146 research outputs found
Convergence to separate variables solutions for a degenerate parabolic equation with gradient source
The large time behaviour of nonnegative solutions to a quasilinear degenerate
diffusion equation with a source term depending solely on the gradient is
investigated. After a suitable rescaling of time, convergence to a unique
profile is shown for global solutions. The proof relies on the half-relaxed
limits technique within the theory of viscosity solutions and on the
construction of suitable supersolutions and barrier functions to obtain optimal
temporal decay rates and boundary estimates. Blowup of weak solutions is also
studied
Positivity, decay, and extinction for a singular diffusion equation with gradient absorption
We study qualitative properties of non-negative solutions to the Cauchy
problem for the fast diffusion equation with gradient absorption \partial_t u
-\Delta_{p}u+|\nabla u|^{q}=0\quad in\;\; (0,\infty)\times\RR^N, where , , and . Based on gradient estimates for the solutions, we
classify the behavior of the solutions for large times, obtaining either
positivity as for , optimal decay estimates as
for , or extinction in finite time for . In addition, we show how the diffusion prevents extinction in finite
time in some ranges of exponents where extinction occurs for the non-diffusive
Hamilton-Jacobi equation
A gradient flow approach to a thin film approximation of the Muskat problem
A fully coupled system of two second-order parabolic degenerate equations
arising as a thin film approximation to the Muskat problem is interpreted as a
gradient flow for the 2-Wasserstein distance in the space of probability
measures with finite second moment. A variational scheme is then set up and is
the starting point of the construction of weak solutions. The availability of
two Liapunov functionals turns out to be a central tool to obtain the needed
regularity to identify the Euler-Lagrange equation in the variational scheme
Weak solutions to lubrication equations in the presence of strong slippage
The existence of global weak solutions is proved for one-dimensional
lubrication models that describe the dewetting process of nanoscopic thin
polymer films on hydrophobyzed substrates and take account of large slippage at
the polymer-substrate interface. The convergence of these solutions as either
the Reynolds number or the capillarity goes to zero, as well as their limiting
behaviour as the slip length goes to zero or infinity are investigated
Global weak solutions for a degenerate parabolic system modeling the spreading of insoluble surfactant
We prove global existence of a nonnegative weak solution to a degenerate
parabolic system, which models the spreading of insoluble surfactant on a thin
liquid film
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