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Consider in a real Hilbert space H the Cauchy problem (P0): u′(t)+Au(t)+Bu(t)=f(t), 0≤t≤T; u(0)=u0, where −A is the infinitesimal generator of a C0-semigroup of contractions, B is a nonlinear monotone operator, and f is a given H-valued function. Inspired by the excellent book on singular perturbations by J.L. Lions, we associate with problem (P0) the following regularization (Pε): −εu″(t)+u′(t)+Au(t)+Bu(t)=f(t), 0≤t≤T; u(0)=u0, u′(T)=uT, where ε>0 is a small parameter. We investigate existence, uniqueness and higher regularity for problem (Pε). Then we establish asymptotic expansions of order zero, and of order one, for the solution of (Pε). Problem (Pε) turns out to be regularly perturbed of order zero, and singularly perturbed of order one, with respect to the norm of C([0,T];H). However, the boundary layer of order one is not visible through the norm of L2(0,T;H). 相似文献
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In this paper, we present a new method for solving singular two-point boundary value problem for certain ordinary differential equation having singular coefficients. Its exact solution is represented in the form of series in reproducing kernel space. In the mean time, the n -term approximation un(x) to the exact solution u(x) is obtained and is proved to converge to the exact solution. Some numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method are compared with the exact solution of each example and are found to be in good agreement with each other. 相似文献
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For systems of retarded functional differential equations with unbounded delay and with finite memory sufficient and necessary conditions of existence of positive solutions on an interval of the form [t0,∞) are derived. A general criterion is given together with corresponding applications (including a linear case, too). Examples are inserted to illustrate the results. 相似文献
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The existence of solutions of degenerate quasilinear pseudoparabolic equations, where the term ∂tu is replace by ∂tb(u), with memory terms and quasilinear variational inequalities is shown. The existence of solutions of equations is proved under the assumption that the nonlinear function b is monotone and a gradient of a convex, continuously differentiable function. The uniqueness is proved for Lipschitz-continuous elliptic parts. The existence of solutions of quasilinear variational inequalities is proved under stronger assumptions, namely, the nonlinear function defining the elliptic part is assumed to be a gradient and the function b to be Lipschitz continuous. 相似文献
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In this paper, we employ the bifurcation theory of planar dynamical systems to study the smooth and nonsmooth traveling wave solutions of the generalized Degasperis-Procesi equation
The parameter condition under which peakons, compactons and periodic cusp wave solutions exist is given. The numerical simulation results show the consistence with the theoretical analysis at the same time. 相似文献
ut-uxxt+4umux=3uxuxx+uuxxx.
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The finite volume method based on stabilized finite element for the stationary Navier–Stokes problem
A finite volume method based on stabilized finite element for the two-dimensional stationary Navier–Stokes equations is investigated in this work. A macroelement condition is introduced for constructing the local stabilized formulation for the problem. We obtain the well-posedness of the FVM based on stabilized finite element for the stationary Navier–Stokes equations. Moreover, for quadrilateral and triangular partition, the optimal H1 error estimate of the finite volume solution uh and L2 error estimate for ph are introduced. Finally, we provide a numerical example to confirm the efficiency of the FVM. 相似文献
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We consider semilinear second-order ordinary differential equations, mainly autonomous, in the form −u″=f(u)+λu, supplied with different sets of standard boundary conditions. Here λ is a real constant or it plays the role of a spectral parameter. Mainly, we study problems in the interval (0,1). It is shown that in this case each problem that we deal with has an infinite sequence of solutions or eigenfunctions. Our aim in the present article is to review recent results on basis properties of sequences of these solutions or eigenfunctions. In a number of cases, it is proved that such a system is a basis in L2 (in addition, a Riesz or Bari basis). In addition, we briefly consider a problem for the half-line (0,∞). In this case, the spectrum of the problem fills a half-line and an analog of the expansions into the Fourier integral is obtained. The proofs are mainly based on the Bari theorem and, in addition, on our general result on sufficient conditions for a sequence of functions to be a Riesz basis in L2. 相似文献
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We study viscous shock waves that are associated with a simple mode (λ,r) of a system ut+f(u)x=uxx of conservation laws and that connect states on either side of an ‘inflection’ hypersurface Σ in state space at whose points r⋅∇λ=0 and (r⋅∇)2λ≠0. Such loss of genuine nonlinearity, the simplest example of which is the cubic scalar conservation law ut+(u3)x=uxx, occurs in many physical systems. We show that such shock waves are spectrally stable if their amplitude is sufficiently small. The proof is based on a direct analysis of the eigenvalue problem by means of geometric singular perturbation theory. Well-chosen rescalings are crucial for resolving degeneracies. By results of Zumbrun the spectral stability shown here implies nonlinear stability of these shock waves. 相似文献
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This note deals with the strongly damped nonlinear wave equation with Dirichlet boundary conditions, where both the nonlinearities f and g exhibit a critical growth, while h is a time-independent forcing term. The existence of an exponential attractor of optimal regularity is proven. As a corollary, a regular global attractor of finite fractal dimension is obtained. 相似文献
utt−Δut−Δu+f(ut)+g(u)=h
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Following Coclite, Holden and Karlsen [G.M. Coclite, H. Holden and K.H. Karlsen, Well-posedness for a parabolic-elliptic system, Discrete Contin. Dyn. Syst. 13 (3) (2005) 659–682] and Tian and Fan [Lixin Tian, Jinling Fan, The attractor on viscosity Degasperis-Procesi equation, Nonlinear Analysis: Real World Applications, 2007], we study the dynamical behaviors of the parabolic–elliptic system
and ut+(f(t,x,u))x+g(t,x,u)+Px−εuxx=0
−Pxx+P=h(t,x,u,ux)+k(t,x,u)
u|t=0=u0.