共查询到20条相似文献,搜索用时 15 毫秒
1.
The existence, uniqueness and multiplicity of positive solutions of the following boundary value problem is considered: u(4)(t)−λf(t,u(t))=0, for 0<t<1,u(0)=u(1)=u″(0)=u″(1)=0, where λ>0 is a constant, f :[0,1]×[0,+∞)→[0,+∞) is continuous. 相似文献
2.
This paper deals with the Cauchy problem ut − uxx + up = 0; − ∞ < x < + ∞, t>0, u( x, 0) = u0( x); − ∞ < x < + ∞, where 0 < p < 1 and u0( x) is continuous, nonnegative, and bounded. In this case, solutions are known to vanish in a finite time T, and interfaces separating the regions where u( x, t) > 0 and u( x, t) = 0 appear when t is close to T. We describe here all possible asymptotic behaviours of solutions and interfaces near an extinction point as the extinction time is approached. We also give conditions under which some of these behaviours actually occur. 相似文献
3.
We investigate the large-time behaviour of solutions to the nonlinear heat-conduction equation with absorption ut = Δ( uσ + 1) − uβ in Q = R N × (0, ∞) (E) with N 1, σ > 0 and critical absorption exponent β = σ + 1 + 2/ N; the initial function u( x, 0) = 0 is assumed to be integrable, nonnegative and compactly supported. We prove that u converges as t → ∞ to a unique self-similar function which is a contracted version of one of the asymptotic profiles of the nonabsorptive problem ut = Δ( uσ + 1), the same for any initial data. The cornerstone of the proof is a result about ω-limits of (infinite-dimensional) asymptotical dynamical systems. Combining this result with an asymptotic evaluation of the mass function as well as typical PDE estimates gives the behaviour of (E) for large times.Similar unusual asymptotic behaviour is obtained for the equation ut = div(¦ Du¦ σ Du) − uβ with same conditions on σ and u( x, 0) and critical value for β = σ + 1 + (σ + 2)/ N. 相似文献
4.
Summary In this paper we examine the first initial boundary value problem for u t=u xx + (1 – u) –, > 0, > 0, on (0, 1) × (0, ) from the point of view of dynamical systems. We construct the set of stationary solutions, determine those which are stable, those which are not and show that there are solutions with initial data arbitrarily close to unstable stationary solutions which quench (reach one in finite time). We also examine the related problem u t=u xx, 0 < x < 1, t > 0; u(0, t)=0, (1 – u(1, t)) –. The set of stationary solutions for this problem, and the dynamical behavior of solutions of the time dependent problem are somewhat different.This research was sponsored by the U.S. Air Force Office of Scientific Research, Air Forse Systems Command Grants 84-0252 and 88-0031. The United States Government is authorized to reproduce and distribute reprints for Governmental purposes not withstanding any copyright notation therein. 相似文献
5.
Consider the equation − ε2Δuε + q( x) uε = f( uε) in , u(∞) < ∞, ε = const > 0. Under what assumptions on q( x) and f( u) can one prove that the solution uε exists and lim ε→0uε = u( x), where u( x) solves the limiting problem q( x) u = f( u)? These are the questions discussed in the paper. 相似文献
6.
Summary
The paper deals with the (n + 1) -point problem u
(n)=f(t, u, u, ..., u
(n–1), u(t
0)= u(t
1)=...= u(t
n), where – < t
0 < t
1 < ... < t
n< + . There are established the sufficient conditions for the existence and uniqueness of solutions of this problem. 相似文献
7.
Let { u0, u1,… un − 1} and { u0, u1,…, un} be Tchebycheff-systems of continuous functions on [ a, b] and let f ε C[ a, b] be generalized convex with respect to { u0, u1,…, un − 1}. In a series of papers ([1], [2], [3]) D. Amir and Z. Ziegler discuss some properties of elements of best approximation to f from the linear spans of { u0, u1,…, un − 1} and { u0, u1,…, un} in the Lp-norms, 1 p ∞, and show (under different conditions for different values of p) that these properties, when valid for all subintervals of [ a, b], can characterize generalized convex functions. Their methods of proof rely on characterizations of elements of best approximation in the Lp-norms, specific for each value of p. This work extends the above results to approximation in a wider class of norms, called “sign-monotone,” [6], which can be defined by the property: ¦ f( x)¦ ¦ g( x)¦, f( x) g( x) 0, a x b, imply f g . For sign-monotone norms in general, there is neither uniqueness of an element of best approximation, nor theorems characterizing it. Nevertheless, it is possible to derive many common properties of best approximants to generalized convex functions in these norms, by means of the necessary condition proved in [6]. For { u0, u1,…, un} an Extended-Complete Tchebycheff-system and f ε C(n)[ a, b] it is shown that the validity of any of these properties on all subintervals of [ a, b], implies that f is generalized convex. In the special case of f monotone with respect to a positive function u0( x), a converse theorem is proved under less restrictive assumptions. 相似文献
8.
Let TR be a time-scale, with a=inf T, b=sup T. We consider the nonlinear boundary value problem | where λR+:=[0,∞), and satisfies the conditions We prove a strong maximum principle for the linear operator defined by the left-hand side of (1), and use this to show that for every solution (λ,u) of (1)–(2), u is positive on T a,b . In addition, we show that there exists λmax>0 (possibly λmax=∞), such that, if 0λ<λmax then (1)–(2) has a unique solution u(λ), while if λλmax then (1)–(2) has no solution. The value of λmax is characterised as the principal eigenvalue of an associated weighted eigenvalue problem (in this regard, we prove a general existence result for such eigenvalues for problems with general, nonnegative weights). 相似文献
9.
In this paper, the two-dimensional generalized complex Ginzburg–Landau equation (CGL)
ut=ρu−Δφ(u)−(1+iγ)Δu−νΔ2u−(1+iμ)|u|2σu+αλ1(|u|2u)+β(λ2)|u|2
is studied. The existence of global attractor for this equation with periodic boundary condition is established and upper bounds of Hausdorff and fractal dimensions of attractor are obtained.
相似文献
10.
Cauchy's problem for the equation
u
xx
+x
–1
u
x
=u
t
( real) was discussed by
D. Colton if –1,–2,–3, ... Now existence and uniqueness theorems and representations of the solutions are given for the cases =–1,–2, –3,... The methods of
D. Colton and of this paper are different but the results are similar.
相似文献
11.
Cauchy's problem for the equation
u
xx
+x
–1
u
x
=u
t
( real) was discussed by
D. Colton if –1,–2,–3, ... Now existence and uniqueness theorems and representations of the solutions are given for the cases =–1,–2, –3,... The methods of
D. Colton and of this paper are different but the results are similar.
相似文献
12.
Let·(σ(
x)
u)= 0 in
D R3, where
D is a bounded domain with a smooth boundary. Suppose that σ ≥
m> 0, σ
H3(
D), where
Hℓ is the Sobolev space. Let the set {
u, σ
uN} be given on Γ for all
u H3/2(Γ), where
uN is the normal derivative of
u on Γ.
相似文献
13.
We consider the optimization problem of minimizing
in the class of functions
W1,G(
Ω), with a constraint on the volume of {
u>0}. The conditions on the function
G allow for a different behavior at 0 and at ∞. We consider a penalization problem, and we prove that for small values of the penalization parameter, the constrained volume is attained. In this way we prove that every solution
u is locally Lipschitz continuous and that the free boundary, ∂{
u>0}∩
Ω is smooth.
相似文献
14.
A
hamiltonian cycle C of a graph
G is an ordered set
u1,
u2,…,
un(G),
u1 of vertices such that
ui≠
uj for
i≠
j and
ui is adjacent to
ui+1 for every
i{1,2,…,
n(
G)−1} and
un(G) is adjacent to
u1, where
n(
G) is the order of
G. The vertex
u1 is the starting vertex and
ui is the
ith vertex of
C. Two hamiltonian cycles
C1=
u1,
u2,…,
un(G),
u1 and
C2=
v1,
v2,…,
vn(G),
v1 of
G are
independent if
u1=
v1 and
ui≠
vi for every
i{2,3,…,
n(
G)}. A set of hamiltonian cycles {
C1,
C2,…,
Ck} of
G is
mutually independent if its elements are pairwise independent. The
mutually independent hamiltonicity IHC(
G) of a graph
G is the maximum integer
k such that for any vertex
u of
G there exist
k mutually independent hamiltonian cycles of
G starting at
u.In this paper, the mutually independent hamiltonicity is considered for two families of Cayley graphs, the
n-dimensional pancake graphs
Pn and the
n-dimensional star graphs
Sn. It is proven that
IHC(
P3)=1,
IHC(
Pn)=
n−1 if
n≥4,
IHC(
Sn)=
n−2 if
n{3,4} and
IHC(
Sn)=
n−1 if
n≥5.
相似文献
15.
Let
u(
r,θ) be biharmonic and bounded in the circular sector ¦θ¦ < π/4, 0 <
r < ρ (ρ > 1) and vanish together with δ
u/δθ when ¦θ¦ = π/4. We consider the transform û(
p,θ) = ∝
01rp − 1u(
r,θ)
dr. We show that for any fixed θ
0 u(
p,θ
0) is meromorphic with no real poles and cannot be entire unless
u(
r, θ
0) ≡ 0. It follows then from a theorem of Doetsch that
u(
r, θ
0) either vanishes identically or oscillates as
r → 0.
相似文献
16.
We consider the semilinear elliptic equation Δ
u=
h(
u) in
Ω{0}, where
Ω is an open subset of
(
N2) containing the origin and
h is locally Lipschitz continuous on [0,∞), positive in (0,∞). We give a complete classification of isolated singularities of positive solutions when
h varies regularly at infinity of index
q(1,
CN) (that is, lim
u→∞h(
λu)/
h(
u)=
λq, for every
λ>0), where
CN denotes either
N/(
N−2) if
N3 or ∞ if
N=2. Our result extends a well-known theorem of Véron for the case
h(
u)=
uq.
相似文献
17.
We study the subcritical problems
P
:–
u=
u
p–,
u>0 on
;
u=0 on ,
being a smooth and bounded domain in
N,
N–3,
p+1=2N/N–2 the critical Sobolev exponent and >0 going to zero — in order to compute the difference of topology that the critical points at infinity induce between the level sets of the functional corresponding to the limit case (P
0).
Résumé Nous étudions les problèmes sous-critiquesP
:–u=u
p–,u > 0 sur;u=0 sur –où est un domaine borné et régulier de N,N–3,p + 1=2N/N –2 est l'exposant critique de Sobolev, et >0 tend vers zéro, afin de calculer la différence de toplogie induite par les points critiques à l'infini entre les ensembles de niveau de la fonctionnelle correspondant au cas limite (P0).
相似文献
18.
We show that for
ε small, there are arbitrarily many
nodal solutions for the following nonlinear elliptic Neumann problem where Ω is a bounded and smooth domain in ℝ
2 and
f grows superlinearly. (A typical
f(
u) is
f(
u)=
a1 u+p –
a1 u-p,
a1,
a2 >0,
p,
q>1.) More precisely, for any positive integer
K, there exists
εK>0 such that for 0<
ε<
εK, the above problem has a nodal solution with
K positive local maximum points and
K negative local minimum points. This solution has at least
K+1 nodal domains. The locations of the maximum and minimum points are related to the mean curvature on ∂Ω. The solutions are constructed as critical points of some finite dimensional reduced energy functional. No assumption on the symmetry, nor the geometry, nor the topology of the domain is needed.
相似文献
19.
An inverse polynomial method of determining the unknown leading coefficient
k=
k(
x) of the linear Sturm–Liouville operator
Au=−(
k(
x)
u′(
x))
′+
q(
x)
u(
x),
x(0,1), is presented. As an additional condition only two measured data at the boundary (
x=0,
x=1) are used. In absence of a singular point (
u′(
x)≠0,
u″(
x)≠0,
x[0,1]) the inverse problem is classified as a
well-conditioned . If there exists at least one singular point, then the inverse problem is classified as
moderately ill-conditioned (
u′(
x0)=0,
x0(0,1);
u′(
x)≠0,
x≠
x0;
u″(
x)≠0,
x[0,1]) and
severely ill-conditioned (
u′(
x0)=
u″(
x0)=0,
x0(0,1);
u′(
x)≠0,
u″(
x)≠0,
x≠
x0). For each of the cases direct problem solution is approximated by corresponding polynomials and the inverse problem is reformulated as a Cauchy problem for to the first order differential equation with respect the unknown function
k=
k(
x). An approximate analytical solution of the each Cauchy problems are derived in explicit form. Numerical simulations all the above cases are given for noise free and noisy data. An accuracy of the presented approach is demonstrated on numerical test solutions.
相似文献
20.
The authors consider the semilinear SchrSdinger equation
-△Au+Vλ(x)u= Q(x)|u|γ-2u in R^N,
where 1 〈 γ 〈 2* and γ≠ 2, Vλ= V^+ -λV^-. Exploiting the relation between the Nehari manifold and fibrering maps, the existence of nontrivial solutions for the problem is discussed.
相似文献