Nonlinear Diffusion Equations and Their Equilibrium States I: Proceedings of a Microprogram held August 25–September 12, 1986 (Mathematical Sciences Research Institute Publications, 12)

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. Front Matter....Pages iii-xiii On the Initial Growth of the Interfaces in Nonlinear Diffusion-Convection Processes....Pages 1-20 Large Time Asymptotics for the Porous Media Equation....Pages 21-34 Regularity of Flows in Porous Media: A Survey....Pages 35-49 Ground States for the Prescribed Mean Curvature Equation: The Supercritical Case....Pages 51-74 Geometric concepts and methods in nonlinear elliptic Euler-Lagrange Equations....Pages 75-107 Nonlinear Parabolic Equations with Sinks and Sources....Pages 109-121 Source-type Solutions of Fourth Order Degenerate Parabolic Equations....Pages 123-146 Nonuniqueness and Irregularity Results for a Nonlinear Degenerate Parabolic Equation....Pages 147-159 Existence and Meyers estimates for solutions of a nonlinear parabolic variational inequality....Pages 161-177 Convergence to Traveling Waves for Systems of Kolmogorov-like Parabolic Equations....Pages 179-190 Symmetry Breaking in Semilinear Elliptic Equations with Critical Exponents....Pages 191-215 Remarks on Saddle Points in the Calculus of Variations....Pages 217-235 On the Elliptic Problem ∆ u - |∇ u | q + λ u p = 0....Pages 237-243 Nonlinear elliptic boundary value problems: Lyusternik-Schnirelman theory, nodal properties and Morse index....Pages 245-265 Harnack-type Inequalities for some Degenerate Parabolic Equations....Pages 267-271 The Inverse Power Method for Semilinear Elliptic Equations....Pages 273-286 Radial Symmetry of the Ground States for a Class of Quasilinear Elliptic Equations....Pages 287-292 Existence and Uniqueness of Ground State Solutions of Quasilinear Elliptic Equations....Pages 293-300 Blow-up of solutions of nonlinear parabolic equations....Pages 301-318 Solutions of Diffusion Equations in Channel Domains....Pages 319-339 A Strong Form of the Mountain Pass Theorem and Application....Pages 341-350 Asymptotic Behaviour of Solutions of the Porous Media Equation with Absorption....Pages 351-359