Long-time behaviour and stability for quasilinear doubly degenerate parabolic equations of higher order

• DOI: 10.48550/arxiv.2204.08231
• Publication Date: 2022-01-01
• Authors: Jonas Jansen; Christina Lienstr...; Katerina Nik
• Keywords: Power-law fluid; Generalized Newtonian fluid; Dilatant

We study the long-time behaviour of solutions to quasilinear doubly degenerate parabolic problems fourth order. The equations model for instance dynamic a non-Newtonian thin-film flow on flat impermeable bottom and with zero contact angle. consider shear-rate dependent fluid rheology which is described by constitutive power-law or Ellis-law viscosity. In all three cases, positive constants (i.e. films) are only steady-state solutions. Moreover, we can give detailed picture respect $H^1(\Omega)$-norm. case shear-thickening fluids, one observes that initially close steady state, converge equilibrium in finite time. shear-thinning case, find states polynomially stable sense that, as time tends infinity, at rate $1/t^{1/\beta}$ some $\beta > 0$. Finally, an Ellis-fluid, exponentially $H^1(\Omega)$.