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Multiplicative asymptotics of solutions of the first boundary value problem on a half-axis for a parabolic equation with a small parameter. (English) Zbl 1077.35011

Let us consider the boundary value problem \[ h \frac{\partial u}{\partial t} = h L_h u, \qquad t>0, \quad x > x_0, \]
\[ u | _{t=0} = 0, \qquad u | _{x=x_0} = \mu (t), \] where \[ L_h = \frac12 h a(x) \frac{\partial^2}{\partial x^2} - b(x) \frac {\partial}{\partial x} \] with \(a(x) \geq \delta >0\), the coefficients \(a(x), \, b(x)\) supposed to be infinitely smooth. Sections 1–4 are devoted to the study of asymptotics of the solution as \(h \to 0+\). It is shown that this asymptotics depends on the sign of \(b(x_0)\), so the three cases \(b(x_0) >0\), \(b(x_0) <0\), \(b(x_0) = 0\) are considered. In section 5 the tunnel canonical operator [V. P. Maslov, Tr. Mat. Inst. Steklova 163, 150–180 (1984; Zbl 0567.35083)] is constructed.
Many references concerning the earlier papers in the topic are given.

MSC:

35B25 Singular perturbations in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
60J60 Diffusion processes

Citations:

Zbl 0567.35083
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References:

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