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Up to now, we only obtain the weak precompactness properties of ((nablaln(n_{varepsilon}+1))_{varepsilonin(0,1)}) and ((nabla w_{varepsilon})_{varepsilonin 0,1)}) in (L^{2}_{mathrm{loc}}(bar {Omega}times[0,infty))), which do not satisfy the strong compact requirement in this space in the cross-diffusion in (2.11 1 by passing to the limit.
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