Nonlinear Diffusion Equations and Their Equilibrium States II

1. Front Matter

   Pages i-xiii

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2. Uniqueness of non-negative solutions of a class of semi-linear elliptic equations

   • Hans G. Kaper, Man Kam Kwong

   Pages 1-17

3. Some qualitative properties of nonlinear partial differential equations

   • Bernhard Kawohl

   Pages 19-31

4. On Existence of Solutions for Non-coercive Problems

   • Philip Korman

   Pages 33-39

5. Diffusion-Reaction Systems in Neutron-Fission Reactors and Ecology

   • Anthony W. Leung

   Pages 41-53

6. Numerical Searches for Ground State Solutions of a Modified Capillary Equation and for Solutions of the Charge Balance Equation

   • Howard A. Levine

   Pages 55-83

7. On Positive Solutions of Semilinear Elliptic Equations in Unbounded Domains

   • P. L. Lions

   Pages 85-122

8. The Behavior of Solutions of a Nonlinear Boundary Layer Equation

   • Chunqing Lu, William C. Troy

   Pages 123-137

9. Asymptotic behavior of solutions of semilinear heat equations on S ¹

   • Hiroshi Matano

   Pages 139-162

10. Some Uniqueness Theorems for Exterior Boundary Value Problems

    • Kevin McLeod

    Pages 163-169

11. Some Aspects of Semilinear Elliptic Equations on ℝn

    • Wei-Ming Ni

    Pages 171-205

12. Global Existence Results for a Strongly Coupled Quasilinear Parabolic System

    • M. A. Pozio, A. Tesei

    Pages 207-216

13. A Survey of Some Superlinear Problems

    • Paul H. Rabinowitz

    Pages 217-233

14. A Priori Estimates for Reaction-Diffusion Systems

    • Franz Rothe

    Pages 235-244

15. Qualitative Behavior for a Class of Reaction-Diffusion-Convection Equations

    • Paul Sacks

    Pages 245-253

16. Resonance and Higher Order Quasilinear Ellipticity

    • Victor L. Shapiro

    Pages 255-272

17. Bifurcation from Symmetry

    • J. Smoller, A. G. Wasserman

    Pages 273-287

18. Positive Solutions of Semilinear Elliptic Equations on General Domains

    • J. Smoller, A. G. Wasserman

    Pages 289-293

19. The Mathematics of Porous Medium Combustion

    • A. M. Stuart

    Pages 295-313

20. Connection Problems Arising from Nonlinear Diffusion Equations

    • D. Terman

    Pages 315-332

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.

• Fusion
• Mathematica
• Scala
• behavior
• curvature
• differential equation
• equation
• form
• function
• genetics
• metrics
• online
• partial differential equation
• stability

• School of Mathematics, University of Minnesota, Minneapolis, USA

  W.-M. Ni, James Serrin

• Mathematical Sciences Research Institute, Berkeley, USA

  W.-M. Ni, James Serrin

• Department of Mathematics and Computer Science, University of Leiden, The Netherlands

  L. A. Peletier

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