The fundamental solution to 1D degenerate diffusion equation with one-sided boundary

Justification of the exact asymptotics of the fundamental solution for a degenerate parabolic equation with a small parameter

“…Recently, many studies have been aimed at studying random diffusion, which correspond to non-strictly parabolic equations with various degeneracies of the main symbol, see [1][2][3] and references therein. Such problems are usually solved in two ways.…”

“…Later, research began on non-strictly parabolic equations. Here [1][2][3] an approach close to the ideas of Molchanov [8] is used. Recently, much attention has been drawn to linear degenerate parabolic equations with a small parameter of the form…”

“…In our work we consider the case 𝛼 = 2, as well as some special cases of Equation ( 1) with the conditions given above. Unlike [1][2][3], we apply the technique of the WKB method, construct a fundamental solution in the form of a series in powers of the small parameter 𝜀, and prove the uniform convergence of this series. It should be noted that, unlike [1][2][3]8], we do not bound ourselves by the construction of the first term of the mentioned series.…”

“…It is interesting, in our opinion, to note the fact that the application of the WKB method under the conditions of the work of [1][2][3] in some cases leads to exact solutions of the WKB type, where the series is cutting off.…”

“…where 𝑆(𝑥, 𝑡) and 𝜑(𝑥, 𝑡, 𝜀) satisfy the same equations as in (2) and it is also assumed that 𝑆 ≥ −𝐶, 𝐶 = 𝑐𝑜𝑛𝑠𝑡 > 0.…”