The Stabilization Rate of Solutions to Parabolic Equations
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Journal of Mathematical Sciences, Vol. 249, No. 6, September, 2020
THE STABILIZATION RATE OF SOLUTIONS TO PARABOLIC EQUATIONS V. N. Denisov ∗ Lomonosov Moscow State University Moscow 119991, Russia [email protected]
E. V. Shelepova Lomonosov Moscow State University Moscow 119991, Russia helen [email protected]
UDC 517.955
We estimate the difference between the solution to the Cauchy problem for an inhomogeneous heat equation and the Riesz means for large time provided that the right-hand side of the heat equation has at most power growth at infinity. Bibliography: 8 titles.
We study the behavior of the solution to the Cauchy problem ∂u(x, t) = u(x, t) + f (x), ∂t u(x, 0) = 0,
x ∈ EN ,
t > 0,
x ∈ EN ,
(1) (2)
for large t, where E N is an Euclidean space of dimension N 1. Regarding results on stabilization of solutions to parabolic equations we refer, for example, to [1]–[4]. In this paper, we consider the problem (1), (2) with constant coefficients. In view of the Duhamel formula, a natural question arises about the stabilization of v(x, t) = u(x, t)/t. We consider the generalized Cauchy problem for the heat equation with source F = θ(t)f (x), where θ(t) is the Heaviside function and f (x) is a measurable locally integrable function in E N , 1 i.e., it is required to find a distribution u ∈ D (E × E N ) such that ∂u = Δu + θ(t)f (x). ∂t
(3)
The convolution u(x, t) = U ∗ (θ(t)f (x)), where U = U (x, t) is the fundamental solution of the heat operator, called the generalized heat potential with density θ(t)f (x). ∗
To whom the correspondence should be addressed.
Translated from Problemy Matematicheskogo Analiza 103, 2020, pp. 85-90. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2496-0911
911
Let f (x) satisfy the estimate 2
|f (x)| < Cε eε|x| ,
x ∈ EN ,
for any ε > 0. Then the convolution u(x, t) = U ∗ (θ(t)f (x)) exists (cf., for example, [5]), has the form t |x−y|2 f (y) − 4(t−τ ) e dydτ, (4) u(x, t) = √ (2 π)N (t − τ )N/2 0 EN
and satisfies the estimate tCε
2
e2ε|x| ,
0 0,
and the initial condition t → +0,
(5)
for any R > 0. Replacing σ = r2 /(4(t − τ )), r2 = |x − y|2 , in the integral with respect to τ in (4), we get +∞ f (y) 1 −σ N 2−4 e σ dσ dy. (6) u(x, t) = N/2 rN −2 4π r2 4t
EN
Then we pass to the spherical coordinates with the origin at x. Denoting 1 f (y)dsy , M (x, ρ) = ωN ρN −1
(7)
|x−y|=ρ
where ωN =
2π N/2 , for the mean of f (y) over the sphere r = ρ, we find Γ(N/2) +∞ ∞ ωN −σ N 2−4 e σ dσ dρ. u(x, t) = N/2 ρM (x, ρ) 4π 0
(8)
ρ2 4t
Integrating by parts on the right-hand side of (8), we extract the mean of (7) with respect to ρ. Denote ρ 2 M1 (x, ρ) = 2 σM (x, σ)dσ, (9) ρ 0
where M (x, σ) is defined by (7). Since the terms outside the integral vanish at ρ = 0 and ρ = +∞, we get ∞ N −1 2 2t ρ − ρ4t √ u(x, t) = e M1 (x, ρ)dρ. (10) Γ(N/2) (2 t)N 0
Setting v(x, t) = u(x, t)/t and using (10), we find 2 v(x, t) = Γ(N/2)
912
∞ 0
ρN −1 − ρ2 √ e 4t M1 (x, ρ)dρ. (2 t)N
(11)
Theorem 1. If the limit lim M1 (x, R) = A(x)
R→+∞
(12)
ex
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