L p -L q -Time decay estimate for solution of the Cauchy problem for hyperbolic partial differential equations of linear thermoelasticity

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1 ANNALES POLONICI MATHEMATICI LIV.2 99) L p -L q -Time decay esimae for soluion of he Cauchy problem for hyperbolic parial differenial equaions of linear hermoelasiciy by Jerzy Gawinecki Warszawa) Absrac. We prove he L p -L q -ime decay esimaes for he soluion of he Cauchy problem for he hyperbolic sysem of parial differenial equaions of linear hermoelasiciy. In our proof based on he marix of fundamenal soluions o he sysem we use Srauss Klainerman s approach 2, 5 o he L p -L q -ime decay esimaes. 0. Inroducion. We consider he Cauchy problem for he hyperbolic sysem of parial differenial equaions of linear hermoelasiciy cf. 3): 0.) 0.2) ρ 2 u µ u λ + µ) grad div u + β grad T = 0, β div u + ρτ 2 T k T = 0, wih iniial condiions u+0, x) = u 0 x), 0.3) T +0, x) = T 0 x), u)+0, x) = u x) T )+0, x) = T x) where u = u, u 2, u 3 ) is he displacemen vecor field of he medium, T he emperaure of he medium, 0, x R 3, = /, = 3 j= 2 j ; ρ, µ, λ, β, τ, k are posiive physical consans; u 0, u, T 0, T are given funcions. R e m a r k 0.. The sysem 0.) 0.2) is he principal par of a hyperbolic sysem of parial differenial equaions describing he evoluion of a hermoelasic medium cf. E. S. Suhubi 3, p. 99, formulae )). For he sake of simpliciy we assume in sysem ) ha γ = and T 0 =. Under he assumpion ha he Cauchy daa u 0, u, T 0, T are smooh enough cf. 4, formulae 5.) 5.3)) he soluion of he problem 0.) 0.3) 985 Mahemaics Subjec Classificaion: 35A07, 35A08, 35E05, 35E5, 35L5, 35L40, 35L45, 73B30, 35B40. Key words and phrases: decay esimaes, parial differenial equaions, Cauchy problem, symmeric hyperbolic sysem of firs order, linear hermoelasiciy.

2 36 J. Gawinecki is given by 0.4) U, x) = u, x), T, x)) = H, ) gx) + H, ) hx) where g ) = g ) + D ) h ), gx) = u x), T x)), hx) = u 0 x), T 0 x)), β β D ) = 2, β 3 β β 2 β 3 0 denoes he hree-dimensional convoluion in R 3 and H, x) is he marix of fundamenal soluions of he sysem 0.) 0.2) consruced in 4 of he form 0.5) H jk, x) A = 32π7!ρ 4 τ) {δ jk δ k4 ) A 2 2 x δ x ) B b 2 x δ A3 + δ jk δ k4 x δ x ) A5 x j x k + δ jj δ k4 ) x 3 δ x x δ x b A 4 2 x x x δ ) ) ) x ) A ) 6 x j x k 2 x 3 δ + B x j x k b 2 x 3 δ x ) b δ jj δ k4 ) A 7 x 3 3x ) jx k x 5 ε x ) ε x ) + A 8 x 3 3x ) jx k x 5 ε x ) ε x ) b + A 9 x 3 3x ) jx k x 5 ε x ) ε x ) b x j + A 0 x 3 ε x ) ε x ) δ 4j δ kj ) x j + A 0 x 2 δ x ) δ x ) } δ 4j δ kj ), j, k =, 2, 3, 4, where δ jk denoes he Kronecker symbol, δ ) is Dirac s disribuion, ε ) is Heaviside s funcion { for > 0, ε) = 0 for < 0,

3 L p -L q -Time decay esimae 37 and,, b, A,..., A 0, B are consans. The aim of his paper is o prove he L -L -ime decay esimaes for he soluion of he problem 0.) 0.3) using he formula 0.4), hen o prove he L 2 -L 2 -ime decay esimaes for his soluion applying some heorems of he heory of symmeric firs order hyperbolic sysems cf. Yu. V. Egorov 3, p ) and finally he L p -L q -ime decay esimaes using inerpolaion inequaliies in Sobolev spaces cf. 0, 7, 4). Such L p -L q -ime decay esimaes play an imporan role in he proof of global in ime) exisence of soluion o he Cauchy problem for nonlinear wave equaions cf. 5, 7). Noaion: Besides oher sandard noaion we use he symbol L p,mr 3 ) = W m,p R 3 ) p, m N {0}) for he well-known Sobolev spaces wih norm L p,m R 3 ) = W m,p R 3 ) cf., ); W 0,p R 3 ) = L p R 3 ) wih norm L p R 3 ). We also wrie f = f, x f, x2 f, x3 f) for he space-ime gradien of a funcion f, and Df = x f, x2 f, x3 f) for he space gradien of f. s denoes he smalles ineger larger han or equal o s for s R.. The L -L -ime decay esimaes. We shall prove he following heorem: Theorem. L -L -ime decay esimaes). Le he Cauchy daa u 0, u, T 0, T be funcions vanishing a infiniy. Moreover, le u, Du 0, T, DT 0 ) L R 3 ). Then he soluion u, T ) of he problem 0.) 0.3) given by he formula 0.4) saisfies he following esimaes:.).2) u, ), T, )) L R 3 ) C + ) u, Du 0, T, DT 0 ) L R 3 ), u, ), T, )) L R 3 ) C + ) u, Du 0, T, DT 0 ) L R 3 ) for 0, where C is a consan independen of u 0, u, T 0, T and. P r o o f. We prove.2). The proof of.) runs in he same way. Wriing he soluion U, x) = u, x), T, x)) given by he formula 0.4) in he form.3) U j, x) = H jk, ) g k x) + H jk, ) h k x), k= k=

4 38 J. Gawinecki j =, 2, 3, 4, where U j, x) = u j, x), j =, 2, 3, U 4, x) = T, x), and differeniaing.3) wih respec o and x l for l =, 2, 3) we ge.4).5).6) U j, x) = U 4, x) = l U j, x) = H jk, ) g k x) k= + k= l= k= 3 µ ρ lh jk, ) + λ + µ j H lk, ) ρ β ρ H jk, ) h k j x), j =, 2, 3, H 4k, ) g k x) k= + 3 k= l= 3 k= l= k ρ lh 4k, ) h k l x) β ρ H lk, ) h k l x), l H jk, ) g k x) + k= h k l x) H jk, ) h k l x), l =, 2, 3, j =, 2, 3, 4, h k m = xm hk, m =, 2, 3. We can wrie.4).6) in vecor form as follows: k=.7) V, x) = R, ) V 0 x) where.8) V, x) = u, T ), V 0 x) = u, Du 0, T, DT 0 ) and R, x) is 6 6 marix wih elemens which are linear combinaions of he erms l H jk, x) and H jk, x) cf..3).6)). From.3).6) and.7) i follows ha in order o prove he esimae.2) i is sufficien o prove he following esimaes:.9).0) H jk, ) f L R 3 ) C + ) f L R 3 ), l H jk, ) f L R 3 ) C + ) f L R 3 ) for j, k =, 2, 3, 4 and any scalar funcion fx) saisfying he assumpions of Theorem.. Taking ino accoun he form of he marix H, x) cf. 0.5)) we ge.) H jk, ) fx)

5 = 32π7! ρ 4 τ) {δ A jk δ k4 ) A 2 + δ jk δ k4 A3 L p -L q -Time decay esimae 39 z = fx + z) ds z fx + z) ds z B b b fx + z) ds z z = z =b fx + z) ds z A 4 fx + z) ds z z = z = A5 + δ jj δ k4 ) z jz k a 3 z =a fx + z) ds z A 6 z jz k a 3 fx + z) ds z + B z =a 2 b 2 z =b δ jj δ k4 ) A 7 z 3 3z jz k z 5 z + A 8 z 3 3z ) jz k z 5 fx + z) dz b z b z + A 9 z 3 3z jz k z 5 ) fx + z) dz z j + δ 4j δ kj )A 0 fx + z) dz z 3 a z + δ 4j δ kj )A 0 z j fx + z) ds z z =a z } j fx + z) ds z z =a 2 2 z jz k b 3 fx + z) ds z ) fx + z) dz where ds z is he area elemen of he sphere z = a j, j =, 2, or of z = b. For simpliciy we consider wo ypical inegrals occurring on he righ hand side of.) oher inegrals in.) are esimaed similarly):.2).3) I = I 2 = fx + y) ds y, fx + y) y 3 dy. Differeniaing he inegrals I and I 2 wih respec o and x l l =, 2, 3)

6 40 J. Gawinecki we obain ).4).5).6).7) I = I 2 = I x l = I 2 x l = fx + y) ds y + fx + y) y 3 dy + xl fx + y) ds y, fx + y) ds y, y 3 x l fx + y) dy. Following S. Klainerman cf. 5, pp ) we ge.8).9) fx + y) = = 2 fx + y) = In view of.8),.9) we have.20) I = = s fx + sy) ds = s ) 2 3 s fx + sy) ds, 2 s fx + sy) ds = for > 0. Taking ino accoun ha.2) s 2 3 fx + sy) = s ) 2 s fx + sy) ds ds y j,k= y 3 fx + y) dy, 2 s fx + sy) ds ds y s ) 2 s fx + sy) ds s ) 2 s fx + sy) ds ds y 2 2 s fx + sy) ds ds y s ) 3 s fx + sy) ds. 2 x j x k fx + sy)y j y k 2 b2 D 2 xfx + sy) ) We use he noaion I j = I j, li j = I j x l, j =, 2.

7 L p -L q -Time decay esimae 4 for y = b and s ) s 2, 2 s 2 for 0 s we ge.22) I b 2 s 2 Dxfx 2 + sy) ds ds y. Using he spherical coordinaes we have.23) I b D 2 xf L R 3 ) for > 0. Acing in he same way we ge.24) Ix l = Since s xl fx + sy) = we have.25) Similarly.26).27) s xl fx + sy) ds ds y. 3 x 2 l x j fx + sy)y j b D 2 x fx + sy) j= I x l b I 2 b 3 b 3 s 2 D 2 xfx + sy) ds ds y D 2 xf L R 3 ) for > 0. fx + y) dy + fx + y) dy + a I 2 x l b 3 fx + y) dy D xfx + y) dy. for y = b Dxfx + y) dy, Changing he variable y o z in he above inegrals we derive.28) I 2 b 3 3 fx + z) dz.29) + a 2 b z a b z a Dxfx + z) dz b 3 3 f L R 3 ) + a 2 D xf L R 3 ) I 2 x l b 3 2 b z a, Dxfx + z) dz b 3 2 D xf L R 3 ).

8 42 J. Gawinecki Noing ha / 3 / 2 / for we ge.30) I 2 + I 2 x l C D xf L R 3 ) for. From.23),.25),.30) we obain.3) H jk, ) f ) L R 3 ) C f L,2 R 3 ) for 0 and j, k =, 2, 3, 4. In order o obain an esimae analogous o.30) for 0 we proceed as above expressing he inegrals I and I 2 cf..8),.9)) in he following form:.32).33) I = 2 I x l = + Afer some calculaions we ge s ) 2 3 s fx + sy) ds ds y s ) 3 s fx + sy) ds ds y, s ) 2 s xl fx + sy) ds ds y..34) I + I x l C f L R 3 ) for 0. I is easy o see ha for 0 fx + y).35) y 3 dy b 3 b 3 f L R 3 ), fx + y) dy.36) fx + y) y 3 dy b 3 j= 3 xj fx + y)y j dy b 3 a D xf L R 3 ),.37) y 3 x l fx + y) dy b 3 xl fx + y) dy b 3 D xf L R 3 ).

9 L p -L q -Time decay esimae 43 Hence.38) I 2 + I 2 x l C f L, R 3 ) for 0. Thus from.38) and.34) we obain.39) H jk, ) f ) L R 3 ) C f L R 3 ) for 0. Now, in view of 2 + ) for 0 and 2 + ) for and aking ino accoun.3),.39) we conclude ha.40) H jk, ) f ) L R 3 ) C + ) f L R 3 ) for The L 2 -L 2 -ime decay esimaes. We derive he L 2 -L 2 -ime decay esimaes for soluion of he Cauchy problem 0.) 0.3). More precisely, we formulae he following heorem: Theorem 2. L 2 -L 2 -ime decay esimaes). Le he Cauchy daa u 0, u, T 0, T be funcions vanishing a infiniy. Moreover, le u, Du 0, T, DT 0 ) L 2 R 3 ). Then he soluion u, T ) of he problem 0.) 0.3) given by he formula 0.4) saisfies he following esimaes: 2.) 2.2) u, ), T, )) L2 R 3 ) u, ), T, )) L 2 R 3 ) C u, Du 0, T, DT 0 ) L 2 R 3 ) for 0, C u, Du 0, T, DT 0 ) L 2 R 3 ) for 0, where C is consan independen of u 0, u, T 0, T and. S k e c h o f p r o o f. Following Yu. V. Egorov cf. 3, pp , ) we reduce he Cauchy problem 0.) 0.3) o an equivalen Cauchy problem for a linear symmeric hyperbolic sysem of firs order. Nex, applying he exisence and uniqueness heorems cf. 3, Theorem 3.2, p. 329) we obain he esimaes 2.), 2.2). 3. The L p -L q -ime decay esimaes. In his secion we express he L p -L q -ime decay esimaes for soluions of he Cauchy problem 0.) 0.3) in erms of heir gradien. We consider he operaor Π defined as follows: 3.) Π fx) = R, ) fx) for any funcion fx) saisfying he assumpions of Theorems. and 2., where R, x) is defined by.7).

10 44 J. Gawinecki 3.2) 3.3) From Theorems. and 2. i follows ha Π 0 :L R 3 ) L R 3 ), Π 0 C + ), Π : L 2 R 3 ) L 2 R 3 ), Π C. By inerpolaion cf. J. Shaah 0, S. Klainerman and G. Ponce 7) we have 3.4) Π θ : L, L 2 θ L, L 2 θ, Π θ = Π 0 θ Π θ wih 0 θ, where X, Y θ 0 θ ) denoes he complex inerpolaion space cf. 8, 4) wih respec o X and Y. In order o obain he L p -L q -ime decay esimaes where q = 2α + 2, p = 2α + 2)/2α + ), /p + /q =, α is a nonnegaive ineger) we noice ha for θ = /α + ) 3α 3.5) L, L 2 /α+) = L p,s 0 where s 0 =, p = 2α + 2 α + 2α +, 3.6) Hence, we have 3.7) 3.8) L, L 2 /α+) = L 2α+2. Π α) : L p,s 0 R 3 ) L 2α+2 R 3 ), Π α) = Π 0 /α+) Π /α+) C + ) α/α+). So, we have proved he following heorem: Theorem 3. L p -L q -ime decay esimaes). Le he Cauchy daa u 0, u, T 0, T be funcions vanishing a infiniy. Moreover, le u, Du 0, T, DT 0 ) L p,s 0 R 3 ) for p = 2α + 2 2α +, s 0 = 3α/α + ) and α a nonnegaive ineger. Then he soluion of he problem 0.) 0.3) given by he formula 0.4) saisfies he following esimaes: 3.9) u, ), T, )) L 2α+2 R 3 ) C + ) α/α+) u, Du 0, T, DT 0 ) L p,s0 R 3 ) for 0 where C is a consan independen of u 0, u, T 0, T and. R e m a r k 3.. In a subsequen paper, we shall apply Theorem 3. in he proof of global in ime) exisence of soluion of he Cauchy problem for he nonlinear hyperbolic sysem of parial differenial equaions describing a hermoelasic medium.

11 L p -L q -Time decay esimae 45 References R. Adams, Sobolev Spaces, Academic Press, New York K. O. Friedrichs, Symmeric hyperbolic linear differenial equaions, Comm. Pure Appl. Mah ), Yu. V. Egorov, Linear Differenial Equaions of Principal Type, Nauka, Moscow 984 in Russian). 4 J. G a w i n e c k i, Marix of fundamenal soluions for he sysem of equaions of hyperbolic hermoelasiciy wih wo relaxaion imes and soluion of he Cauchy problem, Bull. Acad. Polon. Sci., Sér. Sci. Techn. 988 in prin). 5 S. Klainerman, Global exisence for nonlinear wave equaions, Comm. Pure Appl. Mah ), , Long-ime behaviour of soluions o nonlinear evoluion equaions, Arch. Raional Mech. Anal ), S. Klainerman and G. Ponce, Global, small ampliude soluions o nonlinear evoluion equaions, Comm. Pure Appl. Mah ), J. L. Lions e E. Magenes, Problèmes aux limies non homogènes e applicaions, Dunod, Paris A. Piskorek, Fourier and Laplace ransformaion wih heir applicaions, Warsaw Universiy, 988 in Polish). 0 J. Shaah, Global exisence of small soluions o nonlinear evoluion equaions, J. Differenial Equaions ), S. L. S o b o l e v, Applicaions of Funcional Analysis in Mahemaical Physics, 3h ed. Nauka, Moscow 988 in Russian). 2 W. A. S r a u s s, Nonlinear scaering heory a low energy, J. Func. Anal. 4 98), E. S. Suhubi, Thermoelasic solids, in: Coninuum Physics, A. C. Eringen ed.), Academic Press, New York H. T r i e b e l, Inerpolaion Theory, Funcion Spaces, Differenial Operaors, Deuscher Verlag der Wissenschafen, Berlin 978. DEPARTMENT OF MATHEMATICS MILITARY TECHNICAL ACADEMY S. KALISKIEGO WARSZAWA, POLAND Reçu par la Rédacion le Révisé le