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## Über dieses Buch

This IMA Volume in Mathematics and its Applications DEGENERATE DIFFUSIONS is based on the proceedings of a workshop which was an integral part of the 1990- 91 IMA program on "Phase Transitions and Free Boundaries". The aim of this workshop was to provide some focus in the study of degenerate diffusion equations, and by involving scientists and engineers as well as mathematicians, to keep this focus firmly linked to concrete problems. We thank Wei-Ming Ni, L.A. Peletier and J.L. Vazquez for organizing the meet­ ing. We especially thank Wei-Ming Ni for editing the proceedings. We also take this opportunity to thank those agencies whose financial support made the workshop possible: the Army Research Office, the National Science Foun­ dation, and the Office of Naval Research. A vner Friedman Willard Miller, Jr. PREFACE This volume is the proceedings of the IMA workshop "Degenerate Diffusions" held at the University of Minnesota from May 13 to May 18, 1991.

## Inhaltsverzeichnis

### Nonlinear, Nonlocal Problems of Fluid-Solid Interactions

Abstract
In this paper we describe the analysis of several steady-state problems of flow past nonlinearly elastic shells. There is an extensive engineering literature on fluid-solid interactions, encompassing extremely interesting dynamical problems, much of which is based on the use of ad hoc models for the force field exerted by the fluid on the deformable solid. One of our goals is to treat these interactions honestly. To do so, we restrict the class of flows by endowing them with a full panoply of adjectives from fluid dynamics. Specifically, we study steady, planar or axisymmetric, irrotational flows of an inviscid fluid. Most of our results are for incompressible fluids. The flow is assumed to have a prescribed velocity U and pressure P at infinity.
Stuart S. Antman, Massimo Lanza De Cristoforis

### Curvature Dependent Phase Boundary Motion and Parabolic Double Obstacle Problems

Abstract
The use of parabolic double obstacles problems for approximating curvature dependent phase boundary motion is reviewed. It is shown that such problems arise naturally in multi-component diffusion with capillarity. Formal matched asymptotic expansions are employed to show that phase field models with order parameter solving an obstacle problem approximate curvature dependent phase boundary motion. Numerical simulations of surfaces evolving according to their mean curvature are presented.
J. F. Blowey, C. M. Elliott

### On the Harnack Inequality for Non-Negative Solutions of Singular Parabolic Equations

Abstract
This note is to announce some new results and techniques in the theory of singular parabolic equations of the type
$$\left\{ {\begin{array}{*{20}{c}} {u \in {{C}_{{loc}}}\left( {0,T;L_{{loc}}^{2}\left( \Omega \right)} \right),{{{\left| u \right|}}^{m}} \in L_{{loc}}^{2}\left( {0,T;W_{{loc}}^{{1,2}}\left( \Omega \right)} \right)0 < m < 1,} \hfill \\ {{{u}_{t}} - {{{\left( {{{a}_{{ij}}}\left( {x,t} \right){{{\left| u \right|}}^{{m - 1}}}{{u}_{{{{x}_{i}}}}}} \right)}}_{x}}_{{_{j}}} = 0in{{\Omega }_{T}}} \hfill \\ \end{array} } \right.$$
(1.1)
Ya-Zhe Chen, E. Dibenedetto

### A BMO Bound for Eigenfunctions on Riemannian Manifolds

Abstract
In this short note, we prove a BMO bound on eigenfunctions of the Laplacian on Riemannian manifold. This bound is closely related to a conjecture on the Hausdorff measure of the nodal set.
Rui-Tao Dong

### On Some Monotonicity in Time Properties for a Quasilinear Parabolic Equation with Source

Abstract
We consider the Cauchy problem for a one-dimensional quasilinear degenerate heat equation with source. The property of monotone in time behavior of the weak solution at a fixed spatial point x?ϵ R is studied. It is shown that the conditions of a such a behavior depend on “a nonlinear interaction” between the nonlinear heat operator and the source of energy considered. There are two different cases: (i) the solution is monotone in time for any x0 which is far enough from the initial support and (ii) the solution is monotone in time if it becomes large enough. In a general situation the monotone in time behavior at a given point xx0 is proved to depend on the shape of the initial function in some neighborhood of x = x0. Proofs are based on the method of intersection comparison of the solution and the continuous set of stationary solutions of the same equation.
Victor A. Galaktionov, Sergey A. Posashkov

### On the asymptotic Properties of leray’s Solutions to the Exterior Steady Three-Dimensional Navier-Stokes Equations with Zero Velocity at Infinity

Abstract
Let Ω0 be a compact region of the three-dimensional euclidean space R3. Assume that Ω:= R3 — Ω0 is filled with a viscous incompressible fluid whose motion m is governed by the Navier-Stokes equations.
Giovanni P. Galdi

### Some Results on Blow up for Semilinear Parabolic Problems

Abstract
We discuss on recent results concerning the asymptotics near blow-up of nonnega-tive solutions of
$$\begin{array}{*{20}{c}} {{{u}_{t}} = {{u}_{{xx}}} + f\left( u \right); - \infty < x < \infty ,t > 0} \hfill \\ {u\left( {x,0} \right) = {{u}_{0}}\left( x \right); - \infty < x < \infty ,} \hfill \\ \end{array}$$
where f (u)=u p with p > 1 or f (u) = eu, and u 0(x) is continuous, nonnegative and bounded. AMS(MOS) subject classifications. 35B40, 35K55, 35K57
M. A. Herrero, J. J. L. Velázquez

### Long-Time Behaviour of Solutions of Quasilinear Parabolic Equations

Abstract
1. Statement of the problem . In this note, we report on some joint work with Albert Milani concerning the existence and long-time behaviour of solutions to certain quasilinear parabolic initial-boundary value problems.
Kevin McLeod

### Spike-Layers in Semilinear Elliptic Singular Perturbation Problems†

Abstract
The purpose of this expository paper is to describe a new method, introduced in a series of papers [LNT], [NT1,2], [NPT] and [J], in handling “spikes” (or “point-condensation” phenomena) for singularly perturbed semilinear elliptic equations of the form
(1)
where $$\Delta = \sum\limits_{i = 1}^n {\frac{{{\partial ^2}}}{{\partial x_i^2}}}$$ is the Laplace operator in R n , and ε is a small positive number.
Wei-Ming Ni, Izumi Takagi

### Evolution of Nonparametric Surfaces with Speed Depending on Curvature, III. Some Remarks on Mean Curvature and Anisotropic flows

Abstract
This paper is a sequel to our paper [OU] where we investigated questions concerning solvability and asymptotic behavior of solutions to the mean curvature evolution problem
$${u_t} = \sqrt {1 + {{\left| {Du} \right|}^2}} H\left( u \right)\;in\quad \Omega \times \left( {0,\infty } \right),$$
(1.1)
$$u\left( {x,t} \right) = 0\quad on\quad \partial \Omega \times \left[ {0,\infty } \right),$$
(1.2)
$$u\left( {x,0} \right) = {u_0}\left( x \right)\quad in\quad \overline {\Omega ,} \quad {u_0} \in C_0^\infty \left( {\overline \Omega } \right)$$
(1.3)
where Ω is a bounded domain in R n , n ≥ 2, with C boundary ∂Ω, H is the mean curvature operator.
Vladimir I. Oliker, Nina N. Uraltseva

### Continuation and Limit Behavior for Damped Quasi-Variational Systems

Abstract
We shall be concerned with continuation and limit behavior as r → ∞ for solutions of the quasi-variational system
$${\left( {\nabla G\left( {u'} \right)} \right)^\prime } + f\left( {r,u} \right) = Q\left( {r,u,u'} \right),\quad r \in J = \left[ {R,\infty } \right).$$
(1.1)
Patrizia Pucci, James Serrin

### Multibump Solutions of a Semilinear Elliptic PDE on Rn

Abstract
The purpose of this paper is to describe some recent joint work of V. Coti Zelati and the author on semilinear elliptic partial differential equations on R n [1]. This research is in part an outgrowth of earlier work on homoclinic solutions of Hamiltonian systems of ordinary differential equations [2].
Paul H. Rabinowitz

### Einstein/Yang-Mills Equations

To James Serrin on his 65th birthday
Abstract
In this paper, we shall describe some recent results, ([6,7]), concerning the existence of smooth, globally defined static solutions of the Einstein /Yang-Mills (EYM) equations. The EYM equations extend Einstein’s celebrated gravitational field equations, to include other “force” fields: electromagnet ism, and the weak and strong nuclear force fields. It turns out that solutions of such coupled systems of equations, in n = 4 space-time dimensions, have some very interesting mathematical and physical properties. Thus for the non-abelian gauge group G = SU (2), we can prove the existence of infinitely many non-singular globally defined solutions - a result that fails to hold if G = U (1), (the coupling of gravity to electromagnetism). Since the (classical) Yang-Mills equations with G = SU (2) correspond to the weak nuclear force, our result indicates that coupling gravity to the weak nuclear force can prevent singularity formation in space-time. Furthermore, these solutions give rise to constants which correspond to masses. In other words, numbers representing masses are a direct consequence of our theory, and need not be prescribed in an ad-hoc manner - as is usually done in physical theories.
J. Smoller, A. Wasserman

### The Dirichlet Problem for Functions of Least Gradient

Abstract
For a given domain Ω R n, we consider the variational problem of minimizing the L1-norm of the gradient on Ωof a function with prescribed continuous boundary values. Under certain weak conditions on the boundary of the domain Ω, it is shown that the BV solution is continuous and unique.
Peter Sternberg, William P. Ziemer

### Asymptotic behaviour of nonlinear Parabolic Equations. Anomalous Exponents

Abstract
In this lecture I will try to present a certain view of the progress achieved in recent years on the subject of asymptotic behaviour of nonlinear parabolic equations. This is an enormously rich field since we can play with the domain, the initial data, the boundary data and the form of the equation. Therefore, a selection of the topics to treat is necessary. Our interest will be centered around the stabilization of solutions to an equilibrium state, with special emphasis on the calculation of the characteristic exponents, namely the decay rate and the expansion rate.
Juan Luis Vazquez
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