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.
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.
Springer Book Archives
Table of Contents — Volume2.- Uniqueness of Non-negative Solutions of a Class of Semi-linear Elliptic Equations.- Some Qualitative Properties of Nonlinear Partial Differential Equations.- On Existence of Solutions of Non-coercive Problems.- Diffusion-Reaction Systems in Neutron-Fission Reactors and Ecology.- Numerical Searches for Ground State Solutions of a Modified Capillary Equation and for Solutions of the Charge Balance Equation.- On Positive Solutions of Semilinear Elliptic Equations in Unbounded Domains.- The Behavior of Solutions of a Nonlinear Boundary Layer Equation.- Asymptotic Behavior of Solutions of Semilinear Heat Equations on S1.- Some Uniqueness Theorems for Exterior Boundary Value Problems.- Some Aspects of Semi linear Elliptic Equations in ?n.- Global Existence Results for a Strongly Coupled Quasilinear Parabolic System.- A Survey of Some Superlinear Problems.- A Priori Estimates for Reaction-Diffusion Systems.- Qualitative Behavior for a Class of Reaction-Diffusion-Convection Equations.- Resonance and Higher Order Quasilinear Ellipticity.- Bifucation from Symmetry.- Positive Solutions of Semilinear Elliptic Equations on General Domains.- The Mathematics of Porous Medium Combustion.- Connection Problems Arising from Nonlinear Diffusion Equations.- Singularities of some Quasilinear Equations.
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