A nonlinear fourth order diffusion problem: Convergence to the steady state and non-negativity of solutions

AbstractThe paper deals with nonlinear diffusion, both time-dependent and time-independent. The spatial terms in the partial differential equation (p.d.e.) contain a second order nonlinear part (where the non-negative diffusivity depends on the dependent variable) and a fourth order linear part. In the context of non-null, time-independent boundary conditions, convergence of the unsteady to the steady state is established. The second part of the paper discusses criteria on data ensuring non-negativity of the solutions. This is done for the steady state irrespective of the spatial dimension; and it is done for the unsteady state for the one-dimensional rectilinear case only, using a result from the first part of the paper

The evolution to a steady state for a porous medium model

AbstractAn initial boundary value problem is considered for a nonlinear diffusion equation, the diffusivity being a function of the dependent variable. Dirichlet boundary conditions, independent of time, are considered and positive solutions are assumed. This paper is mainly concerned with the rate of convergence, in time, of the unsteady to the steady state. This is done by obtaining an upper estimate for a positive-definite, integral measure of the perturbation (i.e., unsteady–steady state) using differential inequality techniques.A previous result is recalled where the diffusivity k(τ)=τn (n being a positive constant) appropriate to mass transport, or filtration, in a porous medium. The present paper treats an alternative model, sharing some of the characteristics of the previous one: k(τ)=eτ−1, τ being non-negative.The paper concludes by considering a “backwards in time” initial boundary value problem for the perturbation (amenable to the same techniques) and establishes that the solution ceases to exist beyond a critical, computable time