Roberta Dal Passo

On a fourth-order degenerate parabolic equation: global entropy estimates, existence, and qualitative behavior of solutions

By means of energy and entropy estimates, we prove existence and positivity results in higher space dimensions for degenerate parabolic equations of fourth order with nonnegative initial values. We discuss their asymptotic behavior for

Nonuniqueness for the heat flow of harmonic maps on the disk

t\to \infty

Hyperbolic Phenomena in a Strongly Degenerate Parabolic Equation

and give a counterexample to uniqueness.

Discontinuous “viscosity” solutions of a degenerate parabolic equation

Abstract We prove that for suitable initial data the heat flow of harmonic maps, from the disk to the sphere, admits infinitely many solutions, characterised by “backward bubbling” at some arbitrarily large time, all having uniformly bounded energy.

Nonuniqueness of solutions of a degenerate parabolic equation

We consider the equation ut=(ϕ(u) ψ (ux))x, where ϕ>0 and where ψ is a strictly increasing function with lims→∞ψ=ψ∞<∞. We solve the associated Cauchy problem for an increasing initial function, and discuss to what extent the solution behaves qualitatively like solutions of the first-order conservation law ut=ψ∞(ϕ(u))x. Equations of this type arise, for example, in the theory of phase transitions where the corresponding free-energy functional has a linear growth rate with respect to the gradient.

THE THIN FILM EQUATION WITH NONLINEAR DIFFUSION

We study a nonlinear degenerate parabolic equation of the second order. Regularizing the equation by adding some artificial viscosity, we construct a generalized solution. We show that this solution is not necessarily continuous at all points.

Point Singularities and Nonuniqueness for the Heat Flow for Harmonic Maps

SummaryWe give some results about nonuniqueness of the solutions of the Cauchy problem for a class of nonlinear degenerate parabolic equations arising in several applications in biology and physics. This phenomenon is a truly nonlinear one and occurs because of the degeneracy of the equation at the points where u=0. For a given set of values of the parameter involved, we prove that there exists a one parameter family of weak solutions; moreover, restricting the parameter set, nonuniqueness appears even in the class of classical solutions.

A free boundary problem arising in a simplified tumour growth model of contact inhibition

We consider the fourth order degenerate parabolic equation ut þ div ður u uruÞ 1⁄4 0, u 0, ð1:1Þ where m 2 R and n 2 Rþ (precise assumptions on the parameters will be stated in the sequel). Equation (1.1) is introduced to describe the evolution of the height u of a thin liquid film spreading on a solid surface. The fourth-order term accounts for the effects of surface tension. The second-order term accounts, if m 1⁄4 n, for the effect of gravity; it has also been proposed, for different values of m, as a ‘‘porous-media cut-off ’’ of Van der Waals forces. The case n 1⁄4 m 1⁄4 1 describes the extent u of the region occupied by a liquid in the half-space Hele-Shaw cell in the lubrication regime. We refer to the review paper of Oron, Davis and Bankoff [1] for details. The case n 1⁄4 1,

Waiting Time Phenomena for Degenerate Parabolic Equations — A Unifying Approach

Abstract We consider weak solutions of the harmonic map flow between the three-dimensional unit ball B 3and the two-dimensional unit sphere, with as initial and boundary data. In this situation, we show the existence of infinitely many weak solutions. Indeed for any different from the origin we construct a weak solution whose singular set for all positive time is precisely Qand which satisfies a modified energy inequality.

A system of degenerate parabolic nonlinear PDE's: a new free boundary problem

It is observed in vitro and in vivo that when two populations of different types of cells come near to each other, the rate of proliferation of most cells decreases. This phenomenon is often called contact inhibition of growth between two cells. In this paper, we consider a simplified 1-dimensional PDEmodel for normal and abnormal cells, motivated by the paper by Chaplain, Graziano and Preziosi ([5]). We show that if the two populations are initially segregated, then they remain segregated due to the contact inhibition mechanism. In this case the system of PDE’s can be formulated as a free boundary problem.