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Memory driven instability in a diffusion process

Duffy, B.R. and Grinfeld, M. and Freitas, P. (2002) Memory driven instability in a diffusion process. SIAM Journal on Mathematical Analysis, 33 (5). pp. 1090-1106. ISSN 0036-1410

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We consider the ndimensional version of a model proposed by Olmstead et al. [SIAM J. Appl. Math., 46 (1986), pp. 171--188] for the flow of a non-Newtonian fluid in the presence of memory. We prove the existence of a global attractor and obtain conditions for the existence of a Lyapunov functional, which allows us to give a full description of this attractor in a certain region of the parameter space in the bistable case. We then study the stability and bifurcation of stationary solutions and, in particular, prove that for certain values of the parameters it is not possible to stabilize the flow by increasing a Rayleigh-type number. The existence of periodic and homoclinic orbits is also shown by studying the Bogdanov--Takens singularity obtained from a center manifold reduction.