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1.
We consider a parabolic semilinear problem with rapidly oscillating coefficients in a domain Ωε that is ε-periodically perforated by small holes of size O\mathcal {O}(ε). The holes are divided into two ε-periodical sets depending on the boundary interaction at their surfaces, and two different nonlinear Robin boundary conditions σε(u ε) + εκ m (u ε) = εg (m) ε, m = 1, 2, are imposed on the boundaries of holes. We study the asymptotics as ε → 0 and establish a convergence theorem without using extension operators. An asymptotic approximation of the solution and the corresponding error estimate are also obtained. Bibliography: 60 titles. Illustrations: 1 figure.  相似文献   

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We study non-negative solutions of the porous medium equationwith a source and a nonlinear flux boundary condition, ut =(um)xx + up in (0, ), x (0, T); – (um)x (0, t) = uq (0,t) for t (0, T); u (x, 0) = u0 (x) in (0, ), where m > 1,p, q > 0 are parameters. For every fixed m we prove thatthere are two critical curves in the (p, q-plane: (i) the criticalexistence curve, separating the region where every solutionis global from the region where there exist blowing-up solutions,and (ii) the Fujita curve, separating a region of parametersin which all solutions blow up from a region where both globalin time solutions and blowing-up solutions exist. In the caseof blow up we find the blow-up rates, the blow-up sets and theblow-up profiles, showing that there is a phenomenon of asymptoticsimplification. If 2q < p + m the asymptotics are governedby the source term. On the other hand, if 2q > p + m theevolution close to blow up is ruled by the boundary flux. If2q = p + m both terms are of the same order.  相似文献   

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The system of equations
with the boundary conditions
is considered. The solvability of this boundary-value problem and properties of the family of solutions are studied under the condition that the diffusion coefficient is negative. Bibliography: 5 titles. Translated fromProblemy Matematicheskogo Analiza, No. 17, 1997, pp. 72–82.  相似文献   

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The asymptotic behavior of the solution of a boundary-value problem for the equation utxx+ ux =f when the time tends to infinity is investigated. It is proved that the time mean of the solution tends to a stationary solution everywhere except in a boundary region at the left end of the interval.Translated from Matematicheskie Zametki, Vol. 8, No. 3, pp. 273–284, September, 1970.  相似文献   

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We construct and justify the asymptotics (as ε → +0) of a solution of the mixed boundary-value problem for the Poisson equation in the domain obtained by joining two sets Ω+ and Ω- by a large number of thin (of width O (ε)) curvilinear strips (a hub and a rim with a large number of spokes). As a resulting limit problem describing the principal terms of exterior expansions (in Ω± and in the set ω occupied by the strips) we take the problem of conjugating the partial differential equations and an ordinary differential equation depending on a parameter. Bibliography: 16 titles; Illustrations: 1 figure. Translated fromProblemy Matematicheskogo Analiza, No. 14, 1995, pp. 63–90.  相似文献   

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Asymptotic expansions are given for the eigenvalues λn and eigenfunctions un of the following singular Sturm-Liouville problem with indefinite weight: $$\begin{gathered} - ((1 - x^2 )u'(x))' = \lambda xu(x) on ( - 1,1), \hfill \\ lim_{| x | \to 1} u(x) finite \hfill \\ \end{gathered} $$ This eigenvalue problem arises if one separates variables in a partial differential equation which describes electron scattering in a one-dimensional slab configuration. Asymptotic expansions of the normalization constants of the eigenfunctions are also given. The constants in these asymptotic expansions involve complete elliptic integrals. The asymptotic results are compared with the results of numerical calculations.  相似文献   

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A. M. Razmadze Mathematics Institute, Georgian SSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 83, No. 3, pp. 348–357, June, 1990.  相似文献   

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A finite-element algorithm is developed for the problem of headless steady nonlinear seepage (boundary-value problem for a nonlinear elliptic equation in a domain with an unknown boundary) in a multicomponent medium with a piecewise-linear boundary. Numerical solution results are reported for a number of problems. The effects of the form of the nonlinearity on the characteristics of the seepage process are considered.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 61, pp. 83–90, 1987.  相似文献   

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We consider a boundary value problem in a model domain periodically perforated along the boundary. We assume that the homogeneous Neumann condition is posed on the external boundary and the homogeneous Dirichlet condition is posed on the boundary of the cavities. A limit (homogenized) problem is obtained. We prove the convergence of the solutions, eigenvalues, and eigenfunctions of the original problem to the solutions, eigenvalues, and eigenfunctions, respectively, of the limit problem.  相似文献   

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We consider a boundary value problem for the Laplace operator in a model domain periodically perforated along the boundary. We assume that the homogeneous Neumann condition is posed on the exterior boundary and the homogeneous Dirichlet condition is posed on the boundary of the cavities. We construct and justify the asymptotic expansions of eigenelements of the boundary value problem.  相似文献   

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In this paper, the asymptotic behavior of solutions u ε of the Poisson equation in the ε-periodically perforated domain Ωε ? $ {{\mathbb{R}}^n} $ , n ≥ 3, with the third nonlinear boundary condition of the form ? ν u ε + εσ(x, u ε) = ε g(x) on a boundary of cavities, is studied. It is supposed that the diameter of cavities has the order εα with α > 1 and any γ. Here, all types of asymptotic behavior of solutions u ε , corresponding to different relations between parameters α and γ, are studied.  相似文献   

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