L.A. Peletier - Böcker

This volume is the proceedings of the IMA workshop "Degenerate Diffusions" held at the University of Minnesota from May 13-May 18, 1991. The workshop consisted of two parts. The emphasis of the first four days was on current progress or new problems in nonlinear diffusions involving free boundaries or sharp interfaces. Analysts and geometers will find some of the mathematical models described in this volume interesting; and the papers of more pure mathematical nature included here should provide applied mathematicians with powerful methods and useful techniques in handling singular perturbation problems as well as free boundary problems. The last two days of the workshop were a celebration of James Serrin's 65th birthday. A wide range of topics was covered in this part of the workshop. As a consequence, the scope of this book is much broader than what the title Degenerate Diffusions might suggest.

Nonlinear diffusion equations have held a prominent place in the theory of partial differential equations, both for the challenging and deep math­ ematical questions posed by such equations and the important role they play in many areas of science and technology. Examples of current inter­ est are biological and chemical pattern formation, semiconductor design, environmental problems such as solute transport in groundwater flow, phase transitions and combustion theory. Central to the theory is the equation Ut = ~cp(U) + f(u). Here ~ denotes the n-dimensional Laplacian, cp and f are given functions and the solution is defined on some domain n x [0, T] in space-time. FUn­ damental questions concern the existence, uniqueness and regularity of so­ lutions, the existence of interfaces or free boundaries, the question as to whether or not the solution can be continued for all time, the asymptotic behavior, both in time and space, and the development of singularities, for instance when the solution ceases to exist after finite time, either through extinction or through blow up.

This monograph is divided into two parts. The first part presents unifying properties of a family of the new higher order model equations, as well as a new mathematical way of studying them. Topics include the stationary solutions of the Swift-Hohenberg equation and the extended Fisher-Kolmogorov equation which are well established fourth non-linear model equations of parabolic type. In the second part, the exposition centres around applications of the theory presented earlier. Exercises for self-study or classroom use are included.

The study of spatial patterns in extended systems, and their evolution with time, poses challenging questions for physicists and mathematicians alike. Waves on water, pulses in optical fibers, periodic structures in alloys, folds in rock formations, and cloud patterns in the sky: patterns are omnipresent in the world around us. Their variety and complexity make them a rich area of study. In the study of these phenomena an important role is played by well-chosen model equations, which are often simpler than the full equations describing the physical or biological system, but still capture its essential features. Through a thorough analysis of these model equations one hopes to glean a better under­ standing of the underlying mechanisms that are responsible for the formation and evolution of complex patterns. Classical model equations have typically been second-order partial differential equations. As an example we mention the widely studied Fisher-Kolmogorov or Allen-Cahn equation, originally proposed in 1937 as a model for the interaction of dispersal and fitness in biological populations. As another example we mention the Burgers equation, proposed in 1939 to study the interaction of diffusion and nonlinear convection in an attempt to understand the phenomenon of turbulence. Both of these are nonlinear second-order diffusion equations.

Nonlinear diffusion equations have held a prominent place in the theory of partial differential equations, both for the challenging and deep math­ ematical questions posed by such equations and the important role they play in many areas of science and technology. Examples of current inter­ est are biological and chemical pattern formation, semiconductor design, environmental problems such as solute transport in groundwater flow, phase transitions and combustion theory. Central to the theory is the equation Ut = ~cp(U) + f(u). Here ~ denotes the n-dimensional Laplacian, cp and f are given functions and the solution is defined on some domain n x [0, T] in space-time. FUn­ damental questions concern the existence, uniqueness and regularity of so­ lutions, the existence of interfaces or free boundaries, the question as to whether or not the solution can be continued for all time, the asymptotic behavior, both in time and space, and the development of singularities, for instance when the solution ceases to exist after finite time, either through extinction or through blow up.

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.

In recent years considerable interest has been focused on nonlinear diffu­ sion problems, the archetypical equation for these being Ut = D.u + f(u). Here D. denotes the n-dimensional Laplacian, the solution u = u(x, t) is defined over some space-time domain of the form n x [O,T], and f(u) is a given real function whose form is determined by various physical and mathematical applications. These applications have become more varied and widespread as problem after problem has been shown to lead to an equation of this type or to its time-independent counterpart, the elliptic equation of equilibrium D.u + f(u) = o. Particular cases arise, for example, in population genetics, the physics of nu­ clear stability, phase transitions between liquids and gases, flows in porous media, the Lend-Emden equation of astrophysics, various simplified com­ bustion models, and in determining metrics which realize given scalar or Gaussian curvatures. In the latter direction, for example, the problem of finding conformal metrics with prescribed curvature leads to a ground state problem involving critical exponents. Thus not only analysts, but geome­ ters as well, can find common ground in the present work. The corresponding mathematical problem is to determine how the struc­ ture of the nonlinear function f(u) influences the behavior of the solution.