共查询到20条相似文献,搜索用时 31 毫秒
1.
Marina GHISI Massimo GOBBINO 《数学学报(英文版)》2006,22(4):1161-1170
We consider the Cauchy problem εu^″ε + δu′ε + Auε = 0, uε(0) = uo, u′ε(0) = ul, where ε 〉 0, δ 〉 0, H is a Hilbert space, and A is a self-adjoint linear non-negative operator on H with dense domain D(A). We study the convergence of (uε) to the solution of the limit problem ,δu' + Au = 0, u(0) = u0.
For initial data (u0, u1) ∈ D(A1/2)× H, we prove global-in-time convergence with respect to strong topologies.
Moreover, we estimate the convergence rate in the case where (u0, u1)∈ D(A3/2) ∈ D(A1/2), and we show that this regularity requirement is sharp for our estimates. We give also an upper bound for |u′ε(t)| which does not depend on ε. 相似文献
2.
For a bounded linear injectionCon a Banach spaceXand a closed linear operatorA : D(A) X → Xwhich commutes withCwe prove that (1) the abstract Cauchy problem,u″(t) = Au(t),t R,u(0) = Cx,u′(0) = Cy, has a unique strong solution for everyx,y D(A) if and only if (2)A1 = AD(A2) generates aC1-cosine function onX1(D(A) with the graph norm), if (and only if, in caseAhas nonempty resolvent set) (3)Agenerates aC-cosine function onX. HereC1 = CX1. Under the assumption thatAis densely defined andC−1AC = A, statement (3) is also equivalent to each of the following statements: (4) the problemv″(t) = Av(t) + C(x + ty) + ∫t0 Cg(r) dr,t R,v(0) = v′(0) = 0, has a unique strong solution for everyg L1locandx, y X; (5) the problemw″(t) = Aw(t) + Cg(t),t R,w(0) = Cx,w′(0) = Cy, has a unique weak solution for everyg L1locandx, y X. Finally, as an application, it is shown that for any bounded operatorBwhich commutes withCand has range contained in the range ofC,A + Bis also a generator. 相似文献
3.
A. A. Lyashenko 《Mathematical Methods in the Applied Sciences》1995,18(7):549-569
In the present paper we discuss the stability of semilinear problems of the form Aαu + Gα(u) = ? under assumption of an a priori bound for an energy functional Eα(u) ? E, where α is a parameter in a metric space M. Following [11] the problem Aαu + Gα(u) = ?, Eα(u) ? E is called stable in a Hilbert space H at a point α ? M if for any ??H, E, ? > 0 there exists δ > 0 such that for any functions uα1, uα2 satisfying Aαjuαj + Gαj(uαj) = ?αj, Eαj(uαj) ? E, j = 1,2 we have ‖uα1 ? uα2H ? ? provided ρM(αj, α) ? δ, ‖?αj ? ?‖H ? δ, j = 1,2. In the present paper we obtain stability conditions for the problem Aαu + Gα(u) = ?, Eα(u) ? E. 相似文献
4.
Kaouther Ammar 《Central European Journal of Mathematics》2010,8(3):548-568
The paper addresses the existence and uniqueness of entropy solutions for the degenerate triply nonlinear problem: b(v)
t
− div α(v, ▽g(v)) = f on Q:= (0, T) × Ω with the initial condition b(v(0, ·)) = b(v
0) on Ω and the nonhomogeneous boundary condition “v = u” on some part of the boundary (0, T) × ∂Ω”. The function g is continuous locally Lipschitz continuous and has a flat region [A
1, A
2,] with A
1 ≤ 0 ≤ A
2 so that the problem is of parabolic-hyperbolic type. 相似文献
5.
A. K. Aziz M. Schneider R. P. Gilbert 《Mathematical Methods in the Applied Sciences》1980,2(2):168-177
We consider the equation of mixed type (k(y) ? 0 whenever y ? 0) in a region G which is bounded by the curves: A piecewise smooth curve Γ lying in the half-plane y > 0 which intersects the line y = 0 at the points A(-1, 0) and B(0, 0). For y < 0 by a piecewise smooth curve Γ through A which meets the characteristic of (1) issued from B at the point P and the curve Γ which consists of the portion PB of the characteristic through B. We obtain sufficient conditions for the uniqueness of the solution of the problem L[u] = f, dnu: = k(y)uxdy – uydx|γ0 = = Ψ(s) for a “general” function k(y), when r(x, y) is not necessarily zero and Γ1 is of a more general form then in the papers of V. P. Egorov [6], [7]. 相似文献
6.
This paper deals with the initial-value problem of nonlinear evolution inclusions of the form dB(u)/dt + A(u) f, v0 ∈ B(u)(0), where the operator B is induced by a subgradient and A is pseudomonotone. Existence theorem is established via the time discretization technique and the regularization method. In contrast to the previous results, here we impose a weaker coerciveness condition on A and remove the strong monotonicity from B. 相似文献
7.
A. G. Ramm 《Mathematical Methods in the Applied Sciences》2013,36(4):422-426
Consider an abstract evolution problem in a Hilbert space H (1) where A(t) is a linear, closed, densely defined operator in H with domain independent of t ≥ 0 and G(t,u) is a nonlinear operator such that ‖G(t,u)‖a(t) ‖u‖p, p = const > 1, ‖f(t)‖ ≤ b(t). We allow the spectrum of A(t) to be in the right half‐plane Re(λ) < λ0(t), λ0(t) > 0, but assume that limt → ∞λ0(t) = 0. Under suitable assumptions on a(t) and b(t), the boundedness of ‖u(t)‖ as t → ∞ is proved. If f(t) = 0, the Lyapunov stability of the zero solution to problem (1) with u0 = 0 is established. For f ≠ 0, sufficient conditions for the Lyapunov stability are given. The novel point in our study of the stability of the solutions to abstract evolution equations is the possibility for the linear operator A(t) to have spectrum in the half‐plane Re(λ) < λ0(t) with λ0(t) > 0 and limt → ∞λ0(t) = 0 at a suitable rate. The new technique, proposed in the paper, is based on an application of a novel nonlinear differential inequality. Copyright © 2012 John Wiley & Sons, Ltd. 相似文献
8.
For quasilinear equations div A(x, u, ∇u) = 0 with degeneracy ω(x) of the Muckenhoupt A
p
-class, we prove the Harnack inequality, an estimate for the H?lder norm, and a sufficient criterion for the regularity of
boundary points of the Wiener type.
Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 7, pp. 918–936, July, 2008. 相似文献
9.
Patrick S. Hagan 《Studies in Applied Mathematics》1981,64(1):57-88
We consider the class of equations ut=f(uxx, ux, u) under the restriction that for all a,b,c. We first consider this equation over the unbounded domain ? ∞ < x < + ∞, and we show that very nearly every bounded nonmonotonic solution of the form u(t, x)=?(x?ct) is unstable to all nonnegative and all nonpositive perturbations. We then extend these results to nonmonotonic plane wave solutions u(t, x, y)=?(x?ct) of ut = F(uxx, uxy, ux, uy, u). Finally, we consider the class of equations ut=f(uxx, ux, u) over the bounded domain 0 < x < 1 with the boundary conditions u(t, x)=A at x=0 and u(t, x)=B at x=1, and we find the stability of all steady solutions u(t, x)=?(x). 相似文献
10.
Roland Schnaubelt 《Journal of Evolution Equations》2001,1(1):19-37
We study the long-term behaviour of the parabolic evolution equation $\[u'(t)=A(t)u(t)+f(t), t>s,\quad u(s)=x. \]$\[u'(t)=A(t)u(t)+f(t), t>s,\quad u(s)=x. \] If A(t) A(t) converges to a sectorial operator A with s(A)?i \Bbb R = ? \sigma(A)\cap i \Bbb R =\emptyset as t?¥ t\to\infty , then the evolution family solving the homogeneous problem has exponential dichotomy. If also f(t)? f¥ f(t)\to f_\infty , then the solution u converges to the 'stationary solution at infinity', i.e., limt?¥u(t) = -A\sp-1f¥=:u¥, limt?¥u¢(t)=0, limt?¥A(t)u(t)=Au¥. \lim_{t\to\infty}u(t)= -A\sp{-1}f_\infty=:u_\infty, \qquad \lim_{t\to\infty}u'(t)=0, \qquad \lim_{t\to\infty}A(t)u(t)=Au_\infty. . 相似文献
11.
We consider the equation u
tt
+ A(u
t
) + B(u) = 0, where A and B are quasilinear operators with respect to the variable x of the second order and the fourth order, respectively. In a cylindrical domain unbounded with respect to the space variables, we obtain estimates that characterize the minimum growth of any nonzero solution of the mixed problem at infinity.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 2, pp. 239–249, February, 2005. 相似文献
12.
Chanchal Singh 《Journal of Mathematical Analysis and Applications》1984,100(2):409-415
Let V ?H be real separable Hilbert spaces. The abstract wave equation u′' + A(t)u = g(u), where u(t) ?V, A(t) maps V to its dual , and g is a nonlinear map from the ball S(R0) = {u?V: ∥u∥ < R0} into H, is considered. It is assumed that g is locally Lipschitz in S(R0) and possibly singular at the boundary. Local existence and continuation theorems are established for the Cauchy problem u(0) = u0?S(R0), u′(0) = u1?H. Global existence is shown for g(u) = εφ(u), where φ has a potential and ε is small. Global nonexistence is shown for g(u) = εφ(u), where φ satisfies an abstract convexity property and ε is large. 相似文献
13.
The paper discusses the existence of positive solutions, dead core solutions and pseudodead core solutions of the singular
Dirichlet problem (ϕ(u′))′ = λf(t, u, u′), u(0) = u(T) = A. Here λ is the positive parameter, A > 0, f is singular at the value 0 of its first phase variable and may be singular at the value A of its first and at the value 0 of its second phase variable.
This work was supported by grant no. A100190703 of the Grant Agency of the Academy of Sciences of the Czech Republic and by
the Council of Czech Government MSM 6198959214. 相似文献
14.
E. Yu. Bunkova 《Proceedings of the Steklov Institute of Mathematics》2016,294(1):201-214
The article is devoted to the theory of elliptic functions of level n. An elliptic function of level n determines a Hirzebruch genus called an elliptic genus of level n. Elliptic functions of level n are also of interest because they are solutions of the Hirzebruch functional equations. The elliptic function of level 2 is the Jacobi elliptic sine function, which determines the famous Ochanine–Witten genus. It is the exponential of the universal formal group of the form F(u, v) = (u2 ? v2)/(uB(v) ? vB(u)), B(0) = 1. The elliptic function of level 3 is the exponential of the universal formal group of the form F(u, v) = (u2A(v) ? v2A(u))/(uA(v)2 ? vA(u)2), A(0) = 1, A″(0) = 0. In the present study we show that the elliptic function of level 4 is the exponential of the universal formal group of the form F(u, v) = (u2A(v) ? v2A(u))/(uB(v) ? vB(u)), where A(0) = B(0) = 1 and for B′(0) = A″(0) = 0, A′(0) = A1, and B″(0) = 2B2 the following relation holds: (2B(u) + 3A1u)2 = 4A(u)3 ? (3A12 ? 8B2)u2A(u)2. To prove this result, we express the elliptic function of level 4 in terms of the Weierstrass elliptic functions. 相似文献
15.
LetA(u)=–diva(x, u, Du) be a Leray-Lions operator defined onW
0
1,p
() and be a bounded Radon measure. For anyu SOLA (Solution Obtained as Limit of Approximations) ofA(u)= in ,u=0 on , we prove that the truncationsT
k(u) at heightk satisfyA(T
k(u)) A(u) in the weak * topology of measures whenk + .
Résumé SoitA(u)=–diva(x, u, Du) un opérateur de Leray-Lions défini surW 0 1,p () et une mesure de Radon bornée. Pour toutu SOLA (Solution Obtenue comme Limite d'Approximations) deA(u)= dans ,u=0 sur , nous démontrons que les troncaturesT k(u) à la hauteurk vérifientA(T k(u)) A(u) dans la topologie faible * des mesures quandk + .相似文献
16.
Melvin L Heard 《Journal of Mathematical Analysis and Applications》1981,80(1):175-202
We consider semilinear integrodifferential equations of the form u′(t) + A(t) u(t) = ∝0tg(t, s, u(s)) ds + f(t), u(0) = u0. For each t ? 0, the operator A(t) is assumed to be the negative generator of a strongly continuous semigroup in a Banach space X. The domain D(A(t)) of A(t) is allowed to vary with t. Thus our models are Volterra integrodifferential equations of “hyperbolic type.” These types of equations arise naturally in the study of viscoelasticity. Our main results are the proofs of existence, uniqueness, continuation and continuous dependence of the solutions. 相似文献
17.
We obtain the comparison of solutions (formulated in terms, of some functions of them) of two nonlinear evolution problems
with different nonlinear terms. First this is obtained for the equationsu
1 – Δφ
i(u)=0, and then for the abstract equationsdu/dt+A
i
u=0 (i=1,2) on a normal Banach lattice. Different applications of our abstract result are given in the last section. 相似文献
18.
For a triple {V, H, V*} of Hilbert spaces, we consider an evolution inclusion of the form u′(t)+A(t)u(t)+δϕ(t, u(t)) ∋
f(t), u(0) = u0, t ∈ (0, T ], where A(t) and ϕ(t, ·), t ∈ [0, T], are a family of nonlinear operators from V to V * and a family of convex lower semicontinuous functionals with common effective domain D(ϕ) ⊂ V. We indicate conditions on the data under which there exists a unique solution of the problem in the space H
1(0, T; V)∩W
∞1 (0, T;H) and the implicit Euler method has first-order accuracy in the energy norm. 相似文献
19.
In a Banach space E, we consider the inverse problem du(t)/dt = Au(t) + (t)p , u(0) = u
0, u(T) = u1, with an unknown function u(t) and an element p E. The operator A is assumed linear and closed. In this paper, we establish minimal constraints on the function C([0, T]) for which the uniqueness of the solution of the inverse problem is completely described in terms of the eigenvalues of the operator A.Translated from Matematicheskie Zametki, vol. 77, no. 2, 2005, pp. 273–290.Original Russian Text Copyright © 2005 by I. V. Tikhonov, Yu. S. Éidelman.This revised version was published online in April 2005 with a corrected issue number. 相似文献
20.
Lech Zarȩba 《NoDEA : Nonlinear Differential Equations and Applications》2009,16(4):445-467
In this paper we consider the mixed problem for the equation u
tt
+ A
1
u + A
2(u
t
) + g(u
t
) = f(x, t) in unbounded domain, where A
1 is a linear elliptic operator of the fourth order and A
2 is a nonlinear elliptic operator of the second order. Under natural assumptions on the equation coefficients and f we proof existence of a solution. This result contains, as a special case, some of known before theorems of existence. Essentially,
in difference up to previous results we prove theorems of existence without the additional assumption on behavior of solution
at infinity.
相似文献