AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
Classification of radial solutions arising in the study of thermal structures with thermal equilibrium or no flux at the boundary
About this Title
Alfonso Castro, Department of Mathematics, Harvey Mudd College, 301 Platt Boulevard, Claremont, California 91711 and Víctor Padrón, Departamento de Matemáticas, Facultad de Ciencias, Universidad de Los Andes, Mérida 5101, Venezuela
Publication: Memoirs of the American Mathematical Society
Publication Year:
2010; Volume 208, Number 976
ISBNs: 978-0-8218-4726-8 (print); 978-1-4704-0590-8 (online)
DOI: https://doi.org/10.1090/S0065-9266-10-00589-2
Published electronically: April 8, 2010
Keywords: Radial solution,
unstable solution,
stable solution,
thermal structure,
free boundary problem,
ground states,
equilibrium temperature
MSC: Primary 35J60, 85A25, 35K60; Secondary 80A20, 34B16
Table of Contents
Chapters
- Introduction
- 1. Bifurcation diagrams
- 2. Oscillation properties
- 3. Ground states
- 4. Stability of thermal structures
- 5. Proof of main theorems
- 6. The degenerate case, $k=-1$
- 7. Appendix 1. The conservative case ($N=1$)
- 8. Appendix 2. Pohozaev Identity
Abstract
We provide a complete classification of the radial solutions to a class of reaction diffusion equations arising in the study of thermal structures such as plasmas with thermal equilibrium or no flux at the boundary. In particular, our study includes rapidly growing nonlinearities, that is, those where an exponent exceeds the critical exponent. We describe the corresponding bifurcation diagrams and determine existence and uniqueness of ground states, which play a central role in characterizing those diagrams. We also provide information on the stability-unstability of the radial steady states.- Robert A. Adams and John J. F. Fournier, Sobolev spaces, 2nd ed., Pure and Applied Mathematics (Amsterdam), vol. 140, Elsevier/Academic Press, Amsterdam, 2003. MR 2424078
- Donald Aronson, Michael G. Crandall, and L. A. Peletier, Stabilization of solutions of a degenerate nonlinear diffusion problem, Nonlinear Anal. 6 (1982), no. 10, 1001–1022. MR 678053, DOI 10.1016/0362-546X(82)90072-4
- D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math. 30 (1978), no. 1, 33–76. MR 511740, DOI 10.1016/0001-8708(78)90130-5
- Robert Stephen Cantrell and Chris Cosner, Spatial ecology via reaction-diffusion equations, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2003. MR 2191264, DOI 10.1002/0470871296
- Alfonso Castro and Alexandra Kurepa, Radially symmetric solutions to a Dirichlet problem involving critical exponents, Trans. Amer. Math. Soc. 343 (1994), no. 2, 907–926. MR 1207581, DOI 10.1090/S0002-9947-1994-1207581-0
- Alfonso Castro and Alexandra Kurepa, Radial solutions to a Dirichlet problem involving critical exponents when $N=6$, Trans. Amer. Math. Soc. 348 (1996), no. 2, 781–798. MR 1321571, DOI 10.1090/S0002-9947-96-01476-6
- S. Chandrasekhar, An introduction to the study of stellar structure, Dover Publications, Inc., New York, N.Y., 1957. MR 0092663
- R. Courant and D. Hilbert, Methods of mathematical physics. Vol. II, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1989. Partial differential equations; Reprint of the 1962 original; A Wiley-Interscience Publication. MR 1013360
- Michael G. Crandall and Paul H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal. 52 (1973), 161–180. MR 341212, DOI 10.1007/BF00282325
- M.H. Ibañez & F.P. Plachco. On the thermal stability of slabs, cylinders, and spheres. The Astrophysical Journal, Vol. 370, 743–751 (1991).
- M.H. Ibañez, A. Parravano & C.A. Mendoza, On the thermal structure and stability of configurations with heat diffusion and a gain-loss function. I - General results. Astrophysical Journal, Vol. 398, 177-183, (1992).
- M.H. Ibañez, A. Parravano & C.A. Mendoza. On the thermal structure and stability of configurations with heat diffusion and a gain-loss function. II - Application to the interstellar medium. Astrophysical Journal, Vol. 407, 611-619 (1993).
- M.H. Ibañez & P. Rosenzweig. Analytical criteria for nonlinear instability of thermal structures. Phys. Plasmas 2 (11), 4127 (1995).
- M. Ibañez & A. Parravano. On the thermal structure and stability of configurations with heat diffusion and a gain-loss function. III. Molecular gas. The Astrophysical Journal, Vol. 424, 763-771 (1994).
- D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal. 49 (1972/73), 241–269. MR 340701, DOI 10.1007/BF00250508
- Hans G. Kaper and Man Kam Kwong, A free boundary problem arising in plasma physics, Nonlinear diffusion equations and their equilibrium states, 3 (Gregynog, 1989) Progr. Nonlinear Differential Equations Appl., vol. 7, Birkhäuser Boston, Boston, MA, 1992, pp. 265–273. MR 1167844
- Hans G. Kaper and Man Kam Kwong, Uniqueness of nonnegative solutions of a class of semi-linear elliptic equations, Nonlinear diffusion equations and their equilibrium states, II (Berkeley, CA, 1986) Math. Sci. Res. Inst. Publ., vol. 13, Springer, New York, 1988, pp. 1–17. MR 956078, DOI 10.1007/978-1-4613-9608-6_{1}
- Hans G. Kaper and Man Kam Kwong, Ground states of semi-linear diffusion equations, Differential equations with applications in biology, physics, and engineering (Leibnitz, 1989) Lecture Notes in Pure and Appl. Math., vol. 133, Dekker, New York, 1991, pp. 219–226. MR 1171472
- Hans G. Kaper and Man Kam Kwong, Uniqueness for a class of nonlinear initial value problems, J. Math. Anal. Appl. 130 (1988), no. 2, 467–473. MR 929952, DOI 10.1016/0022-247X(88)90322-8
- Patrizia Pucci and James Serrin, Uniqueness of ground states for quasilinear elliptic operators, Indiana Univ. Math. J. 47 (1998), no. 2, 501–528. MR 1647924, DOI 10.1512/iumj.1998.47.1517
- Walter Rudin, Principles of mathematical analysis, 3rd ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York-Auckland-Düsseldorf, 1976. MR 0385023
- D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J. 21 (1971/72), 979–1000. MR 299921, DOI 10.1512/iumj.1972.21.21079
- James Serrin and Moxun Tang, Uniqueness of ground states for quasilinear elliptic equations, Indiana Univ. Math. J. 49 (2000), no. 3, 897–923. MR 1803216, DOI 10.1512/iumj.2000.49.1893
- Joel Smoller, Shock waves and reaction-diffusion equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York-Berlin, 1983. MR 688146