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Asymptotic properties of solutions of semilinear second-order elliptic equations in cylindrical domains. (English. Russian original) Zbl 1173.35367

J. Math. Sci., New York 135, No. 1, 2666-2674 (2006); translation from Tr. Semin. Im. I. G. Petrovskogo 25, 98-111 (2005).
Summary: The equations under consideration have the following structure: \[ \frac {\partial^2u}{\partial x^2_n}+\sum^{n-1}_{t,j=1}\frac {\partial}{\partial x_i} \left(a_{ij}(x)\frac{\partial u}{\partial x_j} \right)+\sum^{n-1}_{i=1}a_i(x) \frac{\partial u}{\partial x_i}-f(u,x_n)=0, \] where \(0<x_n<\infty\), \((x_1,\dots, x_{n-1})\in\Omega\), \(\Omega\) is a bounded Lipschitz domain, \(f(0,x_n)\equiv 0\), \(\frac {\partial f}{\partial u}(0,x_n)\equiv 0\), \(f\) is a function that is continuous and monotonic with respect to \(u\), and all coefficients are bounded measurable functions. Asymptotic formulas are established for solutions of such equations as \(x_n\to +\infty\); the solutions are assumed to satisfy zero Dirichlet or Neumann boundary conditions on \(\partial\Omega\). Previously, such formulas were obtained in the case of \(a_{ij}\), \(a_i\) depending only on \((x_1,\dots,x_{n-1})\).

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35J60 Nonlinear elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
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References:

[1] Yu. V. Egorov, V. A. Kondratiev, and O. A. Oleinik, ”Astmptotic behavior of solutions of nonlinear elliptic and parabolic systems in cylindrical domains,” Mat. Sb., 189, No. 3, 45–68 (1998).
[2] V. A. Kondratiev and O. A. Oleinik, ”Boundary value problems for nonlinear elliptic equations in cylindrical domains,” J. Partial Differential Equations, 6, No. 1, 10–16 (1993). · Zbl 0784.35033
[3] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, New York (1977). · Zbl 0361.35003
[4] O. A. Oleinik, Some Asymptotic Problems of the Theory of Partial Differential Equations, Lezioni Lincei. Acad. Naz. Lincei, Cambridge Univ. Press, Cambridge (1996). · Zbl 1075.35500
[5] O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Equations of Elliptic Type [in Russian], Nauka, Moscow (1973).
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