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1.
We investigate the boundary value problem ?u?t = ?2u?x2 + u(1 ? u ? rv), ?v?t = ?2v?x2 ? buv, 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   

2.
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 Ω ? Rn into Rm which satisfy ψi(x, t) ? vi(x) ? θi(x, t) for all x ? Ω and 1 ? i ? m, where ψiand θi are extended real-valued functions on \?gW × [0, T). 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.
We consider the pure initial value problem for the system of equations νt = νxx + ?(ν) ? w, wt= ε(ν ? γw), ε, γ ? 0, the initial data being (ν(x, 0), w(x, 0)) = (?(x), 0). Here ?(v) = ?v + H(v ? a), where H is the Heaviside step function and a ? (0, 12). 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 ¦x¦ → ∞.  相似文献   

4.
Let V?, W?, W and X be Hilbert spaces (0 < ? ? 1) with V? ? W? ? W ? X algebraically and topologically, each space being dense in the one that follows it. For each t? [0, T] let a?(t; u, v), b?(t; u, v) and b(t; u, v) be continuous sesqui-linear forms on V?, W? and W, respectively, which satisfy certain ellipticity conditions. Consider the two equations a?(t; u?, v) + b?(t; u?, v) = 〈f?, v〉 (v?V?) and (u′, v)x + b(t; u, v) = 〈f, v〉 (v?W). Estimates are obtained on the rate of convergence of u? to u, assuming a?(t; u, v) → (u, v)x and b?(t; u, v) → b(t; u, v) in an appropriate sense. These results are then applied to singular perturbation of a class of parabolic boundary value problems.  相似文献   

5.
We discuss the existence of periodic solutions to the wave equation with variable coefficients utt−div(A(x)∇u)+ρ(x,ut)=f(x,t) with Dirichlet boundary condition. Here ρ(x,v) is a function like ρ(x,v)=a(x)g(v) with g(v)?0 where a(x) is nonnegative, being positive only in a neighborhood of a part of the domain.  相似文献   

6.
Under fairly weak assumptions, the solutions of the system of Volterra equations x(t) = ∝0ta(t, s) x(s) ds + f(t), t > 0, can be written in the form x(t) = f(t) + ∝0tr(t, s) f(s) ds, t > 0, where r is the resolvent of a, i.e., the solution of the equation r(t, s) = a(t, s) + ∝0ta(t, v) r(v, s)dv, 0 < s < t. Conditions on a are given which imply that the resolvent operator f0tr(t, s) f(s) ds maps a weighted L1 space continuously into another weighted L1 space, and a weighted L space into another weighted L space. Our main theorem is used to study the asymptotic behavior of two differential delay equations.  相似文献   

7.
We study the p-system with viscosity given by vt ? ux = 0, ut + p(v)x = (k(v)ux)x + f(∫ vdx, t), with the initial and the boundary conditions (v(x, 0), u(x,0)) = (v0, u0(x)), u(0,t) = u(X,t) = 0. To describe the motion of the fluid more realistically, many equations of state, namely the function p(v) have been proposed. In this paper, we adopt Planck's equation, which is defined only for v > b(> 0) and not a monotonic function of v, and prove the global existence of the smooth solution. The essential point of the proof is to obtain the bound of v of the form b < h(T) ? v(x, t) ? H(T) < ∞ for some constants h(T) and H(T).  相似文献   

8.
The solution of the initial boundary-value problem u?′ ? ?D2u? + u?Du? = f on (a, b) x(0, T), u?(a, t) = u?(b, t) = 0 and u?(x, 0) = 0 on (a, b), is shown to converge to the solution of the limiting equation as the viscosity tends to zero. Estimates on the rate of convergence are given.  相似文献   

9.
The inverse scattering method is used to determine the distribution limit as ? → 0 of the solution u(x, t, ?) of the initial value problem. Ut ? 6uux + ?2uxxx = 0, u(x, 0) = v(x), where v(x) is a positive bump which decays sufficiently fast as x x→±α. The case v(x) ? 0 has been solved by Peter D. Lax and C. David Levermore [8], [9], [10]. The computation of the distribution limit of u(x, t, ?) as ? → 0 is reduced to a quadratic maximization problem, which is then solved.  相似文献   

10.
This paper is concerned with the study of the oscillatory behavior of solutions of the nth-order nonlinear differential equation (a(t)x(n ? v)(t))(v) + q(t)f(x[g(t)]) = 0, where n is even and 1 ? v ? n ? 1. A systematic study is attempted which extends and correlates a number of existing results.  相似文献   

11.
An even-order three-point boundary value problem on time scales   总被引:1,自引:0,他引:1  
We study the even-order dynamic equation (−1)nx(Δ∇)n(t)=λh(t)f(x(t)), t∈[a,c] satisfying the boundary conditions x(Δ∇)i(a)=0 and x(Δ∇)i(c)=βx(Δ∇)i(b) for 0?i?n−1. The three points a,b,c are from a time scale , where 0<β(ba)<ca for b∈(a,c), β>0, f is a positive function, and h is a nonnegative function that is allowed to vanish on some subintervals of [a,c] of the time scale.  相似文献   

12.
In this paper, the fine triangle intersection problem for a pair of maximum kite packings is investigated. Let Fin(v) = {(s,t) : a pair of maximum kite packings of order v intersecting in s blocks and s+t triangles}. Let Adm(v) = {(s, t) : s + t ≤ bv , s,t are non-negative integers}, where b v = v(v 1)/8 . It is established that Fin(v) = Adm(v)\{(bv-1, 0), (bv-1,1)} for any integer v ≡ 0, 1 (mod 8) and v ≥ 8; Fin(v) = Adm(v) for any integer v ≡ 2, 3, 4, 5, 6, 7 (mod 8) and v ≥ 4.  相似文献   

13.
A t-(v, k, λ) covering design is a pair (X, B) where X is a v-set and B is a collection of k-sets in X, called blocks, such that every t element subset of X is contained in at least λ blocks of B. The covering number, Cλ(t, k, v), is the minimum number of blocks a t-(v, k, λ) covering design may have. The chromatic number of (X, B) is the smallest m for which there exists a map φ: XZm such that ∣φ((β)∣ ≥2 for all β ∈ B, where φ(β) = {φ(x): x ∈ β}. The system (X, B) is equitably m-chromatic if there is a proper coloring φ with minimal m for which the numbers ∣φ?1(c)∣ cZm differ from each other by at most 1. In this article we show that minimum, (i.e., ∣B∣ = C λ (t, k, v)) equitably 3-chromatic 3-(v, 4, 1) covering designs exist for v ≡ 0 (mod 6), v ≥ 18 for v ≥ 1, 13 (mod 36), v ≡ 13 and for all numbers v = n, n + 1, where n ≡ 4, 8, 10 (mod 12), n ≥ 16; and n = 6.5a 13b 17c ?4, a + b + c > 0, and n = 14, 62. We also show that minimum, equitably 2-chromatic 3-(v, 4, 1) covering designs exist for v ≡ 0, 5, 9 (mod 12), v ≥ 0, v = 2.5a 13b 17c + 1, a + b + c > 0, and v = 23. © 1993 John Wiley & Sons, Inc.  相似文献   

14.
As shown by an example, the integral function f : n , defined by f(x) = a b[B(x, t)]+ g(t) dt, may not be a strongly semismooth function, even if g(t) 1 and B is a quadratic polynomial with respect to t and infinitely many times smooth with respect to x. We show that f is a strongly semismooth function if g is continuous and B is affine with respect to t and strongly semismooth with respect to x, i.e., B(x, t) = u(x)t + v(x), where u and v are two strongly semismooth functions in n . We also show that f is not a piecewise smooth function if u and v are two linearly independent linear functions, g is continuous and g 0 in [a, b], and n 2. We apply the first result to the edge convex minimum norm network interpolation problem, which is a two-dimensional interpolation problem.  相似文献   

15.
In this paper, we study the “triply” degenerate problem: bt(v)−Δg(v)+divΦ(v)=f on Q:=(0,TΩ, b(v(0,⋅))=b(v0) on Ω and “g(v)=g(a) on some part of the boundary (0,T)×∂Ω,” in the case of continuous nonhomogeneous and nonstationary boundary data a. The functions b,g are assumed to be continuous, locally Lipschitz, nondecreasing and to verify the normalization condition b(0)=g(0)=0 and the range condition R(b+g)=R. Using monotonicity and penalization methods, we prove existence of a weak renormalized entropy solution in the spirit of [K. Ammar, J. Carrillo, P. Wittbold, Scalar conservation laws with general boundary condition and continuous flux function, J. Differential Equations 228 (2006) 111-139].  相似文献   

16.
In this paper, a Galerkin type algorithm is given for the numerical solution of L(x)=(r(t)x'(t))'-p(t)x(t)=g(t); x(a)=xa, x'(a)=x'a, where r (t)>f0, and Spline hat functions form the approximating basis. Using the related quadratic form, a two-step difference equation is derived for the numerical solutions. A discrete Gronwall type lemma is then used to show that the error at the node points satisfies ek=0(h2). If e(t) is the error function on a?t?b; it is also shown (in a variety of norms) that e(t)?Ch2 and e'(t)?C1h. Test case runs are also included. A (one step) Richardson or Rhomberg type procedure is used to show that eRk=0(h4). Thus our results are comparable to Runge-Kutta with half the function evaluations.  相似文献   

17.
The author discusses the asymptotic behavior of the solutions of the functional differential equation x′(t) = Ax(λt) + Bx(t), λ>0 (1) where x(t) is an n-dimensional column vector and A, B are n × n matrices with complex constant entries. He obtains the following results for the case 0 < λ < 1: (i) If B is diagonalizable with eigenvalues bi such that Re bi < 0 for all i, then there is a constant α such that every solution of (1) is O(tα) as t → ∞. (ii) If B is diagonalizable with eigenvalues bi such that 0 < Re b1 ? Re b2 ? ··· ? Re bn and λ times Re bn < Re b1, then every solution of (1) is O(ebnt) as t → ∞. For the case λ>1, he has the following results: (i) If B is diagonalizable with eigenvalues bi such that Re bi>0 for all i, then there is a constant α such that no solution x(t) of (1), except the identically zero solution, is 0(tα) as t → ∞. (ii) If B is diagonalizable with eigenvalues bi such that Re b1 ? Re b2 ? ··· ? Re bn < 0 and λ Re bn < Re b1, then no solution x(t) of (1), except the identically zero solution, is 0(eb1t) as t → ∞.  相似文献   

18.
In this paper, we prove the relation v(t)?u(t,x)?w(t), where u(t,x) is the solution of an impulsive parabolic equations under Neumann boundary condition ∂u(t,x)/∂ν=0, and v(t) and w(t) are solutions of two impulsive ordinary equations. We also apply these estimates to investigate the asymptotic behavior of a model in the population dynamics, and it is shown that there exists a unique solution of the model which converges to the periodic solution of an impulsive ordinary equation asymptotically.  相似文献   

19.
We are concerned with the discrete focal boundary value problem Δ3x(tk) = f(x(t)), x(a) = Δx(t2) = Δ2x(b + 1) = 0. Under various assumptions on f and the integers a, t2, and b we prove the existence of three positive solutions of this boundary value problem. To prove our results we use fixed point theorems concerning cones in a Banach space.  相似文献   

20.
This paper deals with the construction of continuous numerical solutions of mixed problems described by the time-dependent telegraph equation utt + c(t)ut + b(t)u = a(t)uxx, 0 < x < d, t > 0. Here a(t), b(t), and c(t) are positive functions with appropiate additional alternative hypotheses. First, using the separation of variables technique a theoretical series solution is obtained. Then, after truncation using one-step matrix methods and interpolating functions, a continuous numerical solution with a prefixed accuracy in a bounded subdomain is constructed.  相似文献   

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