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On the Lagrangian averaged Euler equations: local well-posedness and blow-up criterion
Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case
1. | School of Mathematics and Information Sciences, Guangzhou University, Guangzhou, 510006 |
2. | College of Mathematics Science, Chongqing Normal University, Chongqing 400047 |
3. | Department of Applied Mathematics, University of Western Ontario, London, Ontario N6A 5B7 |
References:
[1] |
H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach space, SIAM Review, 18 (1976), 620-709.
doi: 10.1137/1018114. |
[2] |
L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, Providence, RI, 1998. |
[3] |
P. Freitas and G. Sweers, Positivity results for a nonlocal elliptic equation, Proceedings of the Royal Society of Edinburgh, 128A (1998), 697-715.
doi: 10.1017/S0308210500021727. |
[4] |
W. S. C. Gurney, S. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21.
doi: 10.1038/287017a0. |
[5] |
P. Hess, On uniqueness of positive solutions to nonlinear elliptic boundary value problems, Math. Z., 154 (1977), 17-18.
doi: 10.1007/BF01215108. |
[6] |
D. Liang, J. W.-H. So, F. Zhang and X. Zou, Population dynamic models with nonlocal delay on bounded fields and their numeric computations, Diff. Eqns. Dynam. Syst., 11 (2003), 117-139. |
[7] |
M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287-289.
doi: 10.1126/science.267326. |
[8] |
C. V. Pao, "Nonlinear Parablic and Elliptic Equations," Plenum, New York, 1992.
doi: 10.1007/978-1-4615-3034-3. |
[9] |
M. H. Protter and H. F. Weinberger, "Maximum Principle in Differential Equations," Springer-Verlag, New York, 1984.
doi: 10.1007/978-1-4612-5282-5. |
[10] |
J. W.-H. So, J. Wu and X. Zou, A reaction diffusion model for a single species with age structure-I. Traveling wave fronts on unbounded domains, Proc. Royal Soc. London. A, 457 (2001), 1841-1853. |
[11] |
D. Xu and X.-Q. Zhao, A nonlocal reaction diffusion population model with stage structure, Canadian Applied Mathematics Quarterly, 11 (2003), 303-319. |
[12] |
X.-Q. Zhao, Global attractivity in a class of nonmonotone reaction diffusion equations with delay, Canad. Appl. Math. Quart., 17 (2009), 271-281. |
show all references
References:
[1] |
H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach space, SIAM Review, 18 (1976), 620-709.
doi: 10.1137/1018114. |
[2] |
L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, Providence, RI, 1998. |
[3] |
P. Freitas and G. Sweers, Positivity results for a nonlocal elliptic equation, Proceedings of the Royal Society of Edinburgh, 128A (1998), 697-715.
doi: 10.1017/S0308210500021727. |
[4] |
W. S. C. Gurney, S. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21.
doi: 10.1038/287017a0. |
[5] |
P. Hess, On uniqueness of positive solutions to nonlinear elliptic boundary value problems, Math. Z., 154 (1977), 17-18.
doi: 10.1007/BF01215108. |
[6] |
D. Liang, J. W.-H. So, F. Zhang and X. Zou, Population dynamic models with nonlocal delay on bounded fields and their numeric computations, Diff. Eqns. Dynam. Syst., 11 (2003), 117-139. |
[7] |
M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287-289.
doi: 10.1126/science.267326. |
[8] |
C. V. Pao, "Nonlinear Parablic and Elliptic Equations," Plenum, New York, 1992.
doi: 10.1007/978-1-4615-3034-3. |
[9] |
M. H. Protter and H. F. Weinberger, "Maximum Principle in Differential Equations," Springer-Verlag, New York, 1984.
doi: 10.1007/978-1-4612-5282-5. |
[10] |
J. W.-H. So, J. Wu and X. Zou, A reaction diffusion model for a single species with age structure-I. Traveling wave fronts on unbounded domains, Proc. Royal Soc. London. A, 457 (2001), 1841-1853. |
[11] |
D. Xu and X.-Q. Zhao, A nonlocal reaction diffusion population model with stage structure, Canadian Applied Mathematics Quarterly, 11 (2003), 303-319. |
[12] |
X.-Q. Zhao, Global attractivity in a class of nonmonotone reaction diffusion equations with delay, Canad. Appl. Math. Quart., 17 (2009), 271-281. |
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