CC BY
62Probl. Anal. Issues Anal. Vol. 7 (25), Special Issue, 2018, pp. 3-11
3
DOI: 10.15393/j3.art.2018.5470
The paper is presented at the conference "Complex analysis and its applications" (COMAN 2018), Gelendzhik - Krasnodar, Russia, June 2-9, 2018.
UDC 517.951
A. BlRYUK, A. SVIDLOV, E. SlLCHENKO
ON THE HEAT INTEGRAL IDENTITY FOR UNBOUNDED
FUNCTIONS
Abstract. The well known heat integral identity in an unbounded strip is extended to a class of unbounded functions both at x near infinity and t near zero. Continuity of derivatives are relaxed to differentiability in the L;1oc-Sobolev sense.
Key words: heat operator, heat kernel
2010 Mathematical Subject Classification: 35K08
1. Introduction. Let L denote the standard heat operator in Euclidean space R x Rn, i. e., Lu = ut — Au. Under some restrictions there is the well-known integral representation identity that express the function u through Lu and the initial conditions u0 = u(0,x). In particular, the classical result states that for u(t,x) G C([0,T) x Rn) n C 1'2((0,T) x Rn) if both u and Lu are bounded in the strip (0,T) x Rn, then there the integral identity (see [2])
t
u(t,x)^y K(t,x — £)u0(£) + y Jk(t — t,x — £)Lu(r,£) d£dr (1)
rn 0 rn
holds. Here K stands for the fundamental solution to the heat operator (the heat kernel) and is given by
K(t,x)= (W eXP (—142) ' t> 0'
[0, t < 0.
© Petrozavodsk State University, 2018
Among many applications, integral identity (1) is useful for analysis of solutions for various parabolic type equations, e.g., the Burgers equation ut — uxx = -uux, the Navier-Stokes equation
Lu = —(u • V)u — Vp,
and others. Unfortunately, the conditions above are too restrictive to cover many practical situations. For example, we are not allowed, formally, to consider the case of unbounded initial conditions. Moreover, for the Burgers, Navier-Stokes and similar equations, the case when the initial conditions are merely in can not be studied directly using (1), since the right-hand side contains the x-derivative of the function u, and, therefore, is unbounded near t = 0.
In this paper we extend identity (1) to a much wider set of functions. In particular, the cases described above will be covered [1].
2. Notations and the main result. Let T be a positive number fixed throughout the paper. Let L(Q) denote a set of measurable functions f : Q ^ R and let
M2(Rn) = {f G L(Rn) : sup |f (x)| exp(-ixf ) < »).
^ xern j
We denote the space of finite sums of the derivatives (in the sense of distributions) of these functions by D'T :
^^ O | ak | .p
D'T(Rn) = {F G D'(Rn) : F = £ , fk G MT(Rn^.
k = 1
Here we have used the multi-index notation. Taking into account that the derivative of zeroth order is the function itself, we have a natural inclusion
M2(Rn) C DT(Rn).
We note that for any F G DT, for any t g (0,T), and for any £ G Rn a function gT£(x) = K(t, x — £) can be considered as a test function, since the value (the action) of F on gT^ is well-defined as follows
(F,gT, > = jt (—h dx.
i»n k = 1
When it is does not lead to ambiguity, we allow the integral notation to denote the value of a distribution F on a test function g:
/ F(x)g(x)dx = (F,g).
Now we are ready to list the assumptions for the main result. Let u : (0, T) x Rn ^ R be a continuous function, such that there exist partial derivatives ut,uxi,ux.x. £ Ljoc((0,T) x Rn) in the Sobolev sense.
Let the functions u and Lu satisfy the following "moderate" growth conditions: for all (t,x) £ (0, T) x Rn we have
|u((,x)K ci(i) ■ exp (M!), |Lu((,x)K C2(t)exp
where c1(t), c2(t) satisfy the following condition for any e £ (0, T):
T T
J c1(t)dt < J c2(t)dt <
e e
Assume also, that there exists an initial condition u0 £ D'T, taken at t = 0 in the following weak sense
lim j K(t — t,£ — x)u(t,x)dx = (uo,gT,^), for each t £ (0,T), £ £ Rn.
Theorem 1. Under the assumptions listed above the integral identity (1) holds true provided that the last integral in the right-hand side of (1) is understood as the improper Lebesgue integral near t = 0:
t
u(t,x)= K(t,x — £)u0(£) d£ + lim / / K(t — t,x — £)Lu(t,£) d£dT.
J e^+0J J
T
If f c2(t)dt < <x>, then the last integral in the right-hand side of (1) exists
0
as the Lebesgue integral.
3. Proof of Theorem 1. Define the following functions
KTj5(t, x) = K(t - t, x - £), (t, x) = K(t - t, £ - x)
and the following differential operators
Lu = ut - Am, L*u = -ut - Am.
In {t € R, x € R"}\|(t, £)}, both functions Kt,5 and are C~-smooth; so we have
£(Kt,5 )=0 and L*(K*5 ) = 0.
In what follows, we fix a point (t, £) € R x Rn, such that t € (0,T).
Let (x) be a Csmooth function of the variable x € Rn. Since the function u and its derivatives ut, ux, and are assumed to be in L^, the following identity holds in the L1oc((0,T) x Rn \ {(t, £)}) sense:
k;5 Zn Lu = (K;5 Zn u)t + k;5 uAZn+
n
+ 2k;5vzn ■ Vu + £ (znu(k;5)Xi - (znukk;5. (2)
¿=1 *
Indeed, for the smooth function u this is a point-wise identity. The general case follows by approximation. We a(so note that i) ZN(x) is compactly supported, then (2) makes sense in L1^(e,T - e) x Rn) for any e € (0,t/2).
We choose smooth functions ZN(x) to satisfy ZN(x) = 1 for |x| < N, ZN(x) = 0 for |x| > N + 1, and the condition that ZN, VZN, AZN are uniformly bounded by N and x.
In a neighborhood of [e, t-e] x {|x| < N+2} the functions ZN and K*^ are infinitely smooth. Since all terms in (2) belong to L1([e,T - e] x {|x| ^ ^ N+2}) and vanish for |x| ^ N +1, we can integrate (2) over [e, t-e] xRn and apply the Fubini theorem.
Omitting the standard mollification argument, we apply the divergence
theorem for the last term in (2), as follows
T-e n
J J £ (Znu(K*,5- (Znu)XiK;j5dxdt =
e Rn ¿=1
T-e n
= / j E(Znu(K*,5k - (ZnukK*j5dxdt =
e |x|<N+2 ¿=1
(Znu(K*,5)x. — (ZnK*^) Pi dadt = 0.
£ d{|x|<N+2} i_1
Here (p1,..., pn) is the outward pointing unit normal field of the sphere {|x| = N + 2}. Thus, the integral of the last term in (2) vanishes.
Integrating by parts the third term in the right-hand side of (2), as follows:
J K*? VZn • Vu dx = - J K*? AZn u + VK*I? • VZn u dx
rn rn
and applying the fundamental theorem of calculus in t variable for the first term in the right-hand side of (2), we finally arrive at
r —£
/ / ^ZN LU dXdt = j ^ (T - ^ X)ZN (X)U(T - ^ X) dX-
£ rn rn
r —£
J K*^ (e,x)ZN (x)u(e, x) dx —J J (K*^uA(N + 2VK^-VZNw) dxdt.
rn £ rn
(3)
First we will take the limit of (3) as N ^ and then as e ^ +0. The following lemmas are needed to pass to the limit in (3) as N ^
Lemma 1. For each (t, £) G (0, T) x Rn there is a number y1 (T, t, £) such that for any t G (0, t) we have
j K*j5 (t,x)exp( M2) dx < 7i(T,t,C).
rn
Proof. An elementary calculations show:
J K*,£(t,x)exP ^^ dx =
rn
= (4»(t — ,))" n J exp (|p — t—)) dx.
T—£
Note that for any a, b € Rn and for any A > 0 we have the inequality:
|a + b|2 < (1 + A)|a|2 + + A) |b|2. Therefore, for any A > 0 we have
|x|2 = |(x - £) + £|2 < (1 + A)|x - £|2 + + A) |£|2. Taking A = T-1, we obtain
|x|2 |x - £|2 (t - t) - T. . T + t
< -TV" |x - £|2 + -T^-r |£|2 <
4T 4(t - t) 4T(t - t) 1 4T(T - t)
T — t 1
^ - 8T(t- t)|x - £|2 + 2(T - t)|£|2.
We have used the inequalities t -1 < t and T + t < 2T for the last step. We obtain
| K;5(t,x)exp(M2) dx <
< (4n(T -1))-? /exP (-|x -£|2 + |£|2) dx
Now the identity
'n\n/2
J exp(-c|x - £|2)dx = j
completes the proof with y1(T, t, £) = (t—t) exp ^^T-Ty) . ^
Lemma 2. For each (t, £) € (0, T) x Rn there is a number 72 (T, t, £) such that for any t € (0, t) we have
|x|2 A , , ^ 1
I I I ^ ( — <*\
f |VK;5 (t,x)| exp( M^) dx < 72 (t,£) •-
t - t
Proof. Using the identity |VK*'(t, x)| = 2(XT—'j) K*-(t,x) and arguing as in the proof of Lemma 1, we obtain
| |VK;5(t,x)| exp(M2) dx <
|y|___z' t — t |e|2
< (4n(T —t«—V ex^—|y|2 + 2T—7)J dy,
rn
where y = x — £. Now the identity
f nn/2r( n+i)
|y| exp(—c|y|2)dy = n+1 ' ), rn c^r(n)
where r(s) = /0° ts—1e—jdt is the Euler gamma function, completes the
n+1
proof with 72 (T, t, 0 = ( J—t) 2 rrlr ex^2(fe)) . D Since |w(t, x)| < c1(t)exp( ■xjr), Lemma 1 implies
r—£ r—£
/ /|K*£uAZNidxdt < Y1(T,T,i)sup|AZN| / „„),«< «.
£ rn £
Using that yT-1 ^ y? for t G [e, t — e], we conclude, by Lemma 2, that
r—£ r—£
J J |VK;i5 • VZnu|dxdt < 72(T,T,e)sup |VZnlj dt < to.
£ rn £
For any fixed (t, x) G [e, t — e] x Rn we have
lim K* -uAZN = 0 and lim VK* f • VZNu = 0,
n—to '' N —^^o ''
because AZN(x) = 0 and VZN(x) =0 for N > |x|. Therefore, we can apply the Lebesgue dominated convergence theorem and conclude that
r — £
lim I I (K* -uAZn + 2VK* - • VZnu)dxdt = 0.
N—TO J J '' ''
Hence, the limit of (3) as N ^ to gives us the following:
K* dxdt
j K* £(t — e,x)u(r — e,x) dx — J K* ^(e, x)u(e, x) dx.
Replacing the domain of integration [e, t — e] x Rn with [ei, t — e2] x Rn and repeating the arguments, we obtain
K* ^ Lu dxdt =
= j K* ^(t — £2,x)w(r — £2,x) dx — J K* ^(ei, x)u(ei, x) dx. (4)
rn rn
The continuity and boundedness assumptions on u allow to conclude, that lim / K* f (t — e2,x)u(T — e2,x) dx = u(t, £).
rn
Taking the limit of (4) and using Lemma 1, we conclude that
T
*,«> = / K*., (ei,x>u(ei,x) dx+ // k;,{ Lud-*
where all integrals are understood in the Lebesgue sense. The assumptions of Theorem 1 allows to take limit as e1 ^ 0 and this completes the proof. In the case /0 c2(t)dt < to Lemma 1 implies that the last term in the right-hand side of (1) exists as the Lebesgue integral.
Acknowledgment. This work was supported by the Dynasty Foundation and the Ministry of Education and Science of the Russian Federation (project 8.2321.2017/PCh).
The authors are grateful to B. E. Levitskii for research support in Kuban State University.
T— fc
References
[1] Biryuk A., Svidlov A., Silchenko E. Use of Holder Norms for Derivatives Bounds for Solutions of Evolution Equations. International Conference "Geometrical analysis and its applications". (Volgograd, May, 30 -June, 3), 2016. pp. 27-29 (in Russian).
[2] Mikhailov V. P. Partial Differential Equations. Second edition. Moscow, Nauka, 1983 (in Russian).
Received June 14, 2018.
Accepted September 21, 2018.
Published online September 26, 2018.
A. Biryuk
Kuban State University
149 Stavropolskaya str., Krasnodar 350040, Russia E-mail: abiryuk@kubsu.ru
A. Svidlov
Kuban State University
149 Stavropolskaya str., Krasnodar 350040, Russia
E. Silchenko
Kuban State University
149 Stavropolskaya str., Krasnodar 350040, Russia
