A New Approach to Equations with Memory

被引：122
作者：
Fabrizio, Mauro [1 ]
Giorgi, Claudio [2 ]
Pata, Vittorino [3 ]
机构：
[1] Univ Bologna, Dipartimento Matemat, I-40126 Bologna, Italy
[2] Univ Brescia, Dipartimento Matemat, I-25133 Brescia, Italy
[3] Politecn Milan, Dipartimento Matemat F Brioschi, I-20133 Milan, Italy
来源：
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS | 2010年 / 198卷 / 01期
关键词：
MINIMUM FREE-ENERGIES; EXPONENTIAL STABILITY; ASYMPTOTIC STABILITY; RELAXATION FUNCTIONS; VISCOELASTICITY; DISSIPATION; ATTRACTORS; SYSTEMS; STATE;
D O I：
10.1007/s00205-010-0300-3
中图分类号：
O29 [应用数学];
学科分类号：
070104 ;
摘要：
We discuss a novel approach to the mathematical analysis of equations with memory, based on a new notion of state. This is the initial configuration of the system at time t = 0 which can be unambiguously determined by the knowledge of the dynamics for positive times. As a model, for a nonincreasing convex function G : R+ R+ such that G(0) = lim s -> 0 G(s) > lim s ->infinity G(s) > 0 we consider an abstract version of the evolution equation partial derivative(tt)u(x, t) - Delta [G(0)u(x, t) + f(0)(infinity) G'(s)u(x, t - s)ds] = 0 arising from linear viscoelasticity.
引用
收藏
页码：189 / 232
页数：44
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