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.
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.
Springer Book Archives
Table of Contents — Volume l.- On the Initial Growth of the Interfaces in Nonlinear Diffusion-Convection Processes.- Large Time Asymptotics for the Porous Media Equation.- Regularity of Flows in Porous Media: A Survey.- Ground States for the Prescribed Mean Curvature Equation: The Supercritical Case.- Geometric Concepts and Methods in Nonlinear Elliptic Euler-Lagrange Equations.- Nonlinear Parabolic Equations with Sinks and Sources.- Source-type Solutions of Fourth Order Degenerate Parabolic Equations.- Nonuniqueness and Irregularity Results for a Nonlinear Degenerate Parabolic Equation.- Existence and Meyers Estimates for Solutions of a Nonlinear Parabolic Variational Inequality.- Convergence to Traveling Waves for Systems of kolmogorov-like parabolic equations.- Symmetry Breaking in Semilinear Elliptic Equations with Critical Exponents.- Remarks on Saddle Points in the Calculus of Variations.- On the Elliptic Problem ?u - |?u|q + ?up = 0.- Nonlinear Elliptic Boundary Value Problems: Lyusternik-Schnirelman Theory, Nodal Properties and Morse Index.- Harnack-type Inequalities for some Degenerate Parabolic Equations.- The Inverse Power Method for Semilinear Elliptic Equations.- Radial Symmetry of the Ground States for a Class of Quasilinear Elliptic Equations.- Existence and Uniqueness of Ground State Solutions of Quasilinear Elliptic Equations.- Blow-up of Solutions of Nonlinear Parabolic Equations.- Solutions of Diffusion Equations in Channel Domains.- A Strong Form of the Mountain Pass Theorem and Application.- Asymptotic Behaviour of Solutions of the Porous Media Equation with Absorption.
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