共查询到20条相似文献,搜索用时 250 毫秒
1.
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
2.
Shuibo Huang Qiaoyu Tian Shengzhi Zhang Jinhua Xi 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(6):2342-2350
We investigate second-term asymptotic behavior of boundary blow-up solutions to the problems Δu=b(x)f(u), x∈Ω, subject to the singular boundary condition u(x)=∞, in a bounded smooth domain Ω⊂RN. b(x) is a non-negative weight function. The nonlinearly f is regularly varying at infinity with index ρ>1 (that is limu→∞f(ξu)/f(u)=ξρ for every ξ>0) and the mapping f(u)/u is increasing on (0,+∞). The main results show how the mean curvature of the boundary ∂Ω appears in the asymptotic expansion of the solution u(x). Our analysis relies on suitable upper and lower solutions and the Karamata regular variation theory. 相似文献
3.
Peng Feng 《Journal of Mathematical Analysis and Applications》2009,356(2):393-1788
In this paper, we show existence, uniqueness and exact asymptotic behavior of solutions near the boundary to a class of semilinear elliptic equations −Δu=λg(u)−b(x)f(u) in Ω, where λ is a real number, b(x)>0 in Ω and vanishes on ∂Ω. The special feature is to consider g(u) and f(u) to be regularly varying at infinity and b(x) is vanishing on the boundary with a more general rate function. The vanishing rate of b(x) determines the exact blow-up rate of the large solutions. And the exact blow-up rate allows us to obtain the uniqueness result. 相似文献
4.
This paper deals with the second term asymptotic behavior of large solutions to the problems Δu=b(x)f(u), x∈Ω, subject to the singular boundary condition u(x)=∞, x∈∂Ω, where Ω is a smooth bounded domain in RN, and b(x) is a non-negative weight function. The absorption term f is regularly varying at infinite with index ρ>1 (that is limu→∞f(ξu)/f(u)=ξρ for every ξ>0) and the mapping f(u)/u is increasing on (0,+∞). Our analysis relies on the Karamata regular variation theory. 相似文献
5.
《Journal of Differential Equations》1987,69(3):368-403
We prove the uniqueness (as well as the existence and regularity) of solutions of the Cauchy problem and of the first and mixed boundary value problems for the equation ut = φ(u)xx + b(u)x. (E) φ and b are assumed to belong to a large class of functions, including, in particular, cases φ(u) = um, b(u) = uλ, m ⩾ 1 and λ > 0. 相似文献
6.
Let Ω⊂RN(N?3) be a bounded domain with smooth boundary. We show the asymptotic behavior of boundary blowup solutions to non-linear elliptic equation Δu±|∇u|q=b(x)f(u) in Ω, subject to the singular boundary condition u(x)=∞ as dist(x,∂Ω)→0,f is Γ-varying at ∞. Our analysis is based on the Karamata regular variation theory combined with the method of lower and supper solution. 相似文献
7.
R.E White 《Journal of Mathematical Analysis and Applications》1979,68(1):157-170
We consider weak solutions to the nonlinear boundary value problem (r, (x, u(x)) u′(x))′ = (Fu)′(x) with r(0, u(0)) u′(0) = ku(0), r(L, u(L)) u′(L) = hu(L) and k, h are suitable elements of [0, ∞]. In addition to studying some new boundary conditions, we also relax the constraints on r(x, u) and (Fu)(x). r(x, u) > 0 may have a countable set of jump discontinuities in u and r(x, u)?1?Lq((0, L) × (0, p)). F is an operator from a suitable set of functions to a subset of Lp(0, L) which have nonnegative values. F includes, among others, examples of the form (Fu)(x) = (1 ? H(x ? x0)) u(x0), (Fu)(x) = ∫xLf(y, u(y)) dy where f(y, u) may have a countable set of jump discontinuities in u or F may be chosen so that (Fu)′(x) = ? g(x, u(x)) u′(x) ? q(x) u(x) ? f(x, u(x)) where q is a distributional derivative of an L2(0, L) function. 相似文献
8.
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
9.
We consider solutions of the singular diffusion equation t, = (um?1 ux)x, m ≦ 0, associated with the flux boundary condition limx→?∞ (um?1ux)x = λ > 0. The evolutions defined by this problem depend on both m and λ. We prove existence and stability of traveling wave solutions, parameterized by λ. Each traveling wave is stable in its appropriate evolution. These traveling waves are in L1 for ?1 < m ≦ 0, but have non-integrable tails for m ≦ ?1. We also show that these traveling waves are the same as those for the scalar conservation law ut = ?[f(u)]x + uxx for a particular nonlinear convection term f(u) = f(u;m, λ). © 1993 John Wiley & Sons, Inc. 相似文献
10.
Kyungkeun Kang 《Journal of Differential Equations》2011,251(9):2466-2493
We study the 3×3 elliptic systems ∇(a(x)∇×u)−∇(b(x)∇⋅u)=f, where the coefficients a(x) and b(x) are positive scalar functions that are measurable and bounded away from zero and infinity. We prove that weak solutions of the above system are Hölder continuous under some minimal conditions on the inhomogeneous term f. We also present some applications and discuss several related topics including estimates of the Green?s functions and the heat kernels of the above systems. 相似文献
11.
Qingbo Huang 《Advances in Mathematics》2006,207(2):599-616
We give interior a priori estimates for the mean oscillation of second derivatives of solutions to the Monge-Ampère equation detD2u=f(x) with zero boundary values, where f(x) is a non-Dini continuous function. If the modulus of continuity of f(x) is φ(r) such that limr→0φ(r)log(1/r)=0, then D2u∈VMO. 相似文献
12.
We study the existence, uniqueness, and asymptotic behavior of blow-up solutions for a general quasilinear elliptic equation of the type ?Δ p u = a(x)u m ?b(x)f(u) with p > 1 and 0 < m < p?1. The main technical tool is a new comparison principle that enables us to extend arguments for semilinear equations to quasilinear ones. Indeed, this paper is an attempt to generalize all available results for the semilinear case with p = 2 to the quasilinear case with p > 1. 相似文献
13.
J. Bourgain 《Israel Journal of Mathematics》1992,77(1-2):1-16
We study the almost everythere convergence to the initial dataf(x)=u(x, 0) of the solutionu(x, t) of the two-dimensional linear Schrödinger equation Δu=i? t u. The main result is thatu(x, t) →f(x) almost everywhere fort → 0 iff ∈H p (R2), wherep may be chosen <1/2. To get this result (improving on Vega’s work, see [6]), we devise a strategy to capture certain cancellations, which we believe has other applications in related problems. 相似文献
14.
Lazhar Bougoffa 《Applied mathematics and computation》2010,216(2):689-8913
The Abel equation of the second kind
[g0(x)+g1(x)u]u′=f0(x)+f1(x)u+f2(x)u2 相似文献
15.
We prove existence, local uniqueness and asymptotic estimates for boundary layer solutions to singularly perturbed equations of the type (ε2(x)u′′(x))=f(x,u(x))+g(x,u(x),ε(x)u′(x)), 0<x<1, with Dirichlet and Neumann boundary conditions. Here the functions ε and g are small and, hence, regarded as singular and regular functional perturbation parameters. The main tool of the proofs is a generalization (to Banach space bundles) of an implicit function theorem of R. Magnus. 相似文献
16.
Let Ω be an open subset of , N ? 3, containing 0. We consider the solutions of ?Δu(x) + g(u(x)) = f(x) in Ω-{0}, where g is nondecreasing and f is bounded and we study the possible singularities at 0: when u(x) = o(|x|1 ? N) we prove that u is isotropic near 0 and show that either it is a C1 function in Ω (removable singularity) or |x|N ? 2u(x) → c, c ≠ 0 (weak singularity) or |x|N ? 2 |u(x) |→ + ∞ (strong singularity). We also characterize the g's for which solutions with a weak singularity exist and improve a previous removability result of H. Brézis and L. Véron (Arch. Rational Mech. Anal.23 (1979), 153–166). 相似文献
17.
18.
A natural class of appropriate viscosity matrices for strictly hyperbolic systems of conservation laws in one space dimension, u1 + f(u)x = 0, u?Rm, is studied. These matrices are admissible in the sense that small-amplitude shock wave solutions of the hyperbolic system are shown to be limits of smooth traveling wave solutions of the parabolic system ut + f(u)x = v(Dux)x as ifv → 0 if D is in this class. The class is determined by a linearized stability requirement: The Cauchy problem for the equation u1 + f′(u0) ux = vDuxx should be well posed in L2 uniformly in v as v → 0. Previous examples of inadmissible viscosity matrices are accounted for through violation of the stability criterion. 相似文献
19.
《Journal of Computational and Applied Mathematics》1998,88(1):103-118
We are concerned with the discrete focal boundary value problem Δ3x(t − k) = f(x(t)), x(a) = Δx(t2) = Δ2x(b + 1) = 0. Under various assumptions on f and the integers a, t2, and b we prove the existence of three positive solutions of this boundary value problem. To prove our results we use fixed point theorems concerning cones in a Banach space. 相似文献