Nonlinear Diffusion Equations and Their Equilibrium States, 3

Nonlinear diffusion equations have held a prominent place in the theory of partial differential equations, both for the challenging and deep math­ ematical questions posed by such equations and the important role they play in many areas of science and technology. Examples of current inter­ est are biological and chemical pattern formation, semiconductor design, environmental problems such as solute transport in groundwater flow, phase transitions and combustion theory. Central to the theory is the equation Ut = ~cp(U) + f(u). Here ~ denotes the n-dimensional Laplacian, cp and f are given functions and the solution is defined on some domain n x [0, T] in space-time. FUn­ damental questions concern the existence, uniqueness and regularity of so­ lutions, the existence of interfaces or free boundaries, the question as to whether or not the solution can be continued for all time, the asymptotic behavior, both in time and space, and the development of singularities, for instance when the solution ceases to exist after finite time, either through extinction or through blow up.

Inhalt

Blow up in Rn for a Parabolic Equation with a Damping Nonlinear Gradient Term.- Shrinking Doughnuts.- Higher Approximations to Eigenvalues for a Nonlinear Elliptic Problem.- Positive Solutions of Emden Equations in Cone-Like Domains.- Nonlinear Parabolic Equations Arising in Semiconductor and Viscous Droplets Models.- A Parabolic Equation with a Mean-Curvature Type Operator.- Heat Flows and Relaxed Energies for Harmonic Maps.- Local Existence and Uniqueness of Positive Solutions of the Equation ?u + (1 + ??(r))u(n+2)/(n-2) = 0, in Rn and a Related Equation.- Singularities of Solutions of a Class of Quasilinear Equations in Divergence Form.- An Existence Result Via Ls-Regularity for Some Nonlinear Elliptic Equations.- Identifying a Time-Dependent Unknown Coefficient in a Nonlinear Heat Equation.- On the Structure of Solutions for Some Semilinear Elliptic Equations.- A Note on Boundary Regularity for Certain Degenerate Parabolic Equations.- The Quenching Problem on the N -dimensional Ball.- Global Solutions for a Class of Monge-Ampère Equations.- The Structure of Solutions near an Extinction Point in a Semilinear Heat Equation with Strong Absorption: A Formal Approach.- On a Conjecture by Hagan and Brenner.- A Nonlinear Diffusion-Absorption Equation with Unbounded Initial Data.- A Free Boundary Problem Arising in Plasma Physics.- Remarks on Quenching, Blow Up and Dead Cores.- Bifurcation at Boundary Points of the Continuous Spectrum.- A Comparison Result and Elliptic Equations Involving Subcritical Exponents.- Advances in Quenching.- On Some Almost Everywhere Symmetry Theorems.- Symmetry Properties of Finite Total Mass Solutions of Matukuma Equation.- An Exact Reduction of Maxwell' s Equations.- A General I-Theorem for Semilinear Elliptic Equations.- On Supercritical Phenomena.- On the Existence and Shape of Solutions to a Semilinear Neumann Problem.- Global Asymptotic Stability for Strongly Nonlinear Second Order Systems.- The Existence and Asymptotic Behaviour of Similarity Solutions to a Quasilinear Parabolic Equation.- Maximal Solutions of Singular Diffusion Equations with General Initial Data.- The Evolution of Harmonic Maps: Existence, Partial Regularity, and Singularities.- Two Dimensional Emden-Fowler Equation with Exponential Nonlinearity.- Global Bifurcation of Positive Solutions in Rn.- Conformai Asymptotics of the Isothermal Gas Spheres Equation.- Chemical Interfacial Reaction Models with Radial Symmetry.