De Sitter space-time is a solution of the vacuum Einstein equation with a positive cosmological constant in Euclidean space. We prove a uniqueness theorem to determine a time-independent source term for the Klein-Gordon equation in de Sitter space-time up to a neighborhood of an open subset when the mass in the equation has a particular value. We establish a new method based on the Duhamel's principle and theory of distributions with compact supports to deal with inverse problems for such equations having time-dependent coefficients

TAKASE, HIROSHI

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RIGHT:URL:CITATION:AUTHOR(S):ISSUE DATE:TITLE:INVERSE SOURCE PROBLEM FOR KLEIN-GORDON EQUATION IN DE SITTER SPACE-TIME (Analysis of inverse problems throughpartial differential equations and relatedtopics)TAKASE, HIROSHITAKASE, HIROSHI. INVERSE SOURCE PROBLEM FOR KLEIN-GORDON EQUATION IN DE SITTER SPACE-TIME (Analysis ofinverse problems through partial differential equations and related topics). 数理解析研究所講究録 2021, 2174: 108-1142021-02http://hdl.handle.net/2433/263960108INVERSE SOURCE PROBLEM FOR KLEIN-GORDON EQUATION IN DE SITTER SPACE-TIME HIROSHI TAKASE ABSTRACT. De Sitter space-time is a solution of the vacuum Einstein equation with a positive cosmological constant in Euclidean space. We prove a unique-ness theorem to determine a time-independent source term for the Klein-Gordon equation in de Sitter space-time up to a neighborhood of an open subset when the mass in the equation has a particular value. We establish a new method based on the Duhamel's principle and theory of distributions with compact supports to deal with inverse problems for such equations having time-dependent coefficients. 1. INTRODUCTION AND MAIN THEOREM Let n E N and T, H, m > 0 be constants. In this paper, we consider the initial value problem (1.1) {Pmu := o;u - e-2Htt:i.u + nHOtU + m 2u = S (I; cHTdT) f(x) u(O, ·) = Otu(O, ·) = 0 in (0, T) X lR.n, where f and S are smooth functions. Let ( t, x) = ( t, x1, ... , xn) E JR Hn be the Cartesian coordinate. If we use de Sitter space-time in Euclidean space (JR.l+n,g) defined by n g = -dt 0 dt + e2Ht L dxj 0 dxj, j=l which satisfies the Einstein equation with a positive cosmological constant, then our differential operator Pm can be written as Pm= • 9 +m2 , where • 9 denotes the d'Alembertian with respect to the metric g. Namely, (1.1) is the initial value problem of the Klein-Gordon equation in de Sitter space-time. This type of equations is dealt by Yagdjian and Galstian [9], [10]. The semi-linear case is also studied by Yagdjian [8] and Nakamura [5]. Let n c ]Rn be an open subset. We consider the inverse source problem to de-termine the source term f up to a neighborhood of n from a data of a solution u to (1.1) in n x (0, T). This inverse problem relates to the inverse source problem for the equations having time-dependent coefficients. This type of inverse problems are studied by Jiang, Liu and Yamamoto [3] by the Bukhgeim-Klibanov method [l]. They prove the local Holder stability under the assumption that there exists Date: April 30, 2020. 109HIROSHI TAKASE the Carleman estimate for a suitable weight function. However, the Bukhgeim-Klibanov method in no longer useful since our goal is to prove uniqueness up to an outer neighborhood of fl and the choice of a weight function is unclear. We must generate a new method to solve this difficulty. To the best knowledge of the author, there are few publications on inverse source problems in the case that the principal part depends on time. Given the situa-tion, we consider a new method to deal with inverse source problems based on the Duhamel's principle and theory of distributions with compact supports. We denote the class of tempered distributions by S' and distributions with compact supports by £,'. Theorem 1.1. Let H, T > 0 be constants and m = y'n~-l H. Let fl C Rn be an open subset, f E C 00 (Rn) nS' and SE C 00 ([0,H- 1 )) with S(0)-/- 0. Assume u E C 00 ([0, T); S') is a solution to the initial value problem (1. 1). If u = 0 in [0, T) x fl, then f = 0 up to nT, where nT is defined by { 1 -HT} nT := X E Rn I dist(x, fl) < - ~ . Remark 1. We should remark on the well-posedness of (1.1). Under the assump-tions in Theorem 1.1, it is well known that there exits a unique solution u E C00 ([0, T); S'). Furthermore, the solution u satisfies Vt E [0, T), u(t, ·) E C 00 (Rn), which is followed by the argument in the proof of Theorem 1.1 by means of the Fourier transform in S' and the fact that a convolution * is a map *: £' X C00 (Rn)---+ C00 (Rn), (e.g., [7, Theorem 27.2]). Thus, the condition u = 0 in [0, T) x fl makes sense. 2. PROOF OF THEOREM 1.1 Proof. Let m = ✓n~-l H. We remark that by defining S(a-) 1 R(r:,) := !!H, r:, E [O, H- ) (l-Hr:,) 2 as a smooth function with R(0) -/- 0, we can write S (late-HT dT) = e-~Ht R (late-HT dT) . For the solution u E C00 ([0, T); S'), by Lemma 3.1, there exists v E C00 ([0, T); S') such that v is a unique solution to the initial value problem {Pmv = 0 v(0, ·) = 0, 8tv(0, ·) = f Furthermore, u and v satisfy in [O, T) X Rn, on Rn. u(t, x) = e- n;;l Ht late n;;3 Hs R (1t e-HT dT) v(s, x)ds. 110The Fourier transform with respect to the space variable x in S' yields { ~'fv + ~-2Htll~2f; + n!!~tV + m 2v = O in [\T) x lRn, v(O, ·) - 0, 8tv(O, ·) - f on JR , where v(t, l) represents the Fourier transform of v in S'. The solution is written by 1 sin (Ill I/ e-Hsds) v(t, l) = e- n, m loll l(l). Define a linear operator Ai : C 00 ([0, T); S')-+ C 00 ([0, T); S') for j EN as Aiw := ilie-n2lHt 1t en;-3HT 1T e-Hsdsw(r,l)dr. Applying Ai to the solution f; yields Aiv = ili e- n;-l Ht 1t e n;-3 Hr 1T e-Hsdsv(r, l)dr - . i _n;- 1 Ht( [- r -Hs cos(lllJ; e-Hsds)lt rt -HTcos(lllJ; e-Hsds) ) A() - il e lo e ds lll2 0 + lo e lll2 dT f l _ . i _ n;-l Ht (- t -Hs cos (Ill 1i e-Hsds) sin (Ill 1i e-Hsds)) A - il e lo e ds lll2 + lll3 f(l), Moreover, define a linear operator £) : C00 ([0, T); S') -+ C00 ([0, T); S') with the help of Ai as rj A ·- Ai A • a A 1..., w .- w + i 8li w. Applying ,Ci to the solution f; yields (2.1) rjA AiA . a A 1..., V = V + Z aliV . n-l li Ill 1i e-Hsds cos (Ill 1i e-Hsds) - li sin (Ill 1i e-Hsds) A = AJf; + ie--2-Ht lll3 f - n-l Ht sin (Ill 1i e-Hsds) . a A +e 2 Ill ialif(l) - n-l Ht sin (Ill 1i e-Hsds) . a A = e 2 l(I i ali J(l) n-1 sin (Ill 1i e-Hsds)--:--= e--2 Ht 1(1 xJ f (l). On the other hand, we define a linear operator Ai : C 00 ([0, T); S')-+ C 00 ([0, T); S') and Li : C 00 ([0, T); S')-+ C 00 ([0, T); S') for j EN as . a ( n-lHt1t n-3H 1T H ) A1w := -. e--2- e-2- 7 e- 8 dsw(r,x)dr 8xJ o o 111HIROSHI TAKASE and Ljw := Ajw + xJw. We then can easily see that Vw = .Ow for all j E N. Let p E P be an arbitrary polynomial, where P denotes a set of all polynomials with variable x. We then have from (2.1) ___ n-i sin (Ill J; e-Hsds )-p(L)v(t, ~) = e--2 Ht Ill pf(l), 1 . n-1 Ht sin(l~I J.' e-H 8 ds) where L := (L , • • • , Ln). The Paley-Wiener theorem shows that e--2- l~I is the Fourier transform of some distribution s(t) E £' contained in the ball of radius J; e-Hsds. More precisely, it is well known that when n is odd number supps(t) = { x E ~n I lxl = lat e-Hsds}, and when n is even number, supps(t) = { x E ~n I lxl :Slat e-Hsds}. The inverse Fourier transform in S' yields p(L)v(t, x) = (s(t) * pf) (x). Our assumption requires (2.2) (s(t) * pf) (x) = 0 in [0, T) x D by the final claim of Lemma 3.1. Let V(t) c ~n be an arbitrary open neighborhood of supps(t). Since the polynomials P form a dense linear subspace of C00 (V(t)) (e.g., [7, Chapter 15]), we have 'vt.p E Co'(V(t)), :l{pk}k=l C P, Pk~ t.p in the sense of C00 (~n). By continuity of multiplication of an element in C 00 (~n) and of convolution as a linear map of C00 (~n) into C 00 (~n) (e.g., [7, Theorem 27.3]), it follows from (2.2) that (s(t) * t.pf) (x) = 0 in [O, T) x D holds for all t.p E C0 (V(t)). Finally, it means from the definition of convolution with elements of£' and C00 (~n) (e.g., [7, Definition 27.1]) that (s(t), t.pf(x - ·)) = 0 in [O, T) x D holds for all t.p E C0 (V(t)). By Lemma 3.2, we conclude that for all (t, x) E [O, T) x D and ally E supps(t), f(x - y) = 0 holds, i.e., f = 0 in LJ D - supps(t). tE[O,T) • 1123. APPENDIX Lemma 3.1. Let fl c Rn be an open subset, H > 0, m = ✓n~-l H be constants, f E C 00 (Rn) n S' and RE C00 ([0, H-1 )). Assume u E C00 ([0, T); S') is a solution to {Pmu = e-~HtR (Ji e-HTdT) f(x) u(0, ·) = 8tu(0, ·) = 0 in [0, T) X Rn, on Rn. There then exists v E C 00 ( [0, T); S') such that v is a unique solution to {Pmv = 0 v(0, ·) = 0, 8tv(0, ·) = f and u and v satisfy the following relation in [0, T) X Rn, on Rn, Moreover, if R(0) =I= 0 and u = 0 in [0, T) x n, then v = 0 in [0, T) x n. Proof. From the argument by means of the Fourier transform in the proof of The-orem 1.1, the existence of such v is proved. It suffices to prove the relation (3.1). For the solution v, we set It then is obvious that w E C 00 ([0, T); S') by the regularity of v. What we have to prove is w satisfies the initial value problem. Some calculations yield a;w = ( - n; 1 He-Ht R(0) + e-2Ht R' (0)) V + e-Ht R(0)OtV (n - 1)2 2 n-lHt rt n-3H (1t H ) + 4 H e--2- lo e-2- s R s e- T dT v(s, •)ds -nHe_ntlHt lot en2 3HsR' (lt e-HTdT) v(s,·)ds + e-~Ht lot e n23 Hs R" (lt e-HT dT) v(s, ·)ds, 113HIROSHI TAKASE e-2Htf:1w = e- nt3 Ht late n:,3 Hs R (1t e-HT dT) l:!.v(s, ·)ds = e- nta Ht late nti Hs R (1t e-HT dT) (8;v + 3H88 v + m 2v)(s, ·)ds = e-HtR(0)8tv - e-"f-Ht R (late-HT dT) f(x) + ( n; 1 He-Ht R(0) + e-2Ht R'(O)) v + e- nta Ht lat a; ( e ntl Hs R (1t e-HT dT)) v(s, ·)ds - nHe- nta Ht lat 8s (e2Hs R (1t e-HT dT)) v(s, ·)ds + n2; 1 H2e- nta Ht late nti Hs R (1t e-Hr dT) v(s, ·)ds = e-Ht R(0)8tv - e- nta Ht R (late-Hr dT) f(x) + (n; 1 He-HtR(0) + e-2HtR'(o)) v + e- nt3 Ht late n:,3 Hs R" (1t e-HT dT) v(s, ·)ds, n(n - 1) 2 n-lHt rt n-3H (1t H ) nH8tw = - 2 H e--2- lo e-2- s R s e- 7 dT v(s, •)ds +nHe-HtR(O)v+nHe_ntim lat en23HsR' (1t e-HrdT) v(s,•)ds and 2 n2 - 1 2 n-1 Ht rt n-3 H (1t H ) m w = - 4-H e--2- lo e-2- s R s e- 7 dT v(s, •)ds. We then have Pmw = a;w - e-2Ht1:1w + nH8tW + m 2w = e-"f-Ht R (late-Hr dT) f (x) and w(0, ·) = 8tw(0, ·) = 0. By the uniqueness of the initial value problem, we can conclude u = w. The final claim follows by differentiating (3.1) with respect tot and applying the Gronwall's inequality to the formula. • Lemma 3.2. Let n EN, f E C 00 (1Rn) and E [' with a compact support supp . If for an arbitrary open neighborhood V of supp , (,cpf) = 0, 'vcp E C0 (V) holds, then f = 0 on supp . 114Proof. Suppose there exists xo E supp , f(xo) -/- 0. Without loss of generality, we can assume f(xo) > 0. Then there exists r > 0 such that f > 0 on Br(xo), which is an open ball centered at x0 of radius r. For a neighborhood V of supp containing Br(xo) and any XE C0 (Br(xo)), define 'P := J E C0 (Br(xo)) c C0 (V). By our assumption (, 1.pf) = (, x) = o holds for all XE C0 (Br(xo)), that is, = 0 on Br(xo)- However since xo E supp it follows that x 0 (/_ U for all open subset U C ]Rn on which = 0. By the contradiction argument, we prove our lemma. • ACKNOWLEDGEMENT The author would like to thank Professor Masahiro Yamamoto (The University of Tokyo) for many valuable discussions and comments. This work was supported by JSPS and RFBR under the Japan-Russia Research Cooperative Program (project No. J19-721) REFERENCES [1] A. L. Bukhgeim and M. V. Klibanov. Global uniqueness of class of multidi-mensional inverse problems. Soviet Math. Dokl., 24:244-247, 1981. [2] V. Isakov. Inverse Problems for Partial Differential Equations. Springer, 3rd edition, 2017. [3] D. Jiang, Y. Liu, and M. Yamamoto. Inverse source problem for the hyper-bolic equation with a time-dependent principal part. Journal of Differential Equations, 262(1):653-681, 2017. [4] M. M. Lavrentiev, V. G. Romanov, and S. P. Shishatskij. Ill-posed Problems of Mathematical Physics and Analysis. American Mathematical Society, 1980. [5] M. Nakamura. The Cauchy problem for semi-linear Klein-Gordon equations in de Sitter spacetime. J. Math. Anal. Appl., 410(1):445-454, 2014. [6] X. S. Raymond. Elementary Introduction to the Theory of Pseudodifferential Operators. 1991. [7] F. Treves. Topological Vector Spaces, Distributions and Kernels. 1967. [8] K. Yagdjian. The semilinear Klein-Gordon equation in de sitter spacetime. Discrete and Continuous Dynamical Systems - Series S, 2(3):679-696, 2009. [9] K. Yagdjian. Integral transform approach to solving Klein-Gordon equation with variable coefficients. Math. Nachr., 288(17-18):2129-2152, 2015. [10] K. Yagdjian and A. Galstian. Fundamental solutions for the Klein-Gordon equation in de Sitter spacetime. Commun. Math. Phys., 285:293-344, 2009. GRADUATE SCHOOL OF MATHEMATICAL SCIENCES, THE UNIVERSITY OF TOKYO, 3-8-1 KOMABA, MEGURO-KU, TOKYO 153-8914, JAPAN E-mail address: htakase©ms. u-tokyo. ac. jp

INVERSE SOURCE PROBLEM FOR KLEIN-GORDON EQUATION IN DE SITTER SPACE-TIME (Analysis of inverse problems through partial differential equations and related topics)

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INVERSE SOURCE PROBLEM FOR KLEIN-GORDON EQUATION IN DE SITTER SPACE-TIME (Analysis of inverse problems through partial differential equations and related topics)

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