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We prove an existence and uniqueness theorem for the solution of a nonclassical boundary value problem of Egorov-Kondrat’ev type for a pseudodifferential equation of variable order.  相似文献   

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A quasilinear boundary-value problem is considered for a degenerate parabolic pseudodifferential equation in a Sobolev space. By the Laplace transformation, this problem is reduced to a boundary-value problem for a degenerate elliptic equation with a parameter. The latter problem is studied with the help of Vishik–Grushin methods and the Rothe theorem. The approach proposed differs from those previously used in relation to similar problems.  相似文献   

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For an elliptic 2lth-order equation with constant (and only leading) real coefficients, we consider the boundary value problem in which the (k j ? 1)st normal derivatives, j = 1,..., l, are specified, where 1 ≤ k 1 < ... < k l . If k j = j, then it becomes the Dirichlet problem; and if k j = j + 1, then it becomes the Neumann problem. We obtain a sufficient condition for this problem to be Fredholm and present a formula for the index of the problem.  相似文献   

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This paper discusses the existence and the blowing-up behaviour of the solution for an initial boundary value problem which arises from the ignition of mixtures of gases. It is shown under the Dirichlet or the third type of boundary condition that for certain a class of initial functions local solutions exist and grow unbounded in finite time, while for another class of initial functions there exist global solutions which converge to a steady state solution of the problem. These results lead to an interesting bifurcation phenomenon on the existence, stability and blowing-up property of the solution in terms of either the strength of the nonlinear function or the size of the diffusion region. Estimates for the stability and instability regions as well as bounds for the finite escape time are explicitly given.  相似文献   

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We study a nonlocal boundary-value problem for a parabolic equation in a two-dimensional domain, establish ana priori estimate in the energy norm, prove the existence and uniqueness of a generalized solution from the classW 2 1,0 (Q T ), and construct a difference scheme for the second-order approximation.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 6, pp. 790–800, June, 1995.  相似文献   

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Statistical estimates of the solutions of boundary value problems for parabolic equations with constant coefficients are constructed on paths of random walks. The phase space of these walks is a region in which the problem is solved or the boundary of the region. The simulation of the walks employs the explicit form of the fundamental solution; therefore, these algorithms cannot be directly applied to equations with variable coefficients. In the present work, unbiased and low-bias estimates of the solution of the boundary value problem for the heat equation with a variable coefficient multiplying the unknown function are constructed on the paths of a Markov chain of random walk on balloids. For studying the properties of the Markov chains and properties of the statistical estimates, the author extends von Neumann-Ulam scheme, known in the theory of Monte Carlo methods, to equations with a substochastic kernel. The algorithm is based on a new integral representation of the solution to the boundary value problem.  相似文献   

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In this paper we consider the mixed problem for the equation u tt  + A 1 uA 2(u t ) + g(u t ) = f(x, t) in unbounded domain, where A 1 is a linear elliptic operator of the fourth order and A 2 is a nonlinear elliptic operator of the second order. Under natural assumptions on the equation coefficients and f we proof existence of a solution. This result contains, as a special case, some of known before theorems of existence. Essentially, in difference up to previous results we prove theorems of existence without the additional assumption on behavior of solution at infinity.   相似文献   

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We study the uniqueness of the solution of a boundary value problem for the biharmonic equation in unbounded domains under the assumption that the generalized solution of this problem has a bounded Dirichlet integral with weight |x|a. Depending on the value of the parameter a, we prove uniqueness theorems or present exact formulas for the dimension of the solution space of this problem in the exterior of a compact set and in a half-space.  相似文献   

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In this paper we consider the discrete anisotropic boundary value problem using critical point theory. Firstly we apply the direct method of the calculus of variations and the mountain pass technique in order to reach the existence of at least one nontrivial solution. Secondly we derive some version of a discrete three critical point theorem which we apply in order to get the existence of at least two nontrivial solutions.  相似文献   

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