Upper semicontinuity for a class of nonlocal evolution equations with Neumann condition

We show the normal hyperbolicity property for the equilibria of the evolution equation∂m(r,t)/∂t=-m(r,t)+g(βJ*m(r,t)+βh), h,β≥0,and using the normal hyperbolicity property we prove the continuity (upper semicontinuity and lower semicontinuity) of the global attractors of the flow generated by this equation, with respect to functional parameterJ.

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On a class of nonlocal evolution equations with the p[u(x,t)]-Laplace operator

Nonlinear Analysis Real World Applications ◽

10.1016/j.nonrwa.2020.103165 ◽

2020 ◽

Vol 56 ◽

pp. 103165

Author(s):

Stanislav Antontsev ◽

Sergey Shmarev

Keyword(s):

Laplace Operator ◽

Evolution Equations ◽

Nonlocal Evolution Equations

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Almost periodic traveling waves of nonlocal evolution equations

Nonlinear Analysis ◽

10.1016/s0362-546x(01)00787-8 ◽

2002 ◽

Vol 50 (6) ◽

pp. 807-838 ◽

Cited By ~ 14

Author(s):

Fengxin Chen

Keyword(s):

Traveling Waves ◽

Evolution Equations ◽

Almost Periodic ◽

Periodic Traveling Waves ◽

Nonlocal Evolution Equations

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Partial differential hemivariational inequalities

Advances in Nonlinear Analysis ◽

10.1515/anona-2016-0102 ◽

2018 ◽

Vol 7 (4) ◽

pp. 571-586 ◽

Cited By ~ 27

Author(s):

Zhenhai Liu ◽

Shengda Zeng ◽

Dumitru Motreanu

Keyword(s):

Evolution Equations ◽

Existence Of Solutions ◽

Mild Solutions ◽

Upper Semicontinuity ◽

Solution Set ◽

Hemivariational Inequality ◽

Hemivariational Inequalities ◽

Well Posedness ◽

Partial Differential ◽

New Class

AbstractThe aim of this paper is to introduce and study a new class of problems called partial differential hemivariational inequalities that combines evolution equations and hemivariational inequalities. First, we introduce the concept of strong well-posedness for mixed variational quasi hemivariational inequalities and establish metric characterizations for it. Then we show the existence of solutions and meaningful properties such as measurability and upper semicontinuity for the solution set of the mixed variational quasi hemivariational inequality associated to the partial differential hemivariational inequality. Relying, on these properties we are able to prove the existence of mild solutions for partial differential hemivariational inequalities. Furthermore, we show the compactness of the set of the corresponding mild trajectories.

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