共查询到20条相似文献,搜索用时 15 毫秒
1.
We consider an inverse boundary value problem for the heat equation ? t u = div (γ? x u) in (0, T) × Ω, u = f on (0, T) × ?Ω, u| t=0 = u 0, in a bounded domain Ω ? ? n , n ≥ 2, where the heat conductivity γ(t, x) is piecewise constant and the surface of discontinuity depends on time: γ(t, x) = k 2 (x ∈ D(t)), γ(t, x) = 1 (x ∈ Ω?D(t)). Fix a direction e* ∈ 𝕊 n?1 arbitrarily. Assuming that ?D(t) is strictly convex for 0 ≤ t ≤ T, we show that k and sup {e*·x; x ∈ D(t)} (0 ≤ t ≤ T), in particular D(t) itself, are determined from the Dirichlet-to-Neumann map : f → ?ν u(t, x)|(0, T)×?Ω. The knowledge of the initial data u 0 is not used in the proof. If we know min0≤t≤T (sup x∈D(t) x·e*), we have the same conclusion from the local Dirichlet-to-Neumann map. Numerical examples of stationary and moving circles inside the unit disk are shown. The results have applications to nondestructive testing. Consider a physical body consisting of homogeneous material with constant heat conductivity except for a moving inclusion with different conductivity. Then the location and shape of the inclusion can be monitored from temperature and heat flux measurements performed at the boundary of the body. Such a situation appears for example in blast furnaces used in ironmaking. 相似文献
2.
Müjdat Kaya 《Mediterranean Journal of Mathematics》2013,10(1):227-239
The problem of determining the initial value u(x, 0) = μ 0(x) in the parabolic equation u t = (k(x)u x (x, t)) x + F(x, t) from the final overdetermination μ T (x) = u(x, T) is formulated. It is proved that the Fréchet derivative of the cost functional ${{J(\mu_0) = \|\mu_T(x) - u(x, T)\|_0^2}}$ can be formulated via the solution of the adjoint parabolic problem. Lipschitz continuity of the gradient is proved. The existence of a quasisolution of the considered inverse problem is proved. A monotone iteration scheme is obtained based on the gradient method. 相似文献
3.
Aurora Llamas 《代数通讯》2013,41(5):1968-1981
We give conditions on the coefficients of a polynomial p(x) so that p(x + t) be log-concave or strictly log-concave. Several applications are given: if p(x) is a polynomial with nonnegative and nondecreasing coefficients, then p(x + t) is strictly log-concave for all t ≥ 1; for any polynomial p(x) with positive leading coefficient, there is t 0 ≥ 0 such that for any t ≥ t 0 it holds that the coefficients of p(x + t) are positive, strictly decreasing, and strictly log-concave; if p(x) is a log-concave polynomial with nonnegative coefficients and no internal zeros, then p(x + t) is strictly log-concave for all t > 0; Betti numbers of lexsegment monomial ideals are strictly log-concave. 相似文献
4.
Mladen Savov 《Probability Theory and Related Fields》2009,144(1-2):79-98
We wish to characterize when a Lévy process X t crosses boundaries b(t), in a two-sided sense, for small times t, where b(t) satisfies very mild conditions. An integral test is furnished for computing the value of sup t→0|X t |/b(t) = c. In some cases, we also specify a function b(t) in terms of the Lévy triplet, such that sup t→0 |X t |/b(t) = 1. 相似文献
5.
Hironobu Sasaki 《偏微分方程通讯》2015,40(11):1959-2004
We study scattering problems for the one-dimensional nonlinear Dirac equation (?t + α?x + iβ)Φ = λ|Φ|p?1Φ. We prove that if p > 3 (resp. p > 3 + 1/6), then the wave operator (resp. the scattering operator) is well-defined on some 0-neighborhood of a weighted Sobolev space. In order to prove these results, we use linear operators D(t)xD(?t) and t?x + x?t ? α/2, where {D(t)}t∈? is the free Dirac evolution group. For the reader's convenience, in an appendix we list and prove fundamental properties of D(t)xD(?t) and t?x + x?t ? α/2. 相似文献
6.
This paper is concerned with the study of the large-time behavior of the solutions u of a class of one-dimensional reaction–diffusion equations with monostable reaction terms f, including in particular the classical Fisher-KPP nonlinearities. The nonnegative initial data u 0(x) are chiefly assumed to be exponentially bounded as x tends to + ∞ and separated away from the unstable steady state 0 as x tends to ? ∞. On the one hand, we give some conditions on u 0 which guarantee that, for some λ > 0, the quantity c λ = λ +f′(0)/λ is the asymptotic spreading speed, in the sense that lim t→+∞ u(t, ct) = 1 (the stable steady state) if c < c λ and lim t→+∞ u(t, ct) = 0 if c > c λ. These conditions are fulfilled in particular when u 0(x) e λx is asymptotically periodic as x → + ∞. On the other hand, we also construct examples where the spreading speed is not uniquely determined. Namely, we show the existence of classes of initial conditions u 0 for which the ω-limit set of u(t, ct + x) as t tends to + ∞ is equal to the whole interval [0, 1] for all x ∈ ? and for all speeds c belonging to a given interval (γ1, γ2) with large enough γ1 < γ2 ≤ + ∞. 相似文献
7.
This article presents a semigroup approach to the mathematical analysis of the inverse parameter problems of identifying the unknown parameters p(t) and q in the linear parabolic equation ut(x, t) = uxx + qux(x, t) + p(t)u(x, t), with Dirichlet boundary conditions u(0, t) = ψ0, u(1, t) = ψ1. The main purpose of this paper is to investigate the distinguishability of the input-output mapping Φ[·]:P→H1,2[0,T], via semigroup theory. In this paper, it is shown that if the nullspace of the semigroup T(t) consists of only zero function, then the input-output mapping Φ[·] has the distinguishability property. It is also shown that the types of the boundary conditions and the region on which the problem is defined play an important role in the distinguishability property of the mapping. Moreover, under the light of the measured output data ux(0, t) = f(t) the unknown parameter p(t) at (x, t) = (0, 0) and the unknown coefficient q are determined via the input data. Furthermore, it is shown that measured output data f(t) can be determined analytically by an integral representation. Hence the input-output mapping Φ[·]:P→H1,2[0,T] is given explicitly interms of the semigroup. 相似文献
8.
We show that a holomorphic map germ ${f : (\mathbb{C}^n,0)\to(\mathbb{C}^{2n-1},0)}$ is finitely determined if and only if the double point scheme D(f) is a reduced curve. If n ≥ 3, we have that μ(D 2(f)) = 2μ(D 2(f)/S 2)+C(f)?1, where D 2(f) is the lifting of the double point curve in ${(\mathbb{C}^n\times \mathbb{C}^n,0)}$ , μ(X) denotes the Milnor number of X and C(f) is the number of cross-caps that appear in a stable deformation of f. Moreover, we consider an unfolding F(t, x) = (t, f t (x)) of f and show that if F is μ-constant, then it is excellent in the sense of Gaffney. Finally, we find a minimal set of invariants whose constancy in the family f t is equivalent to the Whitney equisingularity of F. We also give an example of an unfolding which is topologically trivial, but it is not Whitney equisingular. 相似文献
9.
In this article, we study the semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(x) in the linear parabolic equation ut(x,t)=(k(x)uxx(x,t)), with Dirichlet boundary conditions u(0,t)=ψ0, u(1,t)=ψ1. Main goal of this study is to investigate the distinguishability of the input-output mappings Φ[⋅]:K→C1[0,T], Ψ[⋅]:K→C1[0,T] via semigroup theory. In this paper, we show that if the null space of the semigroup T(t) consists of only zero function, then the input-output mappings Φ[⋅] and Ψ[⋅] have the distinguishability property. Moreover, the values k(0) and k(1) of the unknown diffusion coefficient k(x) at x=0 and x=1, respectively, can be determined explicitly by making use of measured output data (boundary observations) f(t):=k(0)ux(0,t) or/and h(t):=k(1)ux(1,t). In addition to these, the values k′(0) and k′(1) of the unknown coefficient k(x) at x=0 and x=1, respectively, are also determined via the input data. Furthermore, it is shown that measured output dataf(t) and h(t) can be determined analytically, by an integral representation. Hence the input-output mappings Φ[⋅]:K→C1[0,T], Ψ[⋅]:K→C1[0,T] are given explicitly in terms of the semigroup. Finally by using all these results, we construct the local representations of the unknown coefficient k(x) at the end points x=0 and x=1. 相似文献
10.
《Mathematical and Computer Modelling》1999,29(2):1-18
This paper is concerned with the construction of accurate continuous numerical solutions for partial self-adjoint differential systems of the type (P(t) ut)t = Q(t)uxx, u(0, t) = u(d, t) = 0, u(x, 0) = f(x), ut(x, 0) = g(x), 0 ≤ x ≤ d, t >- 0, where P(t), Q(t) are positive definite oRr×r-valued functions such that P′(t) and Q′(t) are simultaneously semidefinite (positive or negative) for all t ≥ 0. First, an exact theoretical series solution of the problem is obtained using a separation of variables technique. After appropriate truncation strategy and the numerical solution of certain matrix differential initial value problems the following question is addressed. Given T > 0 and an admissible error ϵ > 0 how to construct a continuous numerical solution whose error with respect to the exact series solution is smaller than ϵ, uniformly in D(T) = {(x, t); 0 ≤ x ≤ d, 0 ≤ t ≤ T}. Uniqueness of solutions is also studied. 相似文献
11.
Let K be a nonempty closed and convex subset of a real reflexive Banach space X that has weakly sequentially continuous duality mapping J. Let T: K → K be a multivalued non-expansive non-self-mapping satisfying the weakly inwardness condition as well as the condition T(y) = {y} for any y ∈ F(T) and such that for a contraction f: K → K and any t ∈ (0, 1), there exists x t ∈ K satisfying x t ∈ tf(x t ) + (1 ? t)Tx t . Then it is proved that {x t } ? K converges strongly to a fixed point of T, which is also a solution of certain variational inequality. Moreover, the convergence of two explicit methods are also investigated. 相似文献
12.
Changchun Liu 《Monatshefte für Mathematik》2012,165(2):237-249
In this paper, we study the initial-boundary value problem of porous medium equation ρ(x)u t = Δu m + V(x)h(t)u p in a cone D = (0, ∞) × Ω, where \({V(x)\,{\sim}\, |x|^\sigma, h(t)\,{\sim}\, t^s}\). Let ω 1 denote the smallest Dirichlet eigenvalue for the Laplace-Beltrami operator on Ω and let l denote the positive root of l 2 + (n ? 2)l = ω 1. We prove that if \({m < p \leq 1+(m-1)(1+s)+\frac{2(s+1)+\sigma}{n+l}}\), then the problem has no global nonnegative solutions for any nonnegative u 0 unless u 0 = 0; if \({p >1 +(m-1)(1+s)+\frac{2(s+1)+\sigma}{n+l}}\), then the problem has global solutions for some u 0 ≥ 0. 相似文献
13.
K. Tomoeda 《Applied mathematics and computation》1980,7(4):313-320
We are concerned with Friedrichs's scheme for an initial value problem ut(t, x) = A(t, x)ux(t, x), u(0, x) = u0(x), where u0(x) belongs to L∞, not to L2. We show that Friedrichs's scheme is stable in the maximum norm ·L∞, provided that the system is regularly hyperbolic and that the eigenvalues di(t, x) (i = 1,2,..., N) of the N XN matrix A(t, x) satisfy the conditions 1±λdi(t, x)?0 (i = 1,2,..., N), where λ is a mesh ratio. 相似文献
14.
《代数通讯》2013,41(5):2053-2065
Abstract We consider the group G of C-automorphisms of C(x, y) (resp. C[x, y]) generated by s, t such that t(x) = y, t(y) = x and s(x) = x, s(y) = ? y + u(x) where u ∈ C[x] is of degree k ≥ 2. Using Galois's theory, we show that the invariant field and the invariant algebra of G are equal to C. 相似文献
15.
Shu-Yu Hsu 《manuscripta mathematica》2013,140(3-4):441-460
Let n ≥ 3, 0 < m ≤ (n ? 2)/n, p > max(1, (1 ? m)n/2), and ${0 \le u_0 \in L_{loc}^p(\mathbb{R}^n)}$ satisfy ${{\rm lim \, inf}_{R\to\infty}R^{-n+\frac{2}{1-m}} \int_{|x|\le R}u_0\,dx = \infty}$ . We prove the existence of unique global classical solution of u t = Δu m , u > 0, in ${\mathbb{R}^n \times (0, \infty), u(x, 0) = u_0(x)}$ in ${\mathbb{R}^n}$ . If in addition 0 < m < (n ? 2)/n and u 0(x) ≈ A|x|?q as |x| → ∞ for some constants A > 0, q < n/p, we prove that there exist constants α, β, such that the function v(x, t) = t α u(t β x, t) converges uniformly on every compact subset of ${\mathbb{R}^n}$ to the self-similar solution ψ(x, 1) of the equation with ψ(x, 0) = A|x|?q as t → ∞. Note that when m = (n ? 2)/(n + 2), n ≥ 3, if ${g_{ij} = u^{\frac{4}{n+2}}\delta_{ij}}$ is a metric on ${\mathbb{R}^n}$ that evolves by the Yamabe flow ?g ij /?t = ?Rg ij with u(x, 0) = u 0(x) in ${\mathbb{R}^n}$ where R is the scalar curvature, then u(x, t) is a global solution of the above fast diffusion equation. 相似文献
16.
We prove that isentropic gas flow does not admit non-degenerate TVD fields on any invariant set ?(r 0, s 0) = {r 0 < r < s < s 0}, where r, s are Riemann coordinates. A TVD field refers to a scalar field whose spatial variation Var X (?(τ(t, X), u(t, X))) is non-increasing in time along entropic solutions. The result is established under the assumption that the Riemann problem defined by an overtaking shock-rarefaction interaction gives the asymptotic states in the exact solution. Little is known about global existence of large-variation solutions to hyperbolic systems of conservation laws u t + f(u) x = 0. In particular it is not known if isentropic gas flow admits a priori BV bounds which apply to all BV data. In the few cases where such results are available (scalar case, Temple class, systems satisfying Bakhvalov's condition, isothermal gas dynamics) there are TVD fields which play a key role for existence. Our results show that the same approach cannot work for isentropic flow. 相似文献
17.
I. Karatzas 《Journal of Optimization Theory and Applications》1980,31(3):431-440
The following stochastic control problem is considered: to minimize the discounted expected total cost $$J(x;u) = E\int_0^\infty {\exp ( - at)[\phi } (x_l ) + |u_l (x)|]dt,$$ subject todx t =u t (x)dt+dw t ,x 0=x, |u t |≤1, (w t ) a Wiener process, α>0. All bounded by unity, measurable, and nonanticipative functionalsu t (x) of the state processx t are admissible as controls. It is proved that the optimal law is of the form $$\begin{gathered} u_t^* (x) = - 1,x_t > b, \hfill \\ u_t^* (x) = 0,|x_t | \leqslant b, \hfill \\ u_t^* (x) = 1,x_t< - b, \hfill \\ \end{gathered}$$ for some switching pointb > 0, characterized in terms of the function ø(·) through a transcendental equation. 相似文献
18.
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
19.
Juan José Martinez 《代数通讯》2013,41(3):719-732
ABSTRACT Let R be a prime ring with a nonzero derivation d and let f(X 1,…,X t ) be a multilinear polynomial over C, the extended centroid of R. Suppose that b[d(f(x 1,…,x t )), f(x 1,…,x t )] n = 0 for all x i ∈ R, where 0 ≠ b ∈ R and n is a fixed positive integer. Then f(X 1,…,X t ) is centrally valued on R unless char R = 2 and dim C RC = 4. We prove a more generalized version by replacing R with a left ideal. 相似文献
20.
Recently, Philippe et al. (C.R. Acad. Sci. Paris. Ser. I 342, 269–274, 2006; Theory Probab. Appl., 2007, to appear) introduced a new class of time-varying fractionally integrated filters A(d)x t =∑ j=0 ∞ a j (t)x t?j , B(d)x t =∑ j=0 ∞ b j (t)x t?j depending on arbitrary given sequence d=(d t ,t∈?) of real numbers, such that A(d)?1=B(?d), B(d)?1=A(?d) and such that when d t ≡d is a constant, A(d)=B(d)=(1?L) d is the usual fractional differencing operator. Philippe et al. studied partial sums limits of (nonstationary) filtered white noise processes X t =B(d)ε t and Y t =A(d)ε t in the case when (1) d is almost periodic having a mean value $\bar{d}\in (0,1/2)$ , or (2) d admits limits d ±=lim? t→±∞ d t ∈(0,1/2) at t=±∞. The present paper extends the above mentioned results of Philippe et al. into two directions. Firstly, we consider the class of time-varying processes with infinite variance, by assuming that ε t ,t∈? are iid rv’s in the domain of attraction of α-stable law (1<α≤2). Secondly, we combine the classes (1) and (2) of sequences d=(d t ,t∈?) into a single class of sequences d=(d t ,t∈?) admitting possibly different Cesaro limits $\bar{d}_{\pm}\in(0,1-(1/\alpha))$ at ±∞. We show that partial sums of X t and Y t converge to some α-stable self-similar processes depending on the asymptotic parameters $\bar{d}_{\pm}$ and having asymptotically stationary or asymptotically vanishing increments. 相似文献