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.