共查询到20条相似文献,搜索用时 31 毫秒
1.
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
2.
We investigate entire radial solutions of the semilinear biharmonic equation Δ2u=λexp(u) in Rn, n?5, λ>0 being a parameter. We show that singular radial solutions of the corresponding Dirichlet problem in the unit ball cannot be extended as solutions of the equation to the whole of Rn. In particular, they cannot be expanded as power series in the natural variable s=log|x|. Next, we prove the existence of infinitely many entire regular radial solutions. They all diverge to −∞ as |x|→∞ and we specify their asymptotic behaviour. As in the case with power-type nonlinearities [F. Gazzola, H.-Ch. Grunau, Radial entire solutions for supercritical biharmonic equations, Math. Ann. 334 (2006) 905-936], the entire singular solution x?−4log|x| plays the role of a separatrix in the bifurcation picture. Finally, a technique for the computer assisted study of a broad class of equations is developed. It is applied to obtain a computer assisted proof of the underlying dynamical behaviour for the bifurcation diagram of a corresponding autonomous system of ODEs, in the case n=5. 相似文献
3.
A. A. Belolipetskii A. M. Ter-Krikorov 《Computational Mathematics and Mathematical Physics》2016,56(11):1859-1871
The functional equation f(x,ε) = 0 containing a small parameter ε and admitting regular and singular degeneracy as ε → 0 is considered. By the methods of small parameter, a function x n 0(ε) satisfying this equation within a residual error of O(ε n+1) is found. A modified Newton’s sequence starting from the element x n 0(ε) is constructed. The existence of the limit of Newton’s sequence is based on the NK theorem proven in this work (a new variant of the proof of the Kantorovich theorem substantiating the convergence of Newton’s iterative sequence). The deviation of the limit of Newton’s sequence from the initial approximation x n 0(ε) has the order of O(ε n+1), which proves the asymptotic character of the approximation x n 0(ε). The method proposed is implemented in constructing an asymptotic approximation of a system of ordinary differential equations on a finite or infinite time interval with a small parameter multiplying the derivatives, but it can be applied to a wider class of functional equations with a small parameters. 相似文献
4.
This paper studies the propagation of pulse-like solutions of semilinear hyperbolic equations in the limit of short wavelength. The pulses are located at a wavefront Σ?{φ=0} where φ satisfies the eikonal equation and dφ lies on a regular sheet of the characteristic variety. The approximate solutions are uεapprox=U (t, x, φ(t, x)/ε) where U(t, x, r) is a smooth function with compact support in r. When U satisfies a familiar nonlinear transport equation from geometric optics it is proved that there is a family of exact solutions uεexact such that uεapprox has relative error O(ε) as ε→0. While the transport equation is familiar, the construction of correctors and justification of the approximation are different from the analogous problems concerning the propagation of wave trains with slowly varying envelope. 相似文献
5.
R. Bruce Kellogg 《Journal of Differential Equations》2010,248(1):184-208
The semilinear reaction-diffusion equation −ε2Δu+b(x,u)=0 with Dirichlet boundary conditions is considered in a convex polygonal domain. The singular perturbation parameter ε is arbitrarily small, and the “reduced equation” b(x,u0(x))=0 may have multiple solutions. An asymptotic expansion for u is constructed that involves boundary and corner layer functions. By perturbing this asymptotic expansion, we obtain certain sub- and super-solutions and thus show the existence of a solution u that is close to the constructed asymptotic expansion. The polygonal boundary forces the study of the nonlinear autonomous elliptic equation −Δz+f(z)=0 posed in an infinite sector, and then well-posedness of the corresponding linearized problem. 相似文献
6.
William F. Trench 《Linear algebra and its applications》2011,434(7):1631-1637
We consider the asymptotic behavior of solutions of a linear differential system x′=A(t)x, where A is continuous on an interval ([a,∞). We are interested in the situation where the system may not have a desirable asymptotic property such as stability, strict stability, uniform stability, or linear asymptotic equilibrium, but its solutions can be written as x=Pu, where P is continuously differentiable on [a,∞) and u is a solution of a system u′=B(t)u that has the property in question. In this case we say that P preconditions the given system for the property in question. 相似文献
7.
V. F. Gaposhkin 《Mathematical Notes》1996,64(3):316-321
The asymptotic behavior asn → ∞ of the normed sumsσn =n ?1 Σ k =0n?1 Xk for a stationary processX = (X n ,n ∈ ?) is studied. For a fixedε > 0, upper estimates for P(sup k≥n ¦σ k ¦ ≥ε) asn → ∞ are obtained. 相似文献
8.
Paul Sablonnière 《Numerical Algorithms》1992,3(1):401-409
LetL(α, r) denote the class of complex sequences (x n) having an asymptotic expansion of type $$x_n \sim \sum\limits_{i \ge 0} {c_i n^{ - (\alpha + ri)} } , c_0 \ne 0,\alpha > 0,r > 0.$$ We describe the asymptotic behaviour of sequences obtained by applying to (x n) some specific modified versions of the iterated Δ2 transform, the θ2 transform and some combinations of them. In this paper, we study the particular casesr=1 andr=1/2, which are the most useful in practice. The results are also valid for sequences (S n) whose error sequence (x n) defined byx n=S n?S, S=limS n, belongs to someL(α, r). 相似文献
9.
Layered stable equilibria of a reaction-diffusion equation with nonlinear Neumann boundary condition
Arnaldo Simal do Nascimento Renato José de Moura 《Journal of Mathematical Analysis and Applications》2008,347(1):123-135
In this work we investigate the existence and asymptotic profile of a family of layered stable stationary solutions to the scalar equation ut=ε2Δu+f(u) in a smooth bounded domain Ω⊂R3 under the boundary condition εν∂u=δεg(u). It is assumed that Ω has a cross-section which locally minimizes area and limε→0εlnδε=κ, with 0?κ<∞ and δε>1 when κ=0. The functions f and g are of bistable type and do not necessarily have the same zeros what makes the asymptotic geometric profile of the solutions on the boundary to be different from the one in the interior. 相似文献
10.
11.
Paul W. Eloe Johnny Henderson 《Journal of Mathematical Analysis and Applications》2007,331(1):240-247
For the nth order differential equation, y(n)=f(x,y,y′,…,y(n−1)), we consider uniqueness implies existence results for solutions satisfying certain nonlocal (k+2)-point boundary conditions, 1?k?n−1. Uniqueness of solutions when k=n−1 is intimately related to uniqueness of solutions when 1?k?n−2. These relationships are investigated as well. 相似文献
12.
In this paper, we study the existence of multiple positive solutions to some Hamiltonian elliptic systems −Δv=λu+up+εf(x), −Δu=μv+vq+δg(x) in Ω;u,v>0 in Ω; u=v=0 on ∂Ω, where Ω is a bounded domain in RN (N?3); 0?f, g∈L∞(Ω); 1/(p+1)+1/(q+1)=(N−2)/N, p,q>1; λ,μ>0. Using sub- and supersolution method and based on an adaptation of the dual variational approach, we prove the existence of at least two nontrivial positive solutions for all λ,μ∈(0,λ1) and ε,δ∈(0,δ0), where λ1 is the first eigenvalue of the Laplace operator −Δ with zero Dirichlet boundary conditions and δ0 is a positive number. 相似文献
13.
Zhongli Wei 《Journal of Mathematical Analysis and Applications》2005,306(2):619-636
This paper investigates the existence of positive solutions for 2nth-order (n>1) singular sub-linear boundary value problems. A necessary and sufficient condition for the existence of C2n−2[0,1] as well as C2n−1[0,1] positive solutions is given by constructing lower and upper solutions and with the maximal theorem. Our nonlinearity f(t,x1,x2,…,xn) may be singular at xi=0, i=1,2,…,n, t=0 and/or t=1. 相似文献
14.
We look for conditions under which all solutions of the nonlinear ordinary differential equation y(n) + f(t, y) = 0, t ? 0, ?∞ < y < ∞, are oscillatory, as well as consider the asymptotic behaviour of the nonoscillatory solutions. 相似文献
15.
Romain Joly 《Journal of Differential Equations》2006,229(2):588-653
In this paper, we study the convergence of the wave equation with variable internal damping term γn(x)ut to the wave equation with boundary damping γ(x)⊗δx∈∂Ωut when (γn(x)) converges to γ(x)⊗δx∈∂Ω in the sense of distributions. When the domain Ω in which these equations are defined is an interval in R, we show that, under natural hypotheses, the compact global attractor of the wave equation damped on the interior converges in X=H1(Ω)×L2(Ω) to the one of the wave equation damped on the boundary, and that the dynamics on these attractors are equivalent. We also prove, in the higher-dimensional case, that the attractors are lower-semicontinuous in X and upper-semicontinuous in H1−ε(Ω)×H−ε(Ω). 相似文献
16.
We consider an Allen-Cahn type equation of the form ut=Δu+ε−2fε(x,t,u), where ε is a small parameter and fε(x,t,u)=f(u)−εgε(x,t,u) a bistable nonlinearity associated with a double-well potential whose well-depths can be slightly unbalanced. Given a rather general initial data u0 that is independent of ε, we perform a rigorous analysis of both the generation and the motion of interface. More precisely we show that the solution develops a steep transition layer within the time scale of order ε2|lnε|, and that the layer obeys the law of motion that coincides with the formal asymptotic limit within an error margin of order ε. This is an optimal estimate that has not been known before for solutions with general initial data, even in the case where gε≡0.Next we consider systems of reaction-diffusion equations of the form
17.
Wolfgang Wasow 《Linear algebra and its applications》1977,18(2):163-170
Let A(x,ε) be an n×n matrix function holomorphic for |x|?x0, 0<ε?ε0, and possessing, uniformly in x, an asymptotic expansion , as ε→0+. An invertible, holomorphic matrix function P(x,ε) with an asymptotic expansion , as ε→0+, is constructed, such that the transformation y = P(x,ε)z takes the differential equation a positive integer, into , where B(x,ε) is asymptotically equal, to all orders, to a matrix in a canonical form for holomorphic matrices due to V.I. Arnold. 相似文献
18.
Shangbing Ai 《Journal of Mathematical Analysis and Applications》2003,277(2):405-422
We study the existence of well-known singularly perturbed BVP problem ε2y″=1−y2−2b(1−x2)y, y(−1)=y(1)=0 introduced by G.F. Carrier. In particular, we show that there exist multi-spike solutions, and the locations of interior spikes are clustered near x=0 and are separated by an amount of O(ε|lnε|), while only single spikes are allowed near the boundaries x=±1. 相似文献
19.
Masaru Yamaguchi 《Journal of Differential Equations》2004,203(2):255-291
First, we shall deal with the free vibrations of a nonlinear radially symmetric wave equation (∂t2−△)u=f(r,u) in n-dimensional ball Ba with center at the origin and radius a, where f is smooth, monotone decreasing in u, and satisfies f(r,0)=0. f(r,u) has asymptotic properties . For n=1,3 we shall show the existence of infinitely many radially symmetric time-periodic solutions with different periods of irrational multiple of a. Second, we shall deal with BVP for a forced nonlinear wave equation (∂t2−△)u=εg(r,t,u), where g is T-periodic in t and ε is a small parameter. Under some Diophantine condition on a/T we shall show the existence of time-periodic solutions of the BVP. For 1?n?5 we shall construct infinitely many T satisfying the above Diophantine inequality, using asymptotic expansions of the zero points of the Bessel functions. 相似文献
20.
Behzad Djafari Rouhani 《Journal of Differential Equations》2006,229(2):412-425
Let X be a reflexive Banach space. We introduce the notion of weakly almost nonexpansive sequences (xn)n?0 in X, and study their asymptotic behavior by showing that the nonempty weak ω-limit set of the sequence (xn/n)n?1 always lies on a convex subset of a sphere centered at the origin of radius d=limn→∞‖xn/n‖. Subsequently we apply our results to study the asymptotic properties of unbounded trajectories for the quasi-autonomous dissipative system , where A is an accretive (possibly multivalued) operator in X×X, and f−f∞∈Lp((0,+∞);X) for some f∞∈X and 1?p<∞. These results extend recent results of J.S. Jung and J.S. Park [J.S. Jung, J.S. Park, Asymptotic behavior of nonexpansive sequences and mean points, Proc. Amer. Math. Soc. 124 (1996) 475-480], and J.S. Jung, J.S. Park, and E.H. Park [J.S. Jung, J.S. Park, E.H. Park, Asymptotic behaviour of generalized almost nonexpansive sequences and applications, Proc. Nonlinear Funct. Anal. 1 (1996) 65-79], as well as our results cited below containing previous results by several authors. 相似文献