By Hans G. Kaper, Man Kam Kwong (auth.), W.-M. Ni, L. A. Peletier, James Serrin (eds.)

ISBN-10: 1461396085

ISBN-13: 9781461396086

ISBN-10: 1461396107

ISBN-13: 9781461396109

In fresh years significant curiosity has been eager about nonlinear diffu sion difficulties, the archetypical equation for those being Ut = ~U + f(u). the following ~ denotes the n-dimensional Laplacian, the answer u = u(x, t) is outlined over a few space-time area of the shape n x [O,T], and f(u) is a given actual functionality whose shape will depend on a variety of actual and mathematical purposes. those functions became extra diverse and frequent as challenge after challenge has been proven to guide to an equation of this sort or to its time-independent counterpart, the elliptic equation of equilibrium ~u+f(u)=O. specific circumstances come up, for instance, in inhabitants genetics, the physics of nu transparent balance, section transitions among beverages and gases, flows in porous media, the Lend-Emden equation of astrophysics, quite a few simplified com bustion versions, and in choosing metrics which observe given scalar or Gaussian curvatures. within the latter course, for instance, the matter of discovering conformal metrics with prescribed curvature results in a flooring nation challenge related to serious exponents. hence not just analysts, yet geome ters in addition, can locate universal floor within the current paintings. The corresponding mathematical challenge is to figure out how the struc ture of the nonlinear functionality f(u) affects the habit of the solution.

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**Additional info for Nonlinear Diffusion Equations and Their Equilibrium States II: Proceedings of a Microprogram held August 25–September 12, 1986**

**Example text**

On free boundaries satisfying a Bernoulli condition. 3) u = 0, = D \ fl, on an, in G au an - const. on r. This problem occurs in the modelling of potential flow. For its physical background, one can consult [H], among others. If r is sufficiently smooth, then one can infer the uniqueness of (u, r) and the convexity of D, see [T,H]. Here I want to prove the convexity of D without using conformal maps. The initial hope to generalize the convexity result to the analogous problem in higher dimensions (D and n are subsets of Rn with n ;:::: 3) is dampened by several obstacles.

Monographs 23, AMS, Providence, 1968. 5. A. Leung and D. Clark, Bifurcations and large-time asymptotic behavior for prey-predator reaction-diffusion equations with Dirichlet boundary data, J. Diff. Eqs. 35 (1980), 113-127. 6. A. Leung, Equilibria and stabilities for competing-species reaction-diffusion equations with Dirichlet boundary data, Math. Anal. Appl. 73 (1980), 204-218. 7. A. Leung, Stabilities for equilibria of competing-species reaction-diffusion equations with homogeneous Dirichlet conditions, Funk.

One can learn from [TP, Part I, §160 and §162] that as early as 1859 Saint Venant was aware of a counterexample to his apparently general observation. This example of a doubly symmetric nonconvex domain is depicted in Figure 1. It was copied from Filon [F, p. 342] who is incorrectly credited as the first one to discover it by people like Love [L] or Timoshenko and Goodier [TGJ. Nonetheless the counterexample is mentioned by Thomson and Tait [TT] or Todhunter and Pearson [TP] and it was published in [N].

### Nonlinear Diffusion Equations and Their Equilibrium States II: Proceedings of a Microprogram held August 25–September 12, 1986 by Hans G. Kaper, Man Kam Kwong (auth.), W.-M. Ni, L. A. Peletier, James Serrin (eds.)

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