Quantitative asymptotic estimates for evolution problems

Asymptotic stability estimates for some evolution problems with singular convection field

“…We remark that stability and continuity estimates similar to the ones of Theorem 3.1 are also available in [7][8][9]17]. On the other hand, it is also worth to mention that also pointwise estimates of the spatial gradient are available in literature (see e.g.…”

“…3 we will actually prove a result for a general Cauchy-Dirichlet problem, in such a way that Theorem 1.1 is a special case of this statement. The presence of the lower order term does not allow to follow [17]. We establish an estimate of decay of the super-level sets of the solution which is fundamental in order to obtain our result.…”

“…Comparison quantitative estimates between solutions of evolutionary and stationary problems as in (1.9) or (1.10) (see also (3.19) or (3.20) below) are available in [17] for equations not having lower order terms. It should be also worth to mention that recently in [10] new estimates for the behaviour at infinity of solutions to a wide class of parabolic partial differential equations (including also anisotropic type equations) have been considered.…”

“…An essential tool in the study of the time behaviour of our problem relies on the following result, whose proof can be found in [17].…”

On the behavior in time of solutions to motion of Non-Newtonian fluids

“…Thanks to the parabolic embedding, any weak solution u belongs to 12) and the monotonicity condition 13) the first results on existence of solutions were proved by O. Ladyzhenskaya in [8] and by J.L. Lions in [10] under the restriction that…”