共查询到20条相似文献,搜索用时 31 毫秒
1.
Abdelmajid Siai 《Potential Analysis》2006,24(1):15-45
Let Ω be an open bounded set in ℝN, N≥3, with connected Lipschitz boundary ∂Ω and let a(x,ξ) be an operator of Leray–Lions type (a(⋅,∇u) is of the same type as the operator |∇u|p−2∇u, 1<p<N). If τ is the trace operator on ∂Ω, [φ] the jump across ∂Ω of a function φ defined on both sides of ∂Ω, the normal derivative
∂/∂νa related to the operator a is defined in some sense as 〈a(⋅,∇u),ν〉, the inner product in ℝN, of the trace of a(⋅,∇u) on ∂Ω with the outward normal vector field ν on ∂Ω. If β and γ are two nondecreasing continuous real functions everywhere
defined in ℝ, with β(0)=γ(0)=0, f∈L1(ℝN), g∈L1(∂Ω), we prove the existence and the uniqueness of an entropy solution u for the following problem,
in the sense that, if Tk(r)=max {−k,min (r,k)}, k>0, r∈ℝ, ∇u is the gradient by means of truncation (∇u=DTku on the set {|u|<k}) and
, u measurable; DTk(u)∈Lp(ℝN), k>0}, then
and u satisfies,
for every k>0 and every
.
Mathematics Subject Classifications (2000) 35J65, 35J70, 47J05. 相似文献
2.
Jean-René Licois 《Journal d'Analyse Mathématique》1995,66(1):1-36
LetM be a compact riemannian manifold,h an odd function such thath(r)/r is non-decreasing with limit 0 at 0. Letf(r)=h(r)-γr and assume there exist non-negative constantsA andB and a realp>1 such thatf(r)>Ar
P-B. We prove that any non-negative solutionu ofu
tt+Δgu=f(u) onM x ℝ+ satisfying Dirichlet or Neumann boundary conditions on ϖM converges to a (stationary) solution of Δ
g
Φ=f(Φ) onM with exponential decay of ‖u-Φ‖C
2(M). For solutions with non-constant sign, we prove an homogenisation result for sufficiently small λ; further, we show that
for every λ the map (u(0,·),u
t(0,·))→(u(t,·), u
t(t,·)) defines a dynamical system onW
1/2(M)⊂C(M)×L
2(M) which possesses a compact maximal attractor.
相似文献
3.
In this work we prove that the initial value problem of the Benney-Lin equation ut + uxxx + β(uxx + u xxxx) + ηuxxxxx + uux = 0 (x ∈ R, t ≥0 0), where β 〉 0 and η∈R, is locally well-posed in Sobolev spaces HS(R) for s ≥ -7/5. The method we use to prove this result is the bilinear estimate method initiated by Bourgain. 相似文献
4.
We study the continuous as well as the discontinuous solutions of Hamilton-Jacobi equationu
t +H(u,Du) =g in ℝ
n
x ℝ+ withu(x, 0) =u
0(x). The HamiltonianH(s,p) is assumed to be convex and positively homogeneous of degree one inp for eachs in ℝ. IfH is non increasing ins, in general, this problem need not admit a continuous viscosity solution. Even in this case we obtain a formula for discontinuous
viscosity solutions. 相似文献
5.
C. Boldrighini R. A. Minlos A. Pellegrinotti 《Probability Theory and Related Fields》1997,109(2):245-273
Summary We consider a model of random walk on ℤν, ν≥2, in a dynamical random environment described by a field ξ={ξ
t
(x): (t,x)∈ℤν+1}. The random walk transition probabilities are taken as P(X
t
+1= y|X
t
= x,ξ
t
=η) =P
0( y−x)+ c(y−x;η(x)). We assume that the variables {ξ
t
(x):(t,x) ∈ℤν+1} are i.i.d., that both P
0(u) and c(u;s) are finite range in u, and that the random term c(u;·) is small and with zero average. We prove that the C.L.T. holds almost-surely, with the same parameters as for P
0, for all ν≥2. For ν≥3 there is a finite random (i.e., dependent on ξ) correction to the average of X
t
, and there is a corresponding random correction of order to the C.L.T.. For ν≥5 there is a finite random correction to the covariance matrix of X
t
and a corresponding correction of order to the C.L.T.. Proofs are based on some new L
p
estimates for a class of functionals of the field.
Received: 4 January 1996/In revised form: 26 May 1997 相似文献
6.
We study the Cauchy problem for the nonlinear dissipative equations (0.1) uo∂u-αδu + Β|u|2/n
u = 0,x ∃ Rn,t } 0,u(0,x) = u0(x),x ∃ Rn, where α,Β ∃ C, ℜα 0. We are interested in the dissipative case ℜα 0, and ℜδ(α,Β)≥ 0, θ = |∫ u0(x)dx| ⊋ 0, where δ(α, Β) = ##|α|n-1nn/2 / ((n + 1)|α|2 + α2
n/2. Furthermore, we assume that the initial data u0 ∃ Lp are such that (1 + |x|)αu0 ∃ L1, with sufficiently small norm ∃ = (1 + |x|)α u0 1 + u0 p, wherep 1, α ∃ (0,1). Then there exists a unique solution of the Cauchy problem (0.1)u(t, x) ∃ C ((0, ∞); L∞) ∩ C ([0, ∞); L1 ∩ Lp) satisfying the time decay estimates for allt0 u(t)||∞ Cɛt-n/2(1 + η log 〈t〉)-n/2, if hg = θ2/n 2π ℜδ(α, Β) 0; u(t)||∞ Cɛt-n/2(1 + Μ log 〈t〉)-n/4, if η = 0 and Μ = θ4/n 4π)2 (ℑδ(α, Β))2 ℜ((1 + 1/n) υ1-1 υ2) 0; and u(t)||∞ Cɛt-n/2(1 + κ log 〈t〉)-n/6, if η = 0, Μ = 0, κ 0, where υl,l = 1,2 are defined in (1.2), κ is a positive constant defined in (2.31). 相似文献
7.
In this paper we consider abstract equations of the typeK
ν
ν +ν =w
0, in a closed convex subset of a separable Hilbert spaceH. For eachv in the closed convex subset,K
v :H →H is a bounded linear map. As an application of our abstract result we obtain an existence result for nonlinear integral equations
of the typeν(s)+ν(s)∫
0
1
k(s,t)ν(t)dt =W
0(s) in the spaceL
2 [0,1]. 相似文献
8.
S. Staněk 《Ukrainian Mathematical Journal》2008,60(2):277-298
We present existence principles for the nonlocal boundary-value problem (φ(u(p−1)))′=g(t,u,...,u(p−1), αk(u)=0, 1≤k≤p−1, where p ≥ 2, π: ℝ → ℝ is an increasing and odd homeomorphism, g is a Carathéodory function that is either regular or has singularities in its space variables, and α
k: C
p−1[0, T] → ℝ is a continuous functional. An application of the existence principles to singular Sturm-Liouville problems (−1)n(φ(u(2n−)))′=f(t,u,...,u(2n−1)), u(2k)(0)=0, αku(2k)(T)+bku(2k=1)(T)=0, 0≤k≤n−1, is given.
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 2, pp. 240–259, February, 2008. 相似文献
9.
Ricardo A. Sáenz 《Journal of Fourier Analysis and Applications》2012,18(2):240-265
In this paper we study the boundary limit properties of harmonic functions on ℝ+×K, the solutions u(t,x) to the Poisson equation
\frac?2 u?t2 + Du = 0,\frac{\partial^2 u}{\partial t^2} + \Delta u = 0, 相似文献
10.
This paper is concerned with solutions of the Hamiltonian system:
[(u)\dot]=JHu(t,u)\dot{u}=\mathcal{J}H_{u}(t,u)
, where
H(t,u)=\frac12u·Lu+W(t,u)H(t,u)=\frac{1}{2}u\cdot Lu+W(t,u)
with L being a 2N×2N symmetric matrix and W∈C
1(ℝ×ℝ2N
,ℝ) being super quadratic at infinity in u∈ℝ2N
. We use variational methods to study this problem. By virtue of some auxiliary system related to the “limit equation” of
the Hamiltonian system, we constructed linking levels of the variational functional Φ such that the minimax value E
l
based on the linking structure of Φ satisfies
0 < El < El00
11.
P. V. Tsynaiko 《Ukrainian Mathematical Journal》1998,50(9):1478-1482
We study a periodic boundary-value problem for the quasilinear equation u
tt
−u
xx
=F[u, u
t
, u
x
], u(x, 0)=u(x, π)=0, u(x + ω, t) = u(x, t), x ∈ ℝ t ∈ [0, π], and establish conditions that guarantee the validity of a theorem on unique solvability.
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 9, pp. 1293–1296, September, 1998. 相似文献
12.
N. G. Khoma 《Ukrainian Mathematical Journal》1998,50(12):1917-1923
In three spaces, we find exact classical solutions of the boundary-value periodic problem utt - a2uxx = g(x, t) u(0, t) = u(π, t) = 0, u(x, t + T) = u(x, t), x ∈ ℝ, t ∈ ℝ. We study the periodic boundary-value problem for a quasilinear equation whose left-hand side is the d’Alembert operator
and whose right-hand side is a nonlinear operator.
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 12, pp. 1680–1685, December, 1998. 相似文献
13.
V. A. Galaktionov 《Proceedings of the Steklov Institute of Mathematics》2008,260(1):123-143
The celebrated result by Baras and Goldstein (1984) established that the heat equation with the inverse square potential in
the unit ball B
1 ⊂ ℝ
N
, N ≥ 3, u
t
= Δ
u
+ in B
1 × (0,T), u|∂B
1 = 0, in the supercritical range c > c
Hardy = does not have a solution for any nontrivial L
1 initial data u
0(x) ≥ 0 in B
1 (or for a positive measure u
0). More precisely, it was proved that a regular approximation of a possible solution by a sequence {u
n
(x,t)} of classical solutions corresponding to truncated bounded potentials given by V(x) = ↦ V
n
(x) = min{, n} (n ≥ 1) diverges; i.e., as n → ∞, u
n
(x,t) → + ∞ in B
1 × (0, T). Similar features of “nonexistence via approximation” for semilinear heat PDEs were inherent in related results by Brezis-Friedman
(1983) and Baras-Cohen (1987). The main goal of this paper is to justify that this nonexistence result has wider nature and
remains true without the positivity assumption on data u
0(x) that are assumed to be regular and positive at x = 0. Moreover, nonexistence as the impossibility of regular approximations of solutions is true for a wide class of singular
nonlinear parabolic problems as well as for higher order PDEs including, e.g., u
t
= , and , N > 4.
Dedicated to Professor S.I. Pohozaev on the occasion of his 70th birthday 相似文献
14.
Annunziata Esposito 《Rendiconti del Circolo Matematico di Palermo》2004,53(3):437-442
Let u be harmonic in a simply connected domainG ⊂ ℝ2 and letK be a compact subset of G. In this note, it is proved there exists an “elliptic continuation” of u, namely there exist a smooth
functionu
1 and a second order uniformly elliptic operatorL with smooth coefficients in ℝ2, satisfying:u
1=u inK, Lu
1=0 in ℝ2. A similar continuation theorem, with u itself a solution to an elliptic second order equation inG, is also proved. 相似文献
15.
Shu-Yu Hsu 《Mathematische Annalen》2006,334(1):153-197
Let a1,a2, . . . ,am ∈ ℝ2, 2≤f ∈ C([0,∞)), gi ∈ C([0,∞)) be such that 0≤gi(t)≤2 on [0,∞) ∀i=1, . . . ,m. For any p>1, we prove the existence and uniqueness of solutions of the equation ut=Δ(logu), u>0, in satisfying and logu(x,t)/log|x|→−f(t) as |x|→∞, logu(x,t)/log|x−ai|→−gi(t) as |x−ai|→0, uniformly on every compact subset of (0,T) for any i=1, . . . ,m under a mild assumption on u0 where We also obtain similar existence and uniqueness of solutions of the above equation in bounded smooth convex domains of ℝ2 with prescribed singularities at a finite number of points in the domain. 相似文献
16.
Shu-Yu Hsu 《Mathematische Annalen》2003,325(4):665-693
We prove that the solution u of the equation u
t
=Δlog u, u>0, in (Ω\{x
0})×(0,T), Ω⊂ℝ2, has removable singularities at {x
0}×(0,T) if and only if for any 0<α<1, 0<a<b<T, there exist constants ρ0, C
1, C
2>0, such that C
1
|x−x
0|α≤u(x,t)≤C
2|x−x
0|−α holds for all 0<|x−x
0|≤ρ0 and a≤t≤b. As a consequence we obtain a sufficient condition for removable singularities at {∞}×(0,T) for solutions of the above equation in ℝ2×(0,T) and we prove the existence of infinitely many finite mass solutions for the equation in ℝ2×(0,T) when 0≤u
0∉L
1
(ℝ2) is radially symmetric and u
0L
loc
1(ℝ2).
Received: 16 December 2001 / Revised version: 20 May 2002 / Published online: 10 February 2003
Mathematics Subject Classification (1991): 35B40, 35B25, 35K55, 35K65 相似文献
17.
G. I. Laptev 《Journal of Mathematical Sciences》2008,150(5):2384-2394
This paper deals with conditions for the existence of solutions of the equations
18.
In this paper we establish some oscillation or nonoscillation criteria for the second order half-linear differential equation
19.
I. V. Filimonova 《Journal of Mathematical Sciences》2007,143(4):3415-3428
One considers a semilinear parabolic equation u
t
= Lu − a(x)f(u) or an elliptic equation u
tt
+ Lu − a(x)f(u) = 0 in a semi-infinite cylinder Ω × ℝ+ with the nonlinear boundary condition
, where L is a uniformly elliptic divergent operator in a bounded domain Ω ∈ ℝn; a(x) and b(x) are nonnegative measurable functions in Ω. One studies the asymptotic behavior of solutions of such boundary-value problems
for t → ∞.
__________
Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 26, pp. 368–389, 2007. 相似文献
20.
This paper deals with the blow-up properties of solutions to a system of heat equations u
t=Δu, v
t=Δv in B
R×(0, T) with the Neumann boundary conditions εu/εη=e
v, εv/εη=e
u on S
R×[0, T). The exact blow-up rates are established. It is also proved that the blow-up will occur only on the boundary.
This work is supported by the National Natural Science Foundation of China 相似文献
|