JOURNAL OF SCIENCE OF HNUE
Mathematical and Physical Sci., 2014, Vol. 59, No. 7, pp. 9-20 This paper is available online at http://stdb.hnue.edu.vn
REVERSE INEQUALITIES FOR THE FOURIER COSINE CONVOLUTION AND APPLICATIONS TO INVERSE HEAT SOURCE PROBLEMS
Nguyen Xuan Thao and Bui Minh Khoi
School of Applied Mathematics and Informatics, Hanoi University of Science and Technology
Abstract. Convolution inequalities and inverse problems play an important role in mathematical analysis. This paper studies inverse inequalides for the Fourier cosine convolution and the backward heat problem. We give new reverse inequalities for the Fourier cosine convolution and their applications to inverse heat source problems: ut = Uxx + f{x, t), 0 < x < oo, t > 0, our purpose is to evaluate the stability of non-negative heat source f{x, t) from any observations u{x, to), where 0 < *o is a'constant, to < ^ + ^, ^ > 0, ^ > 0, 0<X<X,X>OOT u{xo,t), where 0 < Xo is a constant, 0 < t < T. Applying a new inverse inequality allow us to evaluate heat source through some initial observations with space or time variables.
Keywords: Reverse inequalities, Fourier cosine convolution, one-dimensional inverse, heat source.
1. Introduction
Inverse problems have developed robustly and attracted the attention of many mathematicians in recent decades. Two examples are inverse problems for partial differential equations [2, 4-7] and inverse problems for heat equations [1, 3, 8, 9]. Of these, the problem in [9] is recent research on the reverse heat source problem using reverse inequality for Laplace convolution. We show the following multidimensional heat som"ce equation (see [9]):
dtu{x,t) = Au{x,t) + f[t)'-f{x), x 6 i i " , t > 0, u ( x , 0 ) = 0, xePf, Received March 20, 2014. Accepted September 30, 2014,
Contact Nguyen Xuan Thao, e-mail address: thao,nguyenx uan ©bust edu,vn
for 1^ is a given function and satisfies i^ > 0 in iJ", i^ has compact support, r ^ e C ~ ( R " ) , n > 4
1 ¥> € LiiR"), n < 3.
We then estimate die stabilization of heat source f{t), 0<t<T, fiom the observation u(xo,t), 0<t<T.
where Xo ^ supp^. We have the following theorem:
Theorem 1.1. ([9]) Let if satisfy as above, and Xo i supp V- M* ^«' P > j r j , n 5 3,
1, n > 4.
Then for an arbitrary 5 > 0, there exists a constant C = C{xo,ip>T,p,S,U) > Osuchtha
l{fK(m<CMxo,.)\\Zo,T+5)
for any fe U, U = {f€ C [0, T]; \\f\\c[a,T\ ^ M.f changes the signs at most N-times}, M = const > 0, Af 6 N.
This theorem was proven using reverse inequality for Laplace convolution (see [9J) and heat source conditions that separate variables to f{t)ip(x),x e R", the authors estimated the stabilization oi f(t) according to time variable, i, 0 < i < T.
In this paper, we study one-dimensional inverse heat source problem with heat source f{x, t), x s R+, which does not contain separate variables, as follows:
(M)
(1.2) (U) (I.fl
% = " I I + / ( ^ . '), 0 < X < oo, t > 0, under the marginal condition
Ui(0,f) = 0 , V t > 0 , u^(x,t) -> 0 when x -+ oo, u{x, ^) —^ 0 when x —> oo, and the initial condition
u(x,0) = 0, 05) Here we need to estimate the stability of /(x, t) from any observations v[i,k\
where 0 < <„ = cmist < T-f ,5, T > 0, <S > 0, 0 < x < ;,:, X > 0or«(i„,(),whm 0 < x„ = canst, 0 < < < T. The main finding of this paper proves the reverse incquilily for Fourier cosine convolution (Section 3) and we then apply the new received resollfc one-dimensional inverse heat source problem (I.l) - (1.5).
Reverse inequalities for the Fourier cosine convolution and applications lo inverse heat source...
2. Some known results
We present some convolutions and convolution inequahties used in this article. First of all. The Fourier cosine convolution is defined by [10]:
+00
{f*j){x) = ^ / fiy){9i^ + y) + 9{\^-y\)}dy, x € /?+, (2.i)
0
for which the factorization property holds
Fc(f *^9){y) = (Fof)(y)(F,g)(y), y € R^, f, g e L,(R+), (2.2) where the Fourier cosine transformation is
(FcfKy) = \IIJ /(x)cos(xy)di. (2.3)
0
The Laplace convolution is defined by [10]:
Ul9){x) = jf{t)g{x-t)dt, x>Q, (2.4)
for which the factorization property holds
i{/*s)(p) = (Lf){p){Lg)(v), P€C, Rep > a, f, g are functions of exponential order, where the Laplace transform is of the form
+00
{i/)(p) = / <'-"f(x)dx.
0
Next, the norm of function / on Lp{A x B), where A,BC.
ll/(:>;,«)ilM,4x8)= U j\f{.x,t)Ydxdt
\B A forp> 0.
Moreover, the following reverse convolution inequality holds (see [9]).
j Proposition 2.1. ([9]) Let p > 1, ,5 > 0, 0 < Q < T, and /, j S !,„ (0, T + i5) satisfy ' 0 < / , 9 < M < o o , 0 < i < T + «.
Then
/T+5 / t \ \ ^''P ll/llM.,r)||p||L,(o..)<M'2^-''^M I {jf[s)g{t-s)ds\dA Ll particular, for
t
{fl9m=Jf{t-s)g{s)ds, 0<t<T-\-5
0
and a= 0, we have
mLAo.r,\\g\Km < *f'"""""ll/2fflll(;,„.,-
Although it has an important role the smdy of inverse problems, not many reverse inequalities for convolutions have been studied. In next section, we study a new reverse inequality for the Fourier cosine convolution and apply it to a certain inverse heat problem.
3, A new reverse inequality for the Fourier cosine convolution
In this section, we establish a new reverse inequality for the Fourier cosine convolution of two ftmctions /, g in two-dimensional space.
Theorem 3.L Utp> 1, 6 > 0, T > Q and f,g e Li{D) n Lp{D), satisfy 0 < f,g <
M <oo, {x, t) e D, D = {{x, t) •.0<x<oo,Q<t<T + 6}. Then we have
\\f{^ML,iR,mT))\\9{^MLAR.>^io.6))^M^'^-'y^\\^^^^^
here,
t
u{x, t) = V^j (^f(T, 0 *_ 9{T, t - 0 j (x)dC
0
Proof Since 0 < / , g < M < o o f o r O < x < c o , 0 < f < T - | - 5 a n d formula (2.1), we have
t +0O
j f }'{r,(,)g^{\x-T\,t-i)dTdi
0 0
t -f-oo
= I j f'-\r,(,)g''-\\x -T\,t- i)f{T,i)g(\x - T|,t - £,)drdi
Reverse inequalities for the Fourier cosine convolution and applications to inverse heat source...
t +00
< M^'-' J j /(T, i)9(\x -Tit- Cldrde.
0 0 t +00
<M^^-''fff{r,^){g{x + T,t^^)-\-g{]x-T\,t-0}dTd^
0 0 t
< M^-'^VinJ (f(T,Opj[T.t-i)] (X)de
0
Hence
T+6 +00 / t -1-00 S( T+6 +00
fflf f f'{r,09''{\3:-rlt-C)drdndxdt<M'^''-'^ f j u{x,t) 0 0 \ o 0 / 0 0
dxdt (3.1) On the other hand, by using Fubini's theorem and changing the variables in integrals, we have
T-l-iS +00 / t +00 \
0 0 \o 0 / +00 T+iS T+<5 +00
= J J J j r(T,09'(\x-T\,t-(,)dTdtd(,dX
0 0 ^ 0
+00 T+5 / + 0 0 r+i5 \
= j j [j j nT,(.)g'{\^-r\,t-i)dtdT\ d^do.
0 0 \ 0 4 / +OQ T+5 I +00 / T+6 \ \
= / / I / / " ( T . O I | s ' ' ( | x - r | , t - e ) d t U T
= / / ( / / ' ' ( r , 0 y /(|x-T!.!/)d!/ dr
o o V o Vo / /
' 11 [I ^"'^'^^ ij9''(\^-T\,y]dy dT didx
0 0 Vo Vo / /
T /+00 / 6 +a> \ \
- / ( / ^'^•^"^^ / / ^''^''^ " ^l'^''^^''^ dr de d^dx
de.dx
'I [I ^'^''' ^M //*"''' ^ ^' ^'^'^'"^^ \dT\di
T / +00 / S +CO \ \
0 Vo Vo 0 / /
T + 0 0 5 +00
• J j /"(T,OdTdf J J f(z.y)dzdy. (3.2)
0 0
From (3.1), (3.2) we obtain
T +00 6 -1-00
/ / l'{r, i)dTdi J j 9'(z, y)dzdy < M*-' j j «(x, t) a
x(o,r),||9lU,(«,x(o,.» < MP'-WP MxMl"
„ — . . ) dxdt.
b b b b Thus
ll/l|L,(R+x(0,T))||S||2,,(H+x(0,J» % M " " " l l ^ l ^ , ' ) llr:,-(fi^.x(0,T+«)) '
The proof of the theorem is complete.
Remark 3.1. For fixed 0 < t = <o < T + (i
io
u{x,t„) = V2ij (f{T, i) *^ S(T, to - «)) (x)de,
0
and if ^(r, to - ^) > g{r, to - Co) for 0 < ^ < /3, /? = const, p < to, as in the proof Theorem 3.1, we have
II/l|L,((o,x)x(o,^))||g(x,to - eo)|lMo,-.) ^ M(^^-^)^^ ll«(^>fo)|li(^o,^^.,,.
Indeed, since 0 < / , 5 < M < o o f o r O < x < o o , 0 < ( < T - | - ^ a n d formula (2.1), therefore
to +0O
/ / fir, l)/(N - T1, to - adrdi < M'-~M^, t„).
0 0
X+-I / („ +00 \ X+-,
I [J J f'(''-i^S'{\^-r\.to-OdTdi]dx<M'--^ J u{x,t„)dx, (3.3) 0 Vo 0 / 0
0 0
Hence
Reverse inequalities for the Fourier cosine convolution and applications to inverse heat source...
for J'i" > 0, 7 > 0. On the odier hand, by using Fubini's tiieorem, changing the variables in integrals and since g{T, to - C) > g{T, to - ^o) for 0 < ^ < p, /3 = const, P < to, we have
-f+T / to +00 \
/ / / ^"^^^ ^'^"^'"^" ^1' *°" ^^''^^)''''
0 \o 0 /
£o + O O X H - 7
- I I I '''^''' ^^^'^^"^' '"'• *°" ^' '''"^'"^
0 0 0 (o X X+t
- J J I ^'^^' ^^^^"^ - '^' *o - ^) '^^^^^^
0 0 T to X X+t-T
- I I I f''('''^)s''{z,t,>-CdzdTdi
0 0 0 /3 X 7
> j J J fiT, i)g'(z, to - 0 dzdTdi
0 0 0
> J J J fir, 09'{z, to - ?o) d^drde
0 0 0
0 X -I
> j j fir, i)dTdi j g^{z, to - io) dz. (3.4)
0 0
From (3.3), (3.4), we obtain
ll/lk((o,x)x(o.«)ll9(x,to-&)||vo,7)^MP''-Wp|[„(x,t„) Remark 3.2. For fixed x = XQ > 0 then
<^«,t)= j j f(T,i)[g(xo + T,t-Q+g(\xo-T\,t-i)]d: Wd^,
and if gixa + r, e) > g(xo + ro, e) for 0 < r < a, a = canst, as in the proof Theorem 3.1, we have
i,(Mx(o,T))lls(xo + ro,t)|U,(„,,) < MP''-2>/''||«(xo,f)lli''f„ l , ( 0 , T + i ) -
15
Indeed, since 0 < / , p < M < o o f o r O < a ; < o o , 0 < i < T - l - 5 , therefore t +ca ,
0 0
Hence
xo,t)dt. (3.5)
T+6 / 1 +CO 'v T+S
I [I I f'--'- ^)«'(^» +•''*- OdTd^ dt < M^-'^ J u{:
0 \0 0 / 0 On the other hand, by using Fubini's theorem, changing the variables in integrals and since ^(xo + r,^) > g{xo + TQ, C) for 0 < T < Q we have
T+6 / ( +0O \
I [I I •^''^^' ^^^^""^ "^ ^' * " ^^^'''^^ '^^
0 \0 0 /
T+6 / +O0 T+6 \
= J [I I !'(''' 5)9'(^» + ^' * - OdtdT I di
0 Vo { / T+S / +00 / T+6 \ \
= / (//''('•,e)(/s''(a:o+T,t-e)dtJdTJde
T+S / +00 / T+5-i \ s.
= / / /"(r, e) j g'ixo + T, y)dy j dr j dJ
- / ( / ^'^'''^^ {jg'{^o + r,y)dy\ d r W
> / / r (T, ?) I / / ( ^ o + T, y)dy j dT j df
> y I / / ' ( r , fl I y s'Ca^o + r„, j/)d3/ j dr j df To 5
0 0
From (3.5), (3.6), we obtain
ll/IU,((o,„,x(o,T))ll9(:ro + ro,t)|U,M^M(*-^)/''||«(x„,t)||V^„^^^^,.
16
Reverse inequalities for the Fourier cosine convolution and applications to inverse heat source...
Remark 3.3. In the reverse inequahty for Laplace convolution (Proposition 2.1), the functions / , g are considered in one-dimensional space. As in Theorem 3.1, our reverse inequality for Fourier cosine convolution has a more complicated kernel (see (2.1), (2.4)) with the functions / , g in two-dimensional space.
4. Applications to inverse heat source problems
In this section, we get the result of inverse problem (1.1) - (1.5) when applying Theorem 3.1 with to be specific, an evaluation of the stabilization of the heat source f{x, t) according to space or time.
Theorem 4.1. We consider the heat equation (LI) with marginal conditions (1.2) - (1.4) and an initial condition (1.5) where f G - ^ i ( ^ ) H Lp{D); 0 < f < M < oD,for every (x, t)e D and M is a constant, D = {{x, t) : 0 < x < oo, 0 < t <T + 5},5 > 0, T >
0, p > 1. Then we have
(i) For 0 < to = const <T + 5,T>Q,6>0, 0 < x < X, X > 0,we evaluate the stability of heat source f{x,t) from any observations u{x,to)-'
||/IUp((0,X)x(0,(o-7V2)) < Cl M^^io)\\L,%,X+-y) - for Cl is constant.
(ii) For 0 < Xo = const, 0<t<T,l<p<2, a>Owe evaluate the stability of heat source f{x, t)from any observations u(xo, t):
||/|U,((o,«)x{o,r)) ^ Ca ||u(xo,f)!li''j^(o,r+i)' for C2 is constant.
Proof, (i) Considering t as a parameter, by applying Fourier cosine transformation (2.3) on both sides of (1.1), therefore:
|(Fe«(x,t))(!,) = -y\FM^,tmy) + {FJ{x,tmy), (4.1) for die condition (Fc"(x, 0)){y) = 0.
Equation (4.1) has root as
{FMx, t))(y) = C{y)e-y'' + e'''' J {FJ(x, my^'^di, 0
since {FM^, 0))(y) = 0 dius C{y] = 0.
Then, by using formula (2.2) we have
t t
(F,u(x, t))(!/) = e'"'' J (FJ{x, fl)(!/)6»'«d£ = j (FJ(x, ?))(i/)e-'''<'-"d5
0 0
= j (Fj{x,0){y)F, I , ^ ^ I {y)di -JF.(fiT,i)
lo
s/t-
f{r,i)i ft^(.
e - T ' / 4 ( i - 0
{y)d^
Wdjife)-
Uix,t)^j if{T,0*^^J=-\{x)di
t 4-00
(x-T)V4(l-£) . - ( x + T ) ' / 4 ( l - e ) \
+ 7.—;— dTdl,.
Set
s(^,fl = f ^
\/?
/ r ^ /
then there exists a constant A^ such that 0 < g{T, fl<N<oo{0<r<oo,0<^<
T -\-5). Set Ml = max {M, Af}. We have function g(T, to — {) which does not decrease i n O < 5 < / 3 = f o - 7^/2, 0 < 7^/2 < to, dius, by applying Remark 3.1 we evaluate die stabihty of function f{x, t) from any observations u(i, to), where 0 < to = const <
T + (5, r > 0, (5 > 0, 0 < X < X, X > 0:
lip((0,X)x(0.1o-7V2)) y/ti ^Mr-^'"(2x)V'i'j|„(x,t„)l|^/;^
ll/llL,((0,X)x(0,fe-7V2)) ^ C l | | M ( x , f o ) | | t , ' o , X + 7 ) •
for
Cl = Mf''-""'(27r)V2p v ^
Reverse inequalities for the Fourier cosine convolution and applications to inverse heat source...
(ii)Ontheotherhand,thefunctionc/(xo-i-T,^) does not increase in 0 < r < Q, thus, by applying Remark 3.2 we evaluate the stabihty of function f{x, t) from any observations U{XQ, t), where 0 < xo = const, 0 < t < T, 1 < p < 2, a is an arbitrary non-negative constant:
^Mi'-'^'^"'(2„y'''\\u(xo,t)\\l%,,„,
•ft
l|Lp{(0,a)x(0,T)) ^ Cl ll"(3:0,')|li*o,7-+J) ' for
Cj = M{' (2p-2)/p (27r il/2p The proof of the theorem is complete.
Remark 4.1. For fixed X = Io > 0, 0 < t = to < r + (5flien \ _ g-(io+o)V«
v^
- 1
i.p(o.i)
\
ti(xo, to) • ^ 1 1 /(^' ?){9(^o + T, to - 0 + 9 ( k o - T|, to -'mdrdi,
\Pht J
0 0
for to > (xo + a)^/2, p > 1, as intheproof of Theorem 4.1, we obtain:
ll/IU,((0,o)x(0,io-(io+a)V2)) ^ C ' ( M ( X o , t o ) ) ' ' ' ' ,
for
C = M\ •(2p-2)/p (27r: ,l/2p - ( a : o + Q ) ^ / 4 t o
hi [9], the authors evaluated the stabilization of /(t) according to time variable t (Theorem (1.1)) by using a reverse inequality for the Laplace convolution (Proposition 2.1) and heat source conditions that separate variables to /(t)i^(x), x G it!".
Here, we evaluate the stabilization of heat source /(x, t), x € R+, which does not separate variables, according to space x or time t from some initial observations (Theorem 4.1), or in bofll variables from an initial observation (Remark 4.1) using the reverse inequality for the Fourier cosine convolution which was obtained in Theorem 3.1.
Here is no any numerical example to illustrate the validity/effectiveness of the main result. In the future, we can apply the above results to give a numerical solution for a specific heat source problem and evaluate the advantages of the new me±od compared to old results.
19
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