Nonlinear Diffusion Equations and Their Equilibrium States II: Proceedings of a Microprogram held August 25–September 12, 1986 (Mathematical Sciences Research Institute Publications, 13)

In recent years considerable interest has been focused on nonlinear diffu­ sion problems, the archetypical equation for these being Ut = ~U + f(u). Here ~ 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 ~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 i-xiii Uniqueness of non-negative solutions of a class of semi-linear elliptic equations....Pages 1-17 Some qualitative properties of nonlinear partial differential equations....Pages 19-31 On Existence of Solutions for Non-coercive Problems....Pages 33-39 Diffusion-Reaction Systems in Neutron-Fission Reactors and Ecology....Pages 41-53 Numerical Searches for Ground State Solutions of a Modified Capillary Equation and for Solutions of the Charge Balance Equation....Pages 55-83 On Positive Solutions of Semilinear Elliptic Equations in Unbounded Domains....Pages 85-122 The Behavior of Solutions of a Nonlinear Boundary Layer Equation....Pages 123-137 Asymptotic behavior of solutions of semilinear heat equations on S 1 ....Pages 139-162 Some Uniqueness Theorems for Exterior Boundary Value Problems....Pages 163-169 Some Aspects of Semilinear Elliptic Equations on R n ....Pages 171-205 Global Existence Results for a Strongly Coupled Quasilinear Parabolic System....Pages 207-216 A Survey of Some Superlinear Problems....Pages 217-233 A Priori Estimates for Reaction-Diffusion Systems....Pages 235-244 Qualitative Behavior for a Class of Reaction-Diffusion-Convection Equations....Pages 245-253 Resonance and Higher Order Quasilinear Ellipticity....Pages 255-272 Bifurcation from Symmetry....Pages 273-287 Positive Solutions of Semilinear Elliptic Equations on General Domains....Pages 289-293 The Mathematics of Porous Medium Combustion....Pages 295-313 Connection Problems Arising from Nonlinear Diffusion Equations....Pages 315-332 Singularities of some Quasilinear Equations....Pages 333-365 In recent years considerable interest has been focused on nonlinear diffuƯ sion problems, the archetypical equation for these being Ut = ̃U + f(u). Here ̃ 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 ̃u+f(u)=O. Particular cases arise, for example, in population genetics, the physics of nuƯ clear stability, phase transitions between liquids and gases, flows in porous media, the Lend-Emden equation of astrophysics, various simplified comƯ 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 geomeƯ ters as well, can find common ground in the present work. The corresponding mathematical problem is to determine how the strucƯ ture of the nonlinear function f(u) influences the behavior of the solution