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

• Main Authors: Jansen, Jonas, Lienstromberg, Christina, Nik, Katerina
• Format: Journal Article
• Language: English
• Published: 18-04-2022
• Subjects: Mathematics - Analysis of PDEs
• DOI: 10.48550/arxiv.2204.08231

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