共查询到20条相似文献,搜索用时 62 毫秒
1.
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
2.
This paper presents a semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(ux) in the inhomogenenous quasi‐linear parabolic equation ut(x, t)=(k(ux)ux(x, t))x +F(u), with the Dirichlet boundary conditions u(0, t)=ψ0, u(1, t)=ψ1 and source function F(u). The main purpose of this paper is to investigate the distinguishability of the input–output mappings Φ[·]:??→C1[0, T], Ψ[·]:??→C1[0, T] via semigroup theory. Copyright © 2009 John Wiley & Sons, Ltd. 相似文献
3.
This article presents a semigroup approach to the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(ux) in the quasi‐linear parabolic equation ut(x, t)=(k(ux)ux(x, t))x+F(x, t), with Dirichlet boundary conditions u(0, t)=ψ0, u(1, t)=ψ1 and source function F(x, t). The main purpose of this paper is to investigate the distinguishability of the input–output mappings Φ[·]: ?? → C1[0, T], Ψ[·]: ?? → C1[0, T] via semigroup theory. Copyright © 2008 John Wiley & Sons, Ltd. 相似文献
4.
Ebru Ozbilge 《Mathematical Methods in the Applied Sciences》2008,31(11):1333-1344
In this article, a semigroup approach is presented for the mathematical analysis of the inverse coefficient problems of identifying the unknown diffusion coefficient k(u(x, t)) in the quasi‐linear parabolic equation ut(x, t)=(k(u(x, t))ux(x, t))x, with Dirichlet boundary conditions ux(0, t)=ψ0, u(1, t)=ψ1. The main purpose of this work is to analyze the distinguishability of the input–output mappings Φ[·] : ??→C1[0, T], Ψ[·] : ??→C1[0, T] using semigroup theory. In this article, it is shown that if the null space of semigroups T(t) and S(t) consists of only a zero function, then the input–output mappings Φ[·] and Ψ[·] have the distinguishability property. Copyright © 2008 John Wiley & Sons, Ltd. 相似文献
5.
This article presents a semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(u(x,t)) in the quasi‐linear parabolic equation ut(x,t)=(k(u(x,t))ux(x,t))x, 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 mappings Φ[?]:?? →C1[0,T], Ψ[?]:??→C1[0,T] via semigroup theory. In this paper, it is shown 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. 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 these mappings. Moreover, under the light of measured output data (boundary observations) f(t):=k(u(0,t))ux(0,t) or/and h(t):=k(u(1,t))ux(1,t), the values k(ψ0) and k(ψ1) of the unknown diffusion coefficient k(u(x,t)) at (x,t)=(0,0) and (x,t)=(1,0), respectively, can be determined explicitly. In addition to these, the values ku(ψ0) and ku(ψ1) of the unknown coefficient k(u(x,t)) at (x,t)=(0,0) and (x,t)=(1,0), respectively, are also determined via the input data. Furthermore, it is shown that measured output data f(t) and h(t) can be determined analytically by an integral representation. Hence the input–output mappings Φ[?]:??→ C1[0,T], Ψ[?]:??→C1[0,T] are given explicitly in terms of the semigroup. Copyright © 2007 John Wiley & Sons, Ltd. 相似文献
6.
We study blow-up of radially symmetric solutions of the nonlinear heat equation ut=Δu+|u|p−1u either on RN or on a finite ball under the Dirichlet boundary conditions. We assume and that the initial data is bounded, possibly sign-changing. Our first goal is to establish various characterizations of type I and type II blow-ups. Among many other things we show that the following conditions are equivalent: (a) the blow-up is of type II; (b) the rescaled solution w(y,s) converges to either φ∗(y) or −φ∗(y) as s→∞, where φ∗ denotes the singular stationary solution; (c) u(x,T)/φ∗(x) tends to ±1 as x→0, where T is the blow-up time.Our second goal is to study continuation beyond blow-up. Among other things we show that if a blow-up is of type I and incomplete, then its limit L1 continuation becomes smooth immediately after blow-up, and that type I blow-up implies “type I regularization,” that is, (t−T)1/(p−1)‖u(⋅,t)L∞‖ is bounded as t↘T. We also give various criteria for complete and incomplete blow-ups. 相似文献
7.
Alemdar Hasanov Ali Demir Arzu Erdem 《Journal of Mathematical Analysis and Applications》2007,335(2):1434-1451
This article presents a mathematical analysis of input-output mappings in inverse coefficient and source problems for the linear parabolic equation ut=(kx(x)ux)+F(x,t), (x,t)∈ΩT:=(0,1)×(0,T]. The most experimentally feasible boundary measured data, the Neumann output (flux) data f(t):=−k(0)ux(0,t), is used at the boundary x=0. For each inverse problems structure of the input-output mappings is analyzed based on maximum principle and corresponding adjoint problems. Derived integral identities between the solutions of forward problems and corresponding adjoint problems, permit one to prove the monotonicity and invertibility of the input-output mappings. Some numerical applications are presented. 相似文献
8.
9.
Let {X(t):t∈[0,∞)} be a centered stationary Gaussian process. We study the exact asymptotics of P(sups∈[0,T]X(s)>u), as u→∞, where T is an independent of {X(t)} nonnegative random variable. It appears that the heaviness of T impacts the form of the asymptotics, leading to three scenarios: the case of integrable T, the case of T having regularly varying tail distribution with parameter λ∈(0,1) and the case of T having slowly varying tail distribution. 相似文献
10.
Fred B Weissler 《Journal of Differential Equations》1984,55(2):204-224
The initial value problem on [?R, R] is considered: ut(t, x) = uxx(t, x) + u(t, x)γu(t, ±R) = 0u(0, x) = ?(x), where ? ? 0 and γ is a fixed large number. It is known that for some initial values ? the solution u(t, x) exists only up to some finite time T, and that ∥u(t, ·)∥∞ → ∞ as t → T. For the specific initial value ? = kψ, where ψ ? 0, ψxx + ψγ = 0, ψ(±R) = 0, k is sufficiently large, it is shown that if x ≠ 0, then limt → Tu(t, x) and limt → Tux(t, x) exist and are finite. In other words, blow-up occurs only at the point x = 0. 相似文献
11.
We obtain results of existence and multiplicity of solutions for the second-order equation x″+q(t)g(x)=0, with x(t) defined for all t∈]0,1[ and such that x(t)→+∞ as t→0+ and t→1−. We assume g having superlinear growth at infinity and q(t) possibly changing sign on [0,1]. 相似文献
12.
In this paper, we consider the problem of finding u = u(x, y, t) and p = p(t) which satisfy ut = uxx + uyy + p(t)u + ? in R × [0, T], u(x, y, 0) = f(x, y), (x, y) ∈ R = [0, 1] × [0, 1], u is known on the boundary of R and u(x∗, y∗, t) = E(t), 0 < t ? T, where E(t) is known and (x∗, y∗) is a given point of R. Through a function transformation, the nonlinear two-dimensional diffusion problem is transformed into a linear problem, and a backward Euler scheme is constructed. It is proved by the maximum principle that the scheme is uniquely solvable, unconditionally stable and convergent in L∞ norm. The convergence orders of u and p are of O(τ + h2). The impact of initial data errors on the numerical solution is also considered. Numerical experiments are presented to illustrate the validity of the theoretical results. 相似文献
13.
We introduce a notion of entropy solution for a scalar conservation law on a bounded domain with nonhomogeneous boundary condition: ut+divΦ(u)=f on Q=(0,T)×Ω, u(0,⋅)=u0 on Ω and “u=a on some part of the boundary (0,T)×∂Ω.” Existence and uniqueness of the entropy solution is established for any Φ∈C(R;RN), u0∈L∞(Ω), f∈L∞(Q), a∈L∞((0,T)×∂Ω). In the L1-setting, a corresponding result is proved for the more general notion of renormalised entropy solution. 相似文献
14.
In this paper we study the maximal regularity property for non-autonomous evolution equations t∂u(t)+A(t)u(t)=f(t), u(0)=0. If the equation is considered on a Hilbert space H and the operators A(t) are defined by sesquilinear forms a(t,⋅,⋅) we prove the maximal regularity under a Hölder continuity assumption of t→a(t,⋅,⋅). In the non-Hilbert space situation we focus on Schrödinger type operators A(t):=−Δ+m(t,⋅) and prove Lp−Lq estimates for a wide class of time and space dependent potentials m. 相似文献
15.
Marta Sanz-Solé 《Journal of Functional Analysis》2008,255(1):255-281
We consider a stochastic wave equation in space dimension three driven by a noise white in time and with an absolutely continuous correlation measure given by the product of a smooth function and a Riesz kernel. Let pt,x(y) be the density of the law of the solution u(t,x) of such an equation at points (t,x)∈]0,T]×R3. We prove that the mapping (t,x)?pt,x(y) owns the same regularity as the sample paths of the process {u(t,x),(t,x)∈]0,T]×R3} established in [R.C. Dalang, M. Sanz-Solé, Hölder-Sobolev regularity of the solution to the stochastic wave equation in dimension three, Mem. Amer. Math. Soc., in press]. The proof relies on Malliavin calculus and more explicitly, the integration by parts formula of [S. Watanabe, Lectures on Stochastic Differential Equations and Malliavin Calculus, Tata Inst. Fund. Res./Springer-Verlag, Bombay, 1984] and estimates derived from it. 相似文献
16.
Existence of positive solution for some class of nonlinear fractional differential equations 总被引:1,自引:0,他引:1
Shuqin Zhang 《Journal of Mathematical Analysis and Applications》2003,278(1):136-148
In this paper, we investigate the multiple and infinitely solvability of positive solutions for nonlinear fractional differential equation Du(t)=tνf(u), 0<t<1, where D=t−βδDβγ−δ,δ, β>0, γ?0, 0<δ<1, ν>−β(γ+1). Our main work is to deal with limit case of f(s)/s as s→0 or s→∞ and Φ(s)/s, Ψ(s)/s as s→0 or s→∞, where Φ(s), Ψ(s) are functions connected with function f. In J. Math. Appl. 252 (2000) 804-812, we consider the existence of a positive solution for the particular case of Eq. (1.1), i.e., the Riemann-Liouville type (β=1, γ=0) nonlinear fractional differential equation, using the super-lower solutions method. Here, we devote to the existence of positive solution and multi-positive solutions for Eq. (1.1) by means of the fixed point theorems for the cone. 相似文献
17.
Svatoslav Staněk 《Nonlinear Analysis: Theory, Methods & Applications》2009,71(12):e153
The paper discusses the existence of positive and dead core solutions of the singular differential equation (?(u″))′=λf(t,u,u′,u″) satisfying the boundary conditions u(0)=A, u(T)=A, min{u(t):t∈[0,T]}=0. Here λ is a nonnegative parameter, A is a positive constant and the Carathéodory function f(t,x,y,z) is singular at the value 0 of its space variable y. 相似文献
18.
Goro Akagi 《Journal of Differential Equations》2007,241(2):359-385
The existence of local (in time) solutions of the initial-boundary value problem for the following degenerate parabolic equation: ut(x,t)−Δpu(x,t)−|u|q−2u(x,t)=f(x,t), (x,t)∈Ω×(0,T), where 2?p<q<+∞, Ω is a bounded domain in RN, is given and Δp denotes the so-called p-Laplacian defined by Δpu:=∇⋅(|∇u|p−2∇u), with initial data u0∈Lr(Ω) is proved under r>N(q−p)/p without imposing any smallness on u0 and f. To this end, the above problem is reduced into the Cauchy problem for an evolution equation governed by the difference of two subdifferential operators in a reflexive Banach space, and the theory of subdifferential operators and potential well method are employed to establish energy estimates. Particularly, Lr-estimates of solutions play a crucial role to construct a time-local solution and reveal the dependence of the time interval [0,T0] in which the problem admits a solution. More precisely, T0 depends only on Lr|u0| and f. 相似文献
19.
Fahuai YiYuqing Liu 《Journal of Differential Equations》2002,183(1):189-207
Both one-dimensional two-phase Stefan problem with the thermodynamic equilibrium condition u(R(t),t)=0 and with the kinetic rule uε(Rε(t),t)=εRε′(t) at the moving boundary are considered. We prove, when ε approaches zero, Rε(t) converges to R(t) in C1+δ/2[0,T] for any finite T>0, 0<δ<1. 相似文献
20.
《Comptes Rendus de l'Academie des Sciences Series IIA Earth and Planetary Science》1999,328(4):291-296
The purpose of this paper is to prove the existence of a solution for a nonlinear parabolic equation in the form ut - div(a(t, x, u, Du)) = H(t, x, u, Du) - div(g(t, x)) in QT =]0,T[×Ω, Ω ⊂ RN, with an initial condition u(0) = u0, where u0 is not bounded, |H(t,x, u, ξ)⩽ β|ξ|p + f(t,x) + βeλ1|u|f, |g|p/(p-1) ∈ Lr(QT) for some r = r{N) ⩾ 1, and - div(a(t,x,u, Du)) is the usual Leray-Lions operator. 相似文献