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1.
Let be a smooth bounded domain in . Assume that f?0 is a C1-function on [0,∞) such that f(u)/u is increasing on (0,+∞). Let a be a real number and let b?0, b?0 be a continuous function such that b≡0 on . The purpose of this Note is to establish the asymptotic behaviour of the unique positive solution of the logistic problem Δu+au=b(x)f(u) in , subject to the singular boundary condition u(x)→+∞ as . Our analysis is based on the Karamata regular variation theory. To cite this article: F.-C. Cîrstea, V. R?dulescu, C. R. Acad. Sci. Paris, Ser. I 336 (2003). 相似文献
2.
Jorge Garcí a-Meliá n 《Proceedings of the American Mathematical Society》2007,135(9):2785-2793
In this paper, we prove that for the problem in a bounded domain of has a unique positive solution with on . The nonnegative weight is continuous in , but is only assumed to verify a ``bounded oscillations" condition of local nature near , in contrast with previous works, where a definite behavior of near was imposed.
3.
Quasilinear elliptic equations with boundary blow-up 总被引:2,自引:0,他引:2
Jerk Matero 《Journal d'Analyse Mathématique》1996,69(1):229-247
Assume that Ω is a bounded domain in ℝ
N
withN ≥2, which has aC
2-boundary. We show that forp ∃ (1, ∞) there exists a weak solutionu of the problem δp
u(x) = f(u(x)), x ∃ Ω with boundary blow-up, wheref is a positive, increasing function which meets some natural conditions. The boundary blow-up ofu(x) is characterized in terms of the distance ofx from ∂Ω. For the Laplace operator, our results coincide with those of Bandle and Essén [1]. Finally, for a rather wide subclass
of the class of the admissible functionsf, the solution is unique whenp ∃ (1, 2]. 相似文献
4.
Yujuan Chen Mingxin Wang 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2010,61(2):277-292
Under the proper structure conditions on the nonlinear term f(u) and weight function b(x), the paper shows the uniqueness and asymptotic behavior near the boundary of boundary blow-up solutions to the porous media equations of logistic type ?Δu = a(x)u 1/m ? b(x)f(u) with m > 1. 相似文献
5.
Yujuan Chen Mingxin Wang 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2010,103(1):277-292
Under the proper structure conditions on the nonlinear term f(u) and weight function b(x), the paper shows the uniqueness and asymptotic behavior near the boundary of boundary blow-up solutions to the porous media
equations of logistic type −Δu = a(x)u
1/m
− b(x)f(u) with m > 1. 相似文献
6.
In this article, we investigate the parabolic logistic equation with blow-up initial and boundary values $${u_t} - \Delta u = a(x,t)u - b(x,t){u^p}in\Omega \times (0,T),$$ $$u = \infty on\partial \Omega \times (0,T) \cup \overline \Omega \times \{ 0\} ,$$ where ?? is a smooth bounded domain, T > 0 and p > 1 are constants, and a and b are continuous functions, b > 0 in ?? × [0, T) and b(x, T) ?? 0. We study the existence and uniqueness of positive solutions and their asymptotic behavior near the parabolic boundary. We show that under the extra condition that $b(x,t) \ge c{(T - t)^\theta }d{(x,\partial \Omega )^\beta } on \Omega \times \left[ {0,T} \right)$ for some constants c > 0, ?? > 0, and ?? > ?2, such a solution stays bounded in any compact subset of ?? as t increases to T, and hence solves the equation up to t = T. 相似文献
7.
Laurent Veron 《Journal d'Analyse Mathématique》1992,59(1):231-250
We prove the existence and the uniqueness of a solutionu of−Lu+h|u|
α-1u=f in some open domain ℝd, whereL is a strongly elliptic operator,f a nonnegative function, and α>1, under the assumption that ∂G is aC
2 compact hypersurface, lim
x→∂G
(dist(x, ∂G))2α/(α-1)
f(x)=0, and lim
x→∂G
u(x)=∞. 相似文献
8.
Jorge García-Melin 《Journal of Mathematical Analysis and Applications》2009,360(2):530-536
In this paper we prove the uniqueness of the positive solution for the boundary blow-up problem where Ω is a C2 bounded domain in , under the hypotheses that f(t) is nondecreasing in t>0 and f(t)/tp is increasing for large t and some p>1. We also consider the uniqueness of a related problem when the equation includes a nonnegative weight a(x). 相似文献
9.
J. Garcí a-Meliá n R. Letelier-Albornoz J. Sabina de Lis 《Proceedings of the American Mathematical Society》2001,129(12):3593-3602
In this paper we prove uniqueness of positive solutions to logistic singular problems , , 1$">, 0$"> in , where the main feature is the fact that . More importantly, we provide exact asymptotic estimates describing, in the form of a two-term expansion, the blow-up rate for the solutions near . This expansion involves both the distance function and the mean curvature of .
10.
Huabing Feng Chengkui Zhong 《Nonlinear Analysis: Theory, Methods & Applications》2010,73(10):3472-3478
By the Karamata regular variation theory and constructing comparison function, we show the exact asymptotic behavior of solutions for the degenerate logistic type elliptic problem with boundary blow-up. 相似文献
11.
12.
Continuing our work on the boundary value problem for super-Liouville equation, we study the qualitative behavior of boundary blow-ups. The boundary condition is derived from the chirality conditions in the physics literature, and is geometrically natural. In technical terms, we derive a new Pohozaev type identity and provide a new alternative, which also works at the boundary, to the classical method of Brézis–Merle. 相似文献
13.
王明新 《中国科学A辑(英文版)》2003,46(2):169-175
This paper deals with the blow-up rate estimates of positive solutions for systems of heat equations with nonlinear boundary conditions. The upper and lower bounds of blow-up rate are obtained. 相似文献
14.
15.
Fengjie Li Bingchen Liu Sining Zheng 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2007,58(5):717-735
This paper deals with simultaneous and non-simultaneous blow-up for heat equations coupled via nonlinear boundary fluxes
. It is proved that, if m < q + 1 and n < p + 1, then blow-up must be simultaneous, and that, for radially symmetric and nondecreasing in time solutions, non-simultaneous
blow-up occurs for some initial data if and only if m > q + 1 or n > p + 1. We find three regions: (i) q + 1 < m < p/(p + 1 − n) and n < p+1, (ii) p + 1 < n < q/(q + 1 − m) and m < q+1, (iii) m > q+1 and n > p+1, where both simultaneous and non-simultaneous blow-up are possible. Four different simultaneous blow-up rates are obtained
under different conditions. It is interesting that different initial data may lead to different simultaneous blow-up rates
even for the same values of the exponent parameters.
Supported by the National Natural Science Foundation of China. 相似文献
16.
Fengjie Li Bingchen Liu Sining Zheng 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2007,24(10):717-735
This paper deals with simultaneous and non-simultaneous blow-up for heat equations coupled via nonlinear boundary fluxes
\frac?u?h = um + vp, \frac?v?h = uq + vn\frac{\partial u}{\partial\eta} = u^{m} + v^{p}, \frac{\partial v}{\partial\eta} = u^{q} + v^{n} 相似文献
17.
We consider the system of m linear equations in n integer variables Ax = d and give sufficient conditions for the uniqueness of its integer solution x ∈ {−1, 1}
n
by reformulating the problem as a linear program. Necessary and sufficient uniqueness characterizations of ordinary linear
programming solutions are utilized to obtain sufficient uniqueness conditions such as the intersection of the kernel of A and the dual cone of a diagonal matrix of ±1’s is the origin in R
n
. This generalizes the well known condition that ker(A) = 0 for the uniqueness of a non-integer solution x of Ax = d. A zero maximum of a single linear program ensures the uniqueness of a given integer solution of a linear equation. 相似文献
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19.
THEBLOW┐UPPROPERTYFORASYSTEMOFHEATEQUATIONSWITHNONLINEARBOUNDARYCONDITIONSLINZHIGUI,XIECHUNHONGANDWANGMINGXINAbstract.Thispap... 相似文献
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