarithmic Sobolev inequality. And then we investigate the decay rates and infinite time blow-up for the solutions through the potential well and perturbed energy methods.

scholarly journals Remarks on the critical nonlinear high-order heat equation

2019 ◽  
Vol 26 (1/2) ◽  
pp. 127-152
Author(s):  
Tarek Saanouni
Keyword(s):  
Global Existence ◽  
Heat Equation ◽  
Initial Value Problem ◽  
Existence Of Solutions ◽  
High Order ◽  
Well Posedness ◽  
Potential Well ◽  
Initial Value ◽  
Global Existence Of Solutions ◽  
Nonlinear High Order

The initial value problem for a semi-linear high-order heat equation is investigated. In the focusing case, global well-posedness and exponential decay are obtained. In the focusing sign, global and non global existence of solutions are discussed via the potential well method.

scholarly journals Global and non global solutions for a class of coupled parabolic systems

2020 ◽  
Vol 9 (1) ◽  
pp. 1383-1401 ◽  
Author(s):  
T. Saanouni
Keyword(s):  
Global Existence ◽  
Exponential Decay ◽  
Existence Of Solutions ◽  
Global Solutions ◽  
Parabolic Systems ◽  
Heat Equations ◽  
Well Posedness ◽  
Potential Well ◽  
Global Existence Of Solutions ◽  
Coupled Parabolic Systems

Abstract In the present paper, we investigate the global well-posedness and exponential decay for some coupled non-linear heat equations. Moreover, we discuss the global and non global existence of solutions using the potential well method.

Global and non Global Solutions for Some Fractional Heat Equations With Pure Power Nonlinearity

2017 ◽  
Vol 69 (4) ◽  
pp. 854-872
Author(s):  
Tarek Saanouni
Keyword(s):  
Global Existence ◽  
Heat Equation ◽  
Existence Of Solutions ◽  
Global Solutions ◽  
Heat Equations ◽  
Well Posedness ◽  
Potential Well ◽  
Initial Value ◽  
Fractional Heat Equation ◽  
Global Existence Of Solutions

AbstractThe initial value problem for a semi-linear fractional heat equation is investigated. In the focusing case, global well-posedness and exponential decay are obtained. In the focusing sign, global and non global existence of solutions are discussed via the potential well method.

THE LIFSHITZ–SLYOZOV EQUATION WITH ENCOUNTERS

2001 ◽  
Vol 11 (04) ◽  
pp. 731-748 ◽  
Author(s):  
PHILIPPE LAURENÇOT
Keyword(s):  
Solid Solution ◽  
Mass Transfer ◽  
Distribution Function ◽  
Source Term ◽  
Supersaturated Solid Solution ◽  
Existence Of Solutions ◽  
Volume Distribution ◽  
Nonlocal Nonlinearity ◽  
Global Existence Of Solutions ◽  
New Phase

The Lifshitz–Slyozov theory of coarsening in alloys describes the time evolution of the sizes of the grains of a new phase growing by mass transfer from a supersaturated solid solution. When the encounters between the grains are taken into account, the volume distribution function of the grains obeys a transport equation with a nonlocal nonlinearity and an integral source term. Global existence of solutions is proved for a large class of data including the ones derived by Lifshitz and Slyozov.

scholarly journals Exponential decay and global existence of solutions of a singular nonlocal viscoelastic system with distributed delay and damping terms

Filomat ◽  
2021 ◽  
Vol 35 (3) ◽  
pp. 795-826
Author(s):  
Abdelbaki Choucha ◽  
Salah Boulaaras ◽  
Djamel Ouchenane
Keyword(s):  
Exponential Decay ◽  
Source Term ◽  
Existence Of Solutions ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Distributed Delay ◽  
Viscoelastic System ◽  
Nonlinear Source ◽  
One Dimensional ◽  
Global Existence Of Solutions

We investigate in this work a singular one-dimensional viscoelastic system with a nonlinear source term, distributed delay, nonlocal boundary condition, and damping terms. By the theory of potentialwell, the existence of a global solution is established, and by the energy method and the functional of Lyapunov, we prove the exponential decay result. This work is an extension of Boulaaras? work in ([3] and [27]).

scholarly journals Stability of Ulam–Hyers and Existence of Solutions for Impulsive Time-Delay Semi-Linear Systems with Non-Permutable Matrices

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1493
Author(s):  
Nazim I. Mahmudov ◽  
Amal M. Almatarneh
Keyword(s):  
Time Delay ◽  
Existence Of Solutions ◽  
Linear Time ◽  
Sufficient Conditions ◽  
Representation Formula ◽  
Matrix Exponential ◽  
Delay Systems ◽  
Impulsive Systems ◽  
Impulsive Delay ◽  
The Stability

In this paper, the stability of Ulam–Hyers and existence of solutions for semi-linear time-delay systems with linear impulsive conditions are studied. The linear parts of the impulsive systems are defined by non-permutable matrices. To obtain solution for linear impulsive delay systems with non-permutable matrices in explicit form, a new concept of impulsive delayed matrix exponential is introduced. Using the representation formula and norm estimation of the impulsive delayed matrix exponential, sufficient conditions for stability of Ulam–Hyers and existence of solutions are obtained.

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