Indirect stabilization of coupled abstract evolution equations with memory: Different propagation speeds

2021 ◽  
Author(s):  
Kun‐Peng Jin
Keyword(s):  
Evolution Equations ◽  
Abstract Evolution Equations ◽  
With Memory

Continuous families of exponential attractors for singularly perturbed equations with memory

2010 ◽  
Vol 140 (2) ◽  
pp. 329-366 ◽  
Author(s):  
Stefania Gatti ◽  
Alain Miranville ◽  
Vittorino Pata ◽  
Sergey Zelik
Keyword(s):  
Evolution Equations ◽  
General Theorem ◽  
Perturbation Parameter ◽  
Singular Limit ◽  
Singularly Perturbed ◽  
Exponential Attractors ◽  
Abstract Evolution Equations ◽  
Singularly Perturbed Equations ◽  
Image Position ◽  
With Memory

For a family of semigroups Sε(t) : ℌε → ℌε depending on a perturbation parameter ε ∈ [0, 1], where the perturbation is allowed to become singular at ε = 0, we establish a general theorem on the existence of exponential attractors εε satisfying a suitable Hölder continuity property with respect to the symmetric Hausdorff distance at every ε ∈ [0, 1]. The result is applied to the abstract evolution equations with memorywhere kε(s) = (1/ε)k(s/ε) is the rescaling of a convex summable kernel k with unit mass. Such a family can be viewed as a memory perturbation of the equationformally obtained in the singular limit ε → 0.

Solvability of the abstract evolution equations in L s -spaces with critical temporal weights

2021 ◽  
Vol 19 (1) ◽  
pp. 111-120
Author(s):  
Qinghua Zhang ◽  
Zhizhong Tan
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Evolution Equation ◽  
Nonlinear Operator ◽  
Evolution Equations ◽  
Bounded Function ◽  
Inhomogeneous Term ◽  
Linear Evolution ◽  
Abstract Evolution Equations ◽  
Linear Evolution Equation

Abstract This paper deals with the abstract evolution equations in L s {L}^{s} -spaces with critical temporal weights. First, embedding and interpolation properties of the critical L s {L}^{s} -spaces with different exponents s s are investigated, then solvability of the linear evolution equation, attached to which the inhomogeneous term f f and its average Φ f \Phi f both lie in an L 1 / s s {L}_{1\hspace{-0.08em}\text{/}\hspace{-0.08em}s}^{s} -space, is established. Based on these results, Cauchy problem of the semi-linear evolution equation is treated, where the nonlinear operator F ( t , u ) F\left(t,u) has a growth number ρ ≥ s + 1 \rho \ge s+1 , and its asymptotic behavior acts like α ( t ) / t \alpha \left(t)\hspace{-0.1em}\text{/}\hspace{-0.1em}t as t → 0 t\to 0 for some bounded function α ( t ) \alpha \left(t) like ( − log t ) − p {\left(-\log t)}^{-p} with 2 ≤ p < ∞ 2\le p\lt \infty .

scholarly journals Controllability of Evolution Equations with Memory

2017 ◽  
Vol 55 (4) ◽  
pp. 2437-2459 ◽  
Author(s):  
Felipe W. Chaves-Silva ◽  
Xu Zhang ◽  
Enrique Zuazua
Keyword(s):  
Evolution Equations ◽  
With Memory
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