Memory relaxation of first order evolution equations

Gatti, Stefania; Grasselli, Maurizio; Miranville, Alain; Pata, Vittorino

doi:10.1088/0951-7715/18/4/023

Cited by 34 publications

(51 citation statements)

References 19 publications

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“…Again, we extend the definition to ε = 0 by setting Z 0 = H 1 0 . The following fact has been mentioned in [6,13] without proof. Proof.…”

Section: Higher-order Dissipativitymentioning

confidence: 87%

“…We first note that, for ε > 0, the embedding H 1 ε ⊂ H 0 ε is not compact, due to the lack of compactness of M 1 ε ⊂ M 0 ε (see [38]). Thus, following the lines of [6,13], we introduce the Banach space…”

Section: Higher-order Dissipativitymentioning

confidence: 99%

“…We first report a generalized version of the Gronwall lemma. 1 We take here the occasion to correct the definition of the higher norm of η adopted in [6,13], although the mistake was not influent in those papers. With this choice, the map η(s) → η(εs/ε 0 ) is an isometric isomorphism from L ε into L ε 0 , which is onto if ε > 0.…”

Section: Higher-order Dissipativitymentioning

confidence: 99%

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2010

Quart. Appl. Math.

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Abstract. In a two-dimensional space domain, we consider a reaction-diffusion equation whose diffusion term is a time convolution of the Laplace operator against a nonincreasing summable memory kernel k. This equation models several phenomena arising from many different areas. After rescaling k by a relaxation time ε > 0, we formulate a Cauchy-Dirichlet problem, which is rigorously translated into a similar problem for a semilinear hyperbolic integro-differential equation with nonlinear damping, for a particular choice of the initial data. Using the past history approach, we show that the hyperbolic equation generates a dynamical system which is dissipative provided that ε is small enough, namely, when the equation is sufficiently "close" to the standard reactiondiffusion equation formally obtained by replacing k with the Dirac mass at 0. Then, we provide an estimate of the difference between ε-trajectories and 0-trajectories, and we construct a family of regular exponential attractors which is robust with respect to the singular limit ε → 0. In particular, this yields the existence of a regular global attractor

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“…Again, we extend the definition to ε = 0 by setting Z 0 = H 1 0 . The following fact has been mentioned in [6,13] without proof. Proof.…”

Section: Higher-order Dissipativitymentioning

confidence: 87%

Section: Higher-order Dissipativitymentioning

confidence: 99%

Section: Higher-order Dissipativitymentioning

confidence: 99%

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Untitled

2010

Quart. Appl. Math.

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“…This (deterministic) problem has been studied by many authors. For instance, existence and uniqueness of global solutions to general semilinear integro-differential equations have been analyzed in [1], two kind of equations are compared in [10], being the second a singular perturbation which approaches the first one when the memory kernel collapses into a Dirac mass; this kind of relaxation is also adopted in [13] to prove the existence of a robust family of exponential attractors, and the close behaviour of both problems, which contain, as particular cases, the Allen-Cahn and Cahn-Hiliard equations; trajectory and global (uniform) attractors for such general kind of problems are studied in [6,14,15,16].…”

mentioning

confidence: 99%

“…Following the idea introduced by Dafermos [11] (see also [10,13] and the survey [16]), we introduce the new variable…”

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confidence: 99%

Existence and asymptotic behaviour for stochastic heat equations with multiplicative noise in materials with memory

Caraballo¹,

Čhuešhov²,

Marín-Rubio³

et al. 2007

DCDS-A

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Abstract. The existence and uniqueness of solutions for a stochastic reactiondiffusion equation with infinite delay is proved. Sufficient conditions ensuring stability of the zero solution are provided and a possibility of stabilization by noise of the deterministic counterpart of the model is studied.1. Introduction. Our main goal in this paper is to establish the existence and uniqueness theorem and to analyze the long-time behaviour of a stochastic heat equation with memory subjected to multiplicative white noise. The starting point for our considerations is the following deterministic heat conduction model.Let O be a bounded domain in R d (d ≥ 1). We denote by u = u(x, t) the temperature at position x ∈Ō and time t. Following the theory developed by Coleman & Gurtin [9], Gurtin & Pipkin [17] and Nunziato [20], we assume that the density e(x, t) of the internal energy and the heat flux φ(x, t) are related to the temperature and its gradient by constitutive relations:

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3‐D viscous Cahn–Hilliard equation with memory

Conti

Mola

2008

Math Methods in App Sciences

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We deal with the memory relaxation of the viscous Cahn-Hilliard equation in 3-D, covering the wellknown hyperbolic version of the model. We study the long-term dynamic of the system in dependence of the scaling parameter of the memory kernel ε and of the viscosity coefficient . In particular we construct a family of exponential attractors, which is robust as both ε and go to zero, provided that ε is linearly controlled by .where > 0 is chosen as in (6). Arguing exactly as in Lemmas 4.5 and 4.7, keeping in mind that the initial conditions are null, it is immediate to realize thatThis can be done arguing as in the proof of [9, Lemma 5.5] (see also [8]) by exploiting the exponential decay of ε , the straightforward inequalities M 0 ε can conclude that the global attractor A ε, has finite fractal dimension, which is uniform with respect to ε and . In addition, arguing as in [9, Section 7] with obvious changes (see also [20]), it is possible to prove that the global attractor is upper semicontinuous as (ε, ) → (0, 0), namely lim (ε, )→(0,0) dist(A ε, , A 0,0 ) = 0

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Memory relaxation of first order evolution equations

Cited by 34 publications

References 19 publications

Untitled

Untitled

Existence and asymptotic behaviour for stochastic heat equations with multiplicative noise in materials with memory

3‐D viscous Cahn–Hilliard equation with memory

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