Rita Cirmi, GiuseppaDíaz Díaz, Jesús Ildefonso2023-06-222023-06-222022-12[1] F. Andreu, V. Caselles, J.I. Díaz and J.M. Mazón, Some Qualitative properties for the Total Variation Flow, J. Funct. Anal. 188 (2002), 516-547. [2] S.N. Antontsev, J.I. Díaz and S.I. Shmarev, Energy Methods for Free Boundary Problems: Applications to Nonlinear PDEs and Fluid Mechanics, Progress in Nonlinear Differential Equations and Their Applications, 48, Birkhäuser, Boston, 2002. [3] G. Bellettini, V. Caselles, M. Novaga, The Total Variation Flow in RN. J. Differential Equations, 184, 475-525 (2002). [4] Ph. Benilan, J. I. Díaz. Comparison of solutions of nonlinear evolutions equations with different nonlinear terms. Israel Journal of Mathematics, 42 3 (1982), 241-257. [5] Ph.Bénilan and C. Picard, Quelques aspects non lineaires du principe du maximum. In: Séminaire de Théorie du Potentiel, Paris, No. 4, F. Hirsch and G. 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Hästö, Double phase image restoration, J. Math. Anal. Appl. 501 1 (2021), 123832. [28] J.W. He and R. Glowinski: Steady Bingham fluid flow in cylindrical pipes: a time dependent approach to the iterative solution, Numerical Algebra with Applications 7 (2000), 381-428. [29] S. Kamin, S. and L. Véron, Flat core properties associated to the p-Laplace operator, Proceedings of the American Mathematical Society 118 4,(1993), 1079-1085. [30] G. Mingione and V. Radulescu, Recent developments in problems with nonstandard growth and nonuniform ellipticity, Journal of Mathematical Analysis and Applications 501 1 (2021), 125-197. [31] J. Mossino, and J.M. Rakotoson, Isoperimetric inequalities in parabolic equations, Ann.Scuola Norm.Pisa, b13, (1986), 51-73. [32] P. Mossolov and V. Miasnikov, Variational methods in the theory of the fluidity of a viscous-plastic medium, Journal of Mechanics and Applied Mathematics, 73 (1965), 468-492. [33] L. Orsina and A. C. Ponce, Flat solutions of the 1-Laplacian equation. Rend. Istit. Mat. Univ. Trieste 49 (2017), 41–51. [34] S. Segura de León and C. M. Webler, Global existence and uniqueness for the inhomogeneous 1-Laplace evolution equation, Nonlinear Differential Equations and Applications NoDEA 22 (2015), 1213-1246. [35] G. Talenti, Best constant in Sobolev inequality, Annali di Matematica Pura ed Applicata 110 (1976), 353-372. [36] G. Talenti: Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces, Ann. Mat. Pura Appl. IV, 120 (1977), 159-184. [37] L.Waite and J. M. Fine, Applied Biofluid Mechanics, McGraw-Hill Professional Publishing, New York, 2007.https://hdl.handle.net/20.500.14352/72874We 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.engjournal articleopen access517.9Bingham flowsVariational inequalitiesFinite stopping timeSolid regionRearrangements comparisonFísica matemáticaEcuaciones diferencialesFunciones (Matemáticas)1202.07 Ecuaciones en Diferencias1202 Análisis y Análisis Funcional