Qualitative properties of solutions of some quasilinear equations related to Bingham fluids

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We consider a quasilinear parabolic equation and its associate stationary problem which correspond to a simplified formulation of a Bingham flow and we mainly study two qualitative properties. The first one concerns with the Absence and, respectively, disappearance in finite time, of the movement. We show that there is a suitable balance between the L1-norm of the forcing datum f∞ and the measure of the spatial domain Ω (essentially saying that the forcing daum must be small enough) such that the corresponding solution u∞(x) of the stationary problem is such that u∞ ≡ 0 a.e. in Ω (even if f∞ ≠ 0). Moreover, if f∞ is also the forcing term of the parabolic problem, and if the above mentioned balance is strict, for any u0 ∈ L(Ω) there exists a finite time Tu0,f∞ > 0 such that the unique solution u(t,x) of the parabolic problem globally stops after Tu0,f∞, in the sense that u(t,x) ≡ 0 a.e. in Ω, for any t ≥ Tu0,f∞. The second property concerns with the Formation of a positively measure “solid region”. We show that if the above balance condition fails (i.e., when the forcing datum is large enough) then the solution u∞(x) of the stationary problem satisfies that u∞ ≠ 0 in Ω and its “solid region” (defined as the set S(u∞) = {x ∈ Ω : ∇u∞(x) = 0}) has a positive measure. Similar results are obtained for the symmetric solutions u(t) of the parabolic problem. In addition the convergence u(t) → u∞ in H10 (Ω), as t → +∞, does not take place in any finite time.
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