共查询到20条相似文献,搜索用时 31 毫秒
1.
M Renardy 《Journal of Differential Equations》1983,48(2):280-292
A semigroup approach to differential-delay equations is developed which reduces such equations to ordinary differential equations on a Banach space of histories and seems more suitable for certain partial integro-differential equations than the standard theory. The method is applied to prove a local-time existence theorem for equations of the form utt = g(uxt, uxt)x, where . On a formal level, it is demonstrated that the stretching of filaments of viscoelastic liquids can be described by an equation of this form. 相似文献
2.
Hendrik J Kuiper 《Journal of Mathematical Analysis and Applications》1977,60(1):166-181
In this paper we prove existence, uniqueness, and regularity results for systems of nonlinear second order parabolic equations with boundary conditions of the Dirichlet, Neumann, and regular oblique derivative types. Let K(t) consist of all functions (v1(x), v2(x),…, vm(x)) from into Rm which satisfy ψi(x, t) ? vi(x) ? θi(x, t) for all , where ψiand θi are extended real-valued functions on . We find conditions which will ensure that a solution U(x, t) ≡ (u1(x, t), u2(x, t),…, um(x, t)) which satisfies U(x, 0) ?K(0) will also satisfy U(x, t) ?K(t) for all 0 ? t < T. This result, which has some similarity to the Gronwall Inequality, is then used to prove a global existence theorem. 相似文献
3.
F.J Testa 《Journal of Mathematical Analysis and Applications》1976,55(3):537-548
The integral formulation of the Navier-Stokes initial value problem for boundary-free, incompressible fluid flow is used to establish Volterra integral inequalities on the physical velocity field uμ(x, t). Standard comparison theorems for Volterra equations are then applied to obtain weakly singular, nonlinear Volterra equations of the second kind for upper bounds on . With local existence and uniqueness guaranteed and smoothness of solutions characterized by results on Volterra equations, solutions for bounds may be obtained by standard analytical and numerical methods. A specific example is considered. Existence and uniqueness of local solutions uμ(x, t) to the Navier-Stokes problem is then guaranteed within the radius of convergence of bounds on , thereby precluding local breakdown phenomena. In addition, bounds on are used to obtain improved duration times for convergence of local iteration solutions for uμ(x, t). Finally, a new technique for establishing sufficient conditions for global existence, based on successive application of Volterra comparison theorems, is indicated. 相似文献
4.
We investigate the boundary value problem , , u(?∞, t) = v(∞, t) = 0, u(∞, t) = 1, and v(?∞, t) = γ ?t > 0 where r > 0, b > 0, γ > 0 and x?R. This system has been proposed by Murray as a model for the propagation of wave fronts of chemical activity in the Belousov-Zhabotinskii chemical reaction. Here u and v are proportional to the concentrations of bromous acid and bromide ion, respectively. We determine the global stability of the constant solution (u, v) ≡ (1,0). Furthermore we introduce a moving coordinate and for each fixed x?R we investigate the asymptotic behavior of u(x + ct, t) and v(x + ct, t) as t → ∞ for both large and small values of the wave speed c ? 0. 相似文献
5.
We consider the equation R2, with defined as a divergenceless vector valued function (in the generalized sense) and subject to initial data u0. We prove the existence of a generalized global solution to this equation under assumptions of bounded variation for and u0, and smoothness for ?. 相似文献
6.
Results on partition of energy and on energy decay are derived for solutions of the Cauchy problem . Here the Aj's are constant, k × k Hermitian matrices, x = (x1,…, xn), t represents time, and u = u(t, x) is a k-vector. It is shown that the energy of Mu approaches a limit , where M is an arbitrary matrix; that there exists a sufficiently large subspace of data ?, which is invariant under the solution group U0(t) and such that depending on ? and that the local energy of nonstatic solutions decays as . More refined results on energy decay are also given and the existence of wave operators is established, considering a perturbed equation at infinity. 相似文献
7.
Abraham Boyarsky 《Journal of Mathematical Analysis and Applications》1978,63(2):490-501
Let xtu(w) be the solution process of the n-dimensional stochastic differential equation dxtu = [A(t)xtu + B(t) u(t)] dt + C(t) dWt, where A(t), B(t), C(t) are matrix functions, Wt is a n-dimensional Brownian motion and u is an admissable control function. For fixed ? ? 0 and 1 ? δ ? 0, we say that x?Rn is (?, δ) attainable if there exists an admissable control u such that P{xtu?S?(x)} ? δ, where S?(x) is the closed ?-ball in Rn centered at x. The set of all (?, δ) attainable points is denoted by (t). In this paper, we derive various properties of (t) in terms of K(t), the attainable set of the deterministic control system . As well a stochastic bang-bang principle is established and three examples presented. 相似文献
8.
《Comptes Rendus de l'Academie des Sciences Series IIA Earth and Planetary Science》1998,326(5):595-599
This Note presents a construction of a solution for the nonlinear stochastic differential equation Xt = X0 + ∫0t [u0(X0)|Xs]ds, t ≥ 0. The random variable X0 with values in and the function u0 are given. We denote by Pt the probability distribution of Xt and u(x, t) = [u0(X0)|Xt = x]. We prove that (Pt, u(·, t), t ≥ 0) is a weak solution for system of conservation law arising in adhesion particle dynamics. 相似文献
9.
The existence of a unique strong solution of the nonlinear abstract functional differential equation , (E) is established. X is a Banach space with uniformly convex dual space and, for is m-accretive and satisfies a time dependence condition suitable for applications to partial differential equations. The function F satisfies a Lipschitz condition. The novelty of the paper is that the solution u(t) of (E) is shown to be the uniform limit (as n → ∞) of the sequence un(t), where the functions un(t) are continuously differentiate solutions of approximating equations involving the Yosida approximants. Thus, a straightforward approximation scheme is now available for such equations, in parallel with the approach involving the use of nonlinear evolution operator theory. 相似文献
10.
In this paper we study the existence, uniqueness, and regularity of the solutions for the Cauchy problem for the evolution equation is in (0, 1), 0 ? t ? T, T is an arbitrary positive real number,f(s)?C1, and g(x, t)?L∞(0, T; L2(0, 1)). We prove the existence and uniqueness of the weak solutions for (1) using the Galerkin method and a compactness argument such as that of J. L. Lions. We obtain regular solutions using eigenfunctions of the one-dimensional Laplace operator as a basis in the Galerkin method. 相似文献
11.
Richard E. Ewing 《Journal of Mathematical Analysis and Applications》1979,71(1):167-186
Numerical approximation of the solution of the Cauchy problem for the linear parabolic partial differential equation is considered. The problem: (p(x)ux)x ? q(x)u = p(x)ut, 0 < x < 1,0 < t? T; ; ; p(0) ux(0, t) = g(t), 0 < t0 ? t ? T, is ill-posed in the sense of Hadamard. Complex variable and Dirichlet series techniques are used to establish Hölder continuous dependence of the solution upon the data under the additional assumption of a known uniform bound for ¦ u(x, t)¦ when 0 ? x ? 1 and 0 ? t ? T. Numerical results are obtained for the problem where the data ?1, ?2 and g are known only approximately. 相似文献
12.
Paolo Acquistapace Brunello Terreni 《Journal of Mathematical Analysis and Applications》1984,99(1):9-64
We study existence, uniqueness and regularity of the strict, classical and strong solutions of the non-autonomous evolution equation , with the initial datum u(0) = x, in a Banach space E, under the classical Kato-Tanabe assumptions. The domains of the operators A(t) are not needed to be dense in E. We prove necessary and sufficient conditions for existence and Hölder regularity of the solution and its derivative. 相似文献
13.
David W. Bange 《Journal of Differential Equations》1975,17(1):61-72
This paper treats the quasilinear, parabolic boundary value problem u(0, t) = ?1(t); u(l, t) = ?2(t) on an infinite strip with the functions being periodic in t. The major theorem of the paper gives sufficient conditions on for this problem to have a periodic solution u(x, t) which may be constructed by successive approximations with an integral operator. Some corollaries to this theorem offer more explicit conditions on and indicate a method for determining the initial estimate at which the iteration may begin. 相似文献
14.
R.E OMalley 《Journal of Mathematical Analysis and Applications》1974,45(2):468-484
We shall examine the control problem consisting of the system on the interval 0 ? t ? 1 with the initial values x(0, ?) and z(0, ?) prescribed, where the cost functional J(?) = π(x(1, ?), z(1, ?), ?) + ∝01V(x(t, ?), z(t, ?), u(t, ?), t, ?) dt is to be minimized. We shall restrict attention to the special problem where the fi's are linear in z and u, V is quadratic in z and independent of z when ? = 0, π and V are positive semidefinite functions of x and z, and V is a positive definite function of u. Under appropriate conditions, we shall obtain an asymptotic solution of the problem valid as the small parameter ? tends to zero. The techniques of constructing such asymptotic expansions will be stressed. 相似文献
15.
J.G. Besjes 《Journal of Mathematical Analysis and Applications》1974,48(2):594-609
We consider the first initial-boundary value problem for and L1 are linear elliptic partial differential operators) and investigate the properties of u(x, t, ?) as ? ↓ 0 in the maximum norm. Special attention is paid to approximations obtained by the boundary layer method. We use a priori estimates. 相似文献
16.
David Terman 《Journal of Differential Equations》1983,47(3):406-443
We consider the pure initial value problem for the system of equations , the initial data being (ν(x, 0), w(x, 0)) = (?(x), 0). Here , where H is the Heaviside step function and . This system is of the FitzHugh-Nagumo type and has several applications including nerve conduction and distributed chemical/ biochemical systems. It is demonstrated that this system exhibits a threshold phenomenon. This is done by considering the curve s(t) defined by s(t) = sup{x: v(x, t) = a}. The initial datum, ?(x), is said to be superthreshold if limt→∞ s(t) = ∞. It is proven that the initial datum is superthreshold if ?(x) > a on a sufficiently long interval, ?(x) is sufficiently smooth, and ?(x) decays sufficiently fast to zero as . 相似文献
17.
Václav E. Beneš 《Stochastic Processes and their Applications》1974,2(2):127-140
Girsanov's theorem is a generalization of the Cameron-Martin formula for the derivative of a measure induced by a translation in Wiener space. It states that for ? a nonanticipative Brownian functional with ∫|?|2 ds < ∞ a.s. and with , where , the translated functions are a Wiener process under P?. The Girsanov functionals exp [ζ(?)] have been used in stochastic control theory to define measures corresponding to solutions of stochastic DEs with only measurable control laws entering the right-hand sides. The present aim is to show that these same concepts have direct practical application to final value problems with bounded control. This is done here by an example, the noisy integrator: Make E{x21}∣small, subject to dxt = ut dt + dwt, |u|? 1, xt observed. For each control law there is a definite cost v(1?t, x) of starting at x, t and using that law till t = 1, expressible as an integral with respect to (a suitable) P?. By restricting attention to a dense set of smooth laws, using Itô's lemma, Kac's theorem, and the maximum principle for parabolic equations, it is possible to calculate sgn vx for a critical class of control laws, then to compare control laws, “solve” the Bellman-Hamilton-Jacobi equation, and thus justify selection of the obvious bang-bang law as optimal. 相似文献
18.
The existence of periodic solutions near resonance is discussed using elementary methods for the evolution equation ·u = Au + ?f(t, u) when the linear problem is totally degenerate (e2πA = I) and the period of f is entrained with ? (T = 2π(1 + ?μ)). The approach is to solve the periodicity equation u(T,p,?) = p for an element p(?) in D, the domain of A, as a perturbation from an approximate solution p0. p0 is a solution of the nonlinear boundary value problem 2πμAp + ∝02πe?Asf(s, eAsp) ds = 0 obtained from the periodicity equation by dividing by ?, applying the entrainment assumption, and letting ? → 0. Once p0 is known, the conventional inverse function theorem is applied in a slightly unconventional manner. Two particular cases where results are obtained are ut = ux + ?{g(u) ? h(t, x)} with g strongly monotone and , where in both cases D is a certain class of 2π-periodic functions of x. 相似文献
19.
Ethelbert N Chukwu 《Journal of Mathematical Analysis and Applications》1980,76(1):283-296
Sufficient conditions are developed for the null-controllability of the nonlinear delay process (1) when the values of the control functions u lie in an m-dimensional unit cube Cm of Em. Conditions are placed on f which guarantee that if the uncontrolled system is uniformly asymptotically stable and if the linear control system x(t) = L(t, xt) + B(t) u(t) is proper, then (1) is null-controllable. 相似文献
20.
Karen A Ames 《Journal of Mathematical Analysis and Applications》1984,103(1):172-183
Solutions of Cauchy problems for the singular equations (in a Hilbert space setting) and , ω={(x1,…,xM)εm: 0 < xi < ci for each i=1,…,m} are shown to be unique and to depend Hölder continuously on the initial data in suitably chosen measures for 0?t < T < ∞. Logarithmic convexity arguments are used to derive the inequalities from which such results can be deduced. 相似文献