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On the Lagrangian averaged Euler equations: local wellposedness and blowup criterion
Existence and uniqueness of positive solution to a nonlocal differential equation with homogeneous Dirichlet boundary conditionA nonmonotone 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), 620709. 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), 697715. doi: 10.1017/S0308210500021727. 
[4] 
W. S. C. Gurney, S. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 1721. doi: 10.1038/287017a0. 
[5] 
P. Hess, On uniqueness of positive solutions to nonlinear elliptic boundary value problems, Math. Z., 154 (1977), 1718. 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), 117139. 
[7] 
M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287289. doi: 10.1126/science.267326. 
[8] 
C. V. Pao, "Nonlinear Parablic and Elliptic Equations," Plenum, New York, 1992. doi: 10.1007/9781461530343. 
[9] 
M. H. Protter and H. F. Weinberger, "Maximum Principle in Differential Equations," SpringerVerlag, New York, 1984. doi: 10.1007/9781461252825. 
[10] 
J. W.H. So, J. Wu and X. Zou, A reaction diffusion model for a single species with age structureI. Traveling wave fronts on unbounded domains, Proc. Royal Soc. London. A, 457 (2001), 18411853. 
[11] 
D. Xu and X.Q. Zhao, A nonlocal reaction diffusion population model with stage structure, Canadian Applied Mathematics Quarterly, 11 (2003), 303319. 
[12] 
X.Q. Zhao, Global attractivity in a class of nonmonotone reaction diffusion equations with delay, Canad. Appl. Math. Quart., 17 (2009), 271281. 
show all references
References:
[1] 
H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach space, SIAM Review, 18 (1976), 620709. 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), 697715. doi: 10.1017/S0308210500021727. 
[4] 
W. S. C. Gurney, S. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 1721. doi: 10.1038/287017a0. 
[5] 
P. Hess, On uniqueness of positive solutions to nonlinear elliptic boundary value problems, Math. Z., 154 (1977), 1718. 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), 117139. 
[7] 
M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287289. doi: 10.1126/science.267326. 
[8] 
C. V. Pao, "Nonlinear Parablic and Elliptic Equations," Plenum, New York, 1992. doi: 10.1007/9781461530343. 
[9] 
M. H. Protter and H. F. Weinberger, "Maximum Principle in Differential Equations," SpringerVerlag, New York, 1984. doi: 10.1007/9781461252825. 
[10] 
J. W.H. So, J. Wu and X. Zou, A reaction diffusion model for a single species with age structureI. Traveling wave fronts on unbounded domains, Proc. Royal Soc. London. A, 457 (2001), 18411853. 
[11] 
D. Xu and X.Q. Zhao, A nonlocal reaction diffusion population model with stage structure, Canadian Applied Mathematics Quarterly, 11 (2003), 303319. 
[12] 
X.Q. Zhao, Global attractivity in a class of nonmonotone reaction diffusion equations with delay, Canad. Appl. Math. Quart., 17 (2009), 271281. 
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