Exponential and polynomial decay for first order linear Volterra evolution equations

Mainini, Edoardo; Mola, Gianluca

doi:10.1090/s0033-569x-09-01145-x

Cited by 13 publications

(3 citation statements)

References 14 publications

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“…The thermodynamical admissibility of a singular viscoelastic model characterized by a singular viscoelastic relaxation modulus is analyzed by Giorgi and Morro [49]. The references [50][51][52][53][54][55][56][57][58] are all concerning singular kernel problems both in rigid thermodynamics with memory as well as in viscoelasticity. Specifically, [50-52, 57, 58] study asymptotic behaviour of solutions.…”

Section: Introductionmentioning

confidence: 99%

Singular kernel problems in materials with memory

Carillo

2014

Meccanica

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In recent years the interest on devising and study new materials is growing since they are widely used in different applications which go from rheology to bio-materials or aerospace applications. In this framework, there is also a growing interest in understanding the behaviour of materials with memory, here considered. The name of the model aims to emphasize that the behaviour of such materials crucially depends on time not only through the present time but also through the past history. Under the analytical point of view, this corresponds to model problems represented by integro-differential equations which exhibit a kernel non local in time. This is the case of rigid thermodynamics with memory as well as of isothermal viscoelasticity; in the two different models the kernel represents, in turn, the heat flux relaxation function and the relaxation modulus. In dealing with classical materials with memory these kernels are regular function of both the present time as well as the past history. Aiming to study new materials integrodifferential problems admitting singular kernels are compared. Specifically, on one side the temperature evolution in a rigid heat conductor with memory characterized by a heat flux relaxation function singular at the origin, and, on the other, the displacement evolution within a viscoelastic model characterized by a relaxation modulus which is unbounded at the origin, are considered. One dimensional problems are examined; indeed, even if the results are valid also in three dimensional general cases, here the attention is focussed on pointing out analogies between the two different materials with memory under investigation. Notably, the method adopted has a wider interest since it can be applied in the cases of other evolution problems which are modeled by analogue integrodifferential equations. An initial boundary value problem with homogenous Neumann boundary conditions is studied.

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Section: Introductionmentioning

confidence: 99%

Singular kernel problems in materials with memory

Carillo

2014

Meccanica

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“…Travis and Webb [15] studied the existence of solutions to semilinear second order Volterra integrodifferential equations in Banach spaces by using the theory of strongly continuous cosine families. Mainini and Mola [8] considered in an abstract setting, an instance of the Coleman-Gurtin Key words and phrases. Fractional integrodifferential differention; fractional cosine family; fractional powers of operators.…”

Section: Introductionmentioning

confidence: 99%

Fractional order semilinear Volterra integrodifferential equations in Banach spaces

Li¹

2014

Preprint

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“…We immediately adopt the following lemma from [20] without proof which will be used in the sequel. Lemma 2.3.…”

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confidence: 99%

Decay rates for the Moore-Gibson-Thompson equation with memory

Bounadja¹,

Houari²

2021

EECT

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The main goal of this paper is to investigate the existence and stability of the solutions for the Moore-Gibson-Thompson equation (MGT) with a memory term in the whole spaces R N. The MGT equation arises from modeling high-frequency ultrasound waves as an alternative model to the wellknown Kuznetsov's equation. First, following [8] and [26], we show that the problem is well-posed under an appropriate assumption on the coefficients of the system. Then, we built some Lyapunov functionals by using the energy method in Fourier space. These functionals allows us to get control estimates on the Fourier image of the solution. These estimates of the Fourier image together with some integral inequalities lead to the decay rate of the L 2-norm of the solution. We use two types of memory term here: type I memory term and type III memory term. Decay rates are obtained in both types. More precisely, decay rates of the solution are obtained depending on the exponential or polynomial decay of the memory kernel. More importantly, we show stability of the solution in both cases: a subcritical range of the parameters and a critical range. However for the type I memory we show in the critical case that the solution has the regularity-loss property.

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Exponential and polynomial decay for first order linear Volterra evolution equations

Cited by 13 publications

References 14 publications

Singular kernel problems in materials with memory

Singular kernel problems in materials with memory

Fractional order semilinear Volterra integrodifferential equations in Banach spaces

Decay rates for the Moore-Gibson-Thompson equation with memory

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