Ebook: Nonlinear Diffusion Equations and Their Equilibrium States II: Proceedings of a Microprogram held August 25–September 12, 1986

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.

---

---

Content:
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 ?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

---

Content:
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 ?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
....