The spatial critical points not moving along the heat flow
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
We consider solutions of the heat equation, in domains inR N, and their spatial critical points. In particular, we show that a solutionu has a spatial critical point not moving along the heat flow if and only ifu satisfies some balance law. Furthermore, in the case of Dirichlet, Neumann, and Robin homogeneous initial-boundary value problems on bounded domains, we prove that if the origin is a spatial critical point never moving for sufficiently many compactly supported initial data satisfying the balance law with respect to the origin, then the domain must be a ball centered at the origin.
- S. Angenent,Solutions of the one-dimensional porous medium equation are determined by their five boundary, J. London Math. Soc.42 (1990), 339–353. CrossRef
- I. Chavel,Eigenvalues in Riemannian Geometry, Pure and Applied Math. 115, Academic Press, New York, 1984.
- I. Chavel and L. Karp,Movement of hot spots in Riemannian manifolds, J. Analyse Math.55 (1990), 271–286. CrossRef
- A. Friedman,Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, N.J., 1964.
- D. Gilbarg and N. S. Trudinger,Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1983.
- R. Gulliver and N. B. Willms,A conjectured heat flow problem, inSolutions, SIAM Review37 (1995), 100–104. CrossRef
- P. Hess and P. Polacik,Symmetry and convergence properties for non-negative solutions of nonautonomous reaction-diffusion problems, Proc. Roy. Soc. Edinburgh124A (1994), 573–587.
- S.ItÔ,Diffusion Equations, Transi. Math. Monograph 114, American Math. Soc., 1992.
- S. Jimbo and S. Sakaguchi,Movement of hot spots over unbounded domains in R N, J. Math. Anal. Appl.182 (1994), 810–835. CrossRef
- B. Kawohl,A conjectured heat flow problem, inSolutions, SIAM Review37 (1995), 104–105.
- M. S. Klamkin,A conjectured heat flow problem, inProblems, SIAM Review36 (1994), 107. CrossRef
- The spatial critical points not moving along the heat flow
Journal d’Analyse Mathématique
Volume 71, Issue 1 , pp 237-261
- Cover Date
- Print ISSN
- Online ISSN
- Additional Links