Abstract
In this work we consider the first boundary value problem for a parabolic equation of second order with a small parameter on a half-axis (i.e., we consider the one-dimensional case). We take the zero initial condition. We construct the global (that is, the caustic points are taken into account) asymptotics of a solution for the boundary value problem. The asymptotic solution of this problem has a different structure depending on the sign of the coefficient (the drift coefficient) at the derivative of first order at a boundary point. The constructed asymptotic solutions are justified.
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Frolovitchev, S. Multiplicative asymptotics of solutions of the first boundary value problem on a half-axis for a parabolic equation with a small parameter. Math. Ann. 332, 533–563 (2005). https://doi.org/10.1007/s00208-005-0636-4
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DOI: https://doi.org/10.1007/s00208-005-0636-4