| 摘要(英) |
In the last two decades, significant progress has been made in the theory of nonlinear systems of partial differential equations. There has been much effort dedicated to the developments, motivated by applications to the natural sciences such as physics, chemistry and biology. There is also a substantial amount of consequences related to topical issues for the study of medical and ecological sciences, and other applications involving fluid, plasma, reactor dynamics and so on. In particular, systems of elliptic partial differential equations are mainly encountered in stationary problems of the theory of heat and mass transfer in reacting media, the theory of chemical reactors, combustion theory, mathematical biology and biophysics etc.
This dissertation, consisting of four main parts, is devoted to studying some qualitative properties of solutions, such as the existence, uniqueness and structure of solutions to four specific kinds of nonlinear elliptic systems. In Part 1, sublinear elliptic systems are considered, and the existence, uniqueness and stability of solutions are derived via the bifurcation theory and monotonicity method. Part 2 deals with a cooperative Hamiltonian system, which can also be viewed as a natural extension of the single equation arising from investigating the stationary states of the nonlinear Schrödinger equation. By applying linearization techniques and the implicit function theorem, a complete structure of solutions is clarified. Part 3 studies a Liouville-type system with singularity at the origin. The existence and uniqueness of solutions to the Dirichlet boundary value problem are established. In addition, the structure of solutions in terms of some specific quantities, which are viewed as the total curvature or flux in the single case, will be provided as well. Finally, a singular Bennett-type system coming from modeling dissipative stationary plasmas is introduced in Part 4, and its blow-up phenomena will be investigated.
|
| 參考文獻 |
[1] J.-L. Chern, Z.-Y. Chen and C.-S. Lin, Uniqueness of topological solutions and the structure of solutions for the Chern-Simons system with two Higgs particles, Comm. Math. Phys. 296 (2010), 323-351.
[2] Z.-Y. Chen, J.-L. Chern and Y.-L. Tang, On the solutions to a Liouville-type system involving singularity, Calc. Var. PDE (2011), to appear.
[3] Z.-Y. Chen, J.-L. Chern, J. Shi and Y.-L. Tang, On the uniqueness and structure of solutions to a coupled elliptic system, J. Differential Equations 249 (2010), 3419-3442.
[4] J.-L. Chern, Z.-Y. Chen, Y.-L. Tang and C.-S. Lin, Uniqueness and structure of solutions to the Dirichlet problem for an elliptic system, J. Differential Equations 246 (2009), 3704-3714.
[5] S. Chanillo and M. K.-H. Kiessling, Conformally invariant systems of nonlinear PDE of Liouville type, Geom. Funct. Anal. 5 (1995), no. 6, 924-947.
[6] S. Childress and J. K. Percus, Nonlinear aspects of Chemotaxis, Math. Biosci. 56 (1981), 217-237.
[7] M. Chipot, I. Shafrir and G. Wolansky, On the solutions of Liouville systems, J. Differential Equations 140 (1997), 59-105.
[8] M. Chipot, I. Shafrir and G. Wolansky, Erratum: "On the solutions of Liouville systems" [J. Differential Equations 140 (1997), no. 1, 59-105], J. Differential Equations 178 (2002), 630.
[9] S. A. Chang and P. Yang, Conformal deformations of metrics on $S^2$, J. Diff. Geom. 27 (1988), 256-296.
[10] G. Dunne, Self-dual Chern-Simons Theories, Lecture Notes in Physics, vol. m36, Berlin: Springer-Verlag, 1995.
[11] P. Debye and E. Huckel, Zur Theorie der Electrolyte, Phys. Zft. 24 (1923), 305-325.
[12] J. Jost, C. S. Lin and G. Wang, Analytic aspects of the Toda system. II. Bubbling behavior and existence of solutions, Comm. Pure Appl. Math. 59 (2006), no. 4, 526-558.
[13] J. Jost and G. Wang, Classification of solutions of a Toda system in $R^2$, Int. Math. Res. Not. 2002, no. 6, 277-290.
[14] J. Jost and G. Wang, Analytic aspects of the Toda system. I. A Moser-Trudinger inequality, Comm. Pure Appl. Math. 54 (2001), no. 11, 1289-1319.
[15] M. K.-H. Kiessling, Symmetry results for finite-temperature, relativistic Thomas-Fermi equations, Comm. Math. Phys. 226 (2002), 607-626.
[16] M. K.-H. Kiessling and J. L. Lebowitz, Dissipative stationary Plasmas: Kinetic Modeling Bennet Pinch, and generalizations, Phys. Plasmas 1 (1994), 1841-1849.
[17] E. F. Keller and L. A. Segel, Traveling bands of Chemotactic Bacteria: A theoretical analysis, J. Theor. Biol. 30 (1971), 235-248.
[18] C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math. 123 (1996), 221-231.
[19] C.-S. Lin, Uniqueness of solutions to the mean field equations for the spherical Onsager vortex, Arch. Rational Mech. Anal. 153 (2000), 153-176.
[20] C.-S. Lin and L. Zhang, Profile of bubbling solutions to a Liouville system, Ann. I. H. Poincare-AN 27 (2010), 117-143.
[21] M. S. Mock, Asymptotic behavior of solutions of transport equations for semiconductor devices, J. Math. Anal. Appl. 49 (1975), 215-225.
[22] I. Rubinstein, Electro diffusion of Ions, SIAM, Stud. Appl. Math. 11 (1990).
|
[Endnote RIS 格式]
[相關文章]
[文章引用]
[完整記錄]
[館藏目錄]
[檢視]
plurk
funp
live
udn
HD
myshare
netvibes
friend
youpush
delicious
baidu
Google bookmarks
del.icio.us
hemidemi
facebook
twitter
google
reddit