Ebook: Nonlinear Diffusion Equations and Their Equilibrium States, 3: Proceedings from a Conference held August 20–29, 1989 in Gregynog, Wales

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.

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Content:
Front Matter....Pages i-x
Blow Up in ?n for a Parabolic Equation with a Damping Nonlinear Gradient Term....Pages 1-20
Shrinking Doughnuts....Pages 21-38
Higher Approximations to Eigenvalues for a Nonlinear Elliptic Problem....Pages 39-69
Positive Solutions of Emden Equations in Cone-Like Domains....Pages 71-75
Nonlinear Parabolic Equations Arising in Semiconductor and Viscous Droplets Models....Pages 77-88
A Parabolic Equation with a Mean-Curvature Type Operator....Pages 89-97
Heat Flows and Relaxed Energies for Harmonic Maps....Pages 99-109
Singularities of Solutions of a Class of Quasilinear Equations in Divergence Form....Pages 111-128
Identifying a Time Dependent Unknown Coefficient in a Nonlinear Heat Equation....Pages 129-144
On the Structure of Solutions for Some Semilinear Elliptic Equations....Pages 145-152
A Note on Boundary Regularity for Certain Degenerate Parabolic Equations....Pages 153-169
The Quenching Problem on the N-dimensional Ball....Pages 171-176
Global Solutions for a Class of Monge-Amp?re Equations....Pages 177-182
The Structure of Solutions near an Extinction Point in a Semilinear Heat Equation with Strong Absorption: A Formal Approach....Pages 183-196
On a conjecture by Hagan and Brenner....Pages 197-214
A Nonlinear Diffusion-Absorption Equation with Unbounded Initial Data....Pages 215-236
A Free Boundary Problem Arising in Plasma Physics....Pages 237-241
Remarks on Quenching, Blow Up and Dead Cores....Pages 243-263
Bifurcation at boundary points of the continuous spectrum....Pages 265-273
A Comparison Result and Elliptic Equations Involving Subcritical Exponents....Pages 275-286
Advances in Quenching....Pages 287-297
On Some Almost Everywhere Symmetry Theorems....Pages 299-318
Symmetry Properties of Finite Total Mass Solutions of Matukuma Equation....Pages 319-346
An Exact Reduction of Maxwell’s Equations....Pages 347-374
A General I-Theorem for Semilinear Elliptic Equations....Pages 375-389
On Supercritical Phenomena....Pages 391-405
On the Existence and Shape of Solutions to a Semilinear Neumann Problem....Pages 407-416
Global Asymptotic Stability for Strongly Nonlinear Second Order Systems....Pages 417-424
The Existence and Asymptotic Behaviour of Similarity Solutions to a Quasilinear Parabolic Equation....Pages 425-436
Maximal Solutions of Singular Diffusion Equations with General Initial Data....Pages 437-449
The Evolution of Harmonic Maps: Existence, Partial Regularity, and Singularities....Pages 451-469
Two Dimensional Emden-Fowler Equation with Exponential Nonlinearity....Pages 471-484
Conformal Asymptotics of the Isothermal Gas Spheres Equation....Pages 485-491
Chemical Interfacial Reaction Models with Radial Symmetry....Pages 493-512
Back Matter....Pages 513-535
....Pages 537-559