Fundamental Solution for Cauchy Initial Value Problem for Parabolic PDEs with Discontinuous Unbounded First-Order Coeffi
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Fundamental Solution for Cauchy Initial Value Problem for Parabolic PDEs with Discontinuous Unbounded First-Order Coefficient at the Origin. Extension of the Classical Parametrix Method Maria Rosaria Formica1 Leonid Sirota2
· Eugeny Ostrovsky2 ·
Received: 15 June 2019 / Accepted: 11 June 2020 © Springer Nature B.V. 2020
Abstract We prove the existence of a fundamental solution of the Cauchy initial boundary value problem on the whole space for a parabolic partial differential equation with discontinuous unbounded first-order coefficient at the origin. We establish also non-asymptotic, rapidly decreasing at infinity, upper and lower estimates for the fundamental solution. We extend the classical parametrix method of E.E. Levi. Keywords Partial Differential Equation of parabolic type · Fundamental solution · Generalized Mittag-Leffler function · Chapman-Kolmogorov equation · Neumann series · Volterra’s integral equation Mathematics Subject Classification (2010) 35A08 · 35K15 · 35K20
1 Introduction We consider in this article the Partial Differential Equation (PDE) of (uniform) parabolic type ∂u 1 ∂ 2 u b(t, x) ∂u = , + ∂t 2 ∂x 2 |x|γ ∂x
(1.1)
where u = u(t, x), t ≥ 0, x ∈ R, γ = const ∈ (0, 1) and the coefficient b = b(t, x) is a continuous bounded function such that b(t, 0) = 0, together with the initial value problem
B M.R. Formica
[email protected] E. Ostrovsky [email protected] L. Sirota [email protected]
1
Parthenope University of Naples, via Generale Parisi 13, Palazzo Pacanowsky, 80132, Naples, Italy
2
Department of Mathematics and Statistics, Bar-Ilan University, 59200, Ramat Gan, Israel
M.R. Formica et al.
(Cauchy statement) lim u(t, x) = f (x),
t→s+
u = u(t, x) = u[f ](t, x),
s ≥ 0,
(1.2)
where f (·) is a measurable function satisfying some growth condition at |x| → ∞; for the definiteness one can suppose its continuity and boundedness, i.e. ess supx |f (x)| < ∞. Briefly, Lt,x u = L[b]t,x u = 0, where L[b]t,x is the linear parabolic partial differential operator of the form def
L[b]t,x = Lt,x =
1 ∂2 ∂ b(t, x) ∂ − . − ∂t 2 ∂x 2 |x|γ ∂x
(1.3)
Let us discuss our restriction γ ∈ (0, 1). The case γ = 0 corresponds to the well-known “regular”equation and it has been widely investigated. The case −1 < γ < 0 has been considered in the article [5]. If γ > 1, then the considered problem is incorrect; when γ = 1 several additional restrictions are required, still in the case when b(x, t) is constant (see [30]). Let us recall a classical definition. Definition 1 Let 0 < s < t and (x, y) ∈ R2 . The (measurable) function p = p(t, x, s, y) is said a Fundamental Solution (F.S.) for the equation (1.1), subject to the initial condition (1.2), iff for all the fixed values 0 < s < t , (x, y) ∈ R2 , it satisfies the equation (1.1) and, for all the bounded continuous functions f = f (x), p(t, x, s, y)f (y)dy = f (x), x ∈ R, (1.4) lim t→s+ R
if, of course, there exists. In this case the F.S. p(t, x, s, y) may be interpreted as the transfer density of probability for diffusion random n
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