Propagation of Singularities of Solutions to Hyperbolic Parabolic Coupled Systems. 1. Introduction

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1 Math. Nachr. 4 00, Propagation of Singularities of Solutions to Hyperbolic Parabolic Coupled Systems By Shuxing Chen and Ya Guang Wang of Shanghai Received October 5, 000; accepted March 3, 00 Abstract. In this paper, the authors study the propagation of singlarities for a semilinear hyperbolic parabolic coupled system, which comes from the model of thermoelasticity. Both of the Cauchy problem and the problem inside of a domain are considered. We obtain that the microlocal singularities of solutions to the semilinear hyperbolic parabolic coupled system are propagated along null bicharacteristics of the hyperbolic operator by using the theory of paradifferential operators. Furthermore, for the Cauchy problem of the semilinear coupled system, if the initial data have singularities at the origin, we prove that the solutions have the same order regularity with respect to spatial variables as in hyperbolic problems in the forward characteristic cone issuing from the origin, which improves the previous results for semilinear systems in thermoelasticity.. Introduction This paper is devoted to the study of propagation of singularities of solutions to hyperbolic parabolic coupled systems. It is well known in the theory of microlocal analysis that singularities of solutions to partial differential equations of principal type propagate along bicharacteristics on cotangent bundle of a given domain in the base space [, 4, 6, 7, 8]. Even for nonlinear partial differential equations of principal type a corresponding theorem on singularity propagation is also established e. g. refer to [, 4]. These conclusions have been also extended to equations or systems with multiple characteristics. On the other hand, because heat equations or more general parabolic equations are hypoelliptic, there is no singularity propagation for their solutions. However, in many physical problems, like in thermoelasticity, viscous fluid etc., people often meets hyperbolic parabolic coupled systems. Since the hyperbolic part in the system still describes the phenomena of wave propagation, then it is very natural to study the phenomena of singularity propagation for solutions to hyperbolic parabolic coupled systems. 000 Mathematics Subject Classification. Primary: 35M0, 35B65, 35A7; Secondary: 73B30. Keywords and phrases. Semilinear hyperbolic parabolic systems, thermoelasticity, Cauchy problems, propagation of singularities, microlocal analysis. Corresponding author/ygwang@online.sh.cn c WILEY-VCH Verlag Berlin GmbH, 3086 Berlin, X/0/ $ /0

2 Chen and Wang, Singularities in Hyperbolic Parabolic Coupled Systems 47 We are mainly going to study the singularity propagation of multidimensional semilinear thermoelastic system, which is a typical model of hyperbolic parabolic coupled system. In this paper this problem will be studied in the cases inside of a domain and for Cauchy problems as well. As it is known that for these two cases the singularity is described by different ways. In the first case, the singularity is described by the wave front set on the cotangent bundle T IR t IR n x, while in the Cauchy problem case the singularity is described by the wave front set on the cotangent bundle T IR n, which is moving along with the parameter t. These two ways to describe the process of singularity propagation are essentially equivalent, but the expressions of the result as well as the corresponding proof are different. The details will be given in the sequel. Since the system taken into account is nonlinear, a good linearization is necessary. To this end we resort to the theory of paradifferential operators. After paralinearization of nonlinear terms in the system we obtain a paradifferential system, which can be similarly treated as linear system with smoother error terms. Therefore, some technique used in [, 5] can also be applied in our case. Consider the following semilinear hyperbolic parabolic coupled system in IR t IR n x : t n a ij x ix j u = fu, θ, t u, u, θ,. t b ij x ix j θ + c j tx j u + d ij x ix j u = gu, θ, t u, u, j= where all coefficients are smooth functions of t, xwith a ij t, x n n and b ij t, x n n being positive definite, and nonlinear functions f and g are smooth with respect to their arguments with u and θ denoting gradients of u and θ with respect to the space variables x =x,...,x n. The system. comes from the semilinear model of thermoelasticity in higher dimensions, which takes the form in three space variables as:. u tt µ + λ div u + µ rotrot u+γ θ = fu, u t, u, θ, θ t β θ + γ div u t = gu, u t, u, θ, where µ, λ, β, γ,γ are smooth functions of t, x withµ µ 0 > 0, λ + µ>0and β β 0 > 0. Particularly, the system. in one space variable case is reduced to u tt αu xx + γ θ x = fu, u t,u x,θ,.3 θ t βθ xx + γ u tx = gu, u t,u x,θ α, β > 0. By using the Fourier analysis, the Cauchy problem for.3 with piecewise H s initial data was studied in [0, 3] for the case of f, g being independent of u t,u x. The purpose of this paper is to study the propagation of singularities for the semilinear hyperbolic parabolic coupled system.. Our method can also be applied to the fully nonlinear case. The details and the corresponding applications will be given in future. The main results of this paper are stated as follows, here the definition of microlocal Sobolev spaces can be found in [, 4, 7].

3 48 Math. Nachr Theorem.. For the hyperbolic parabolic coupled system., let Γ be a null bicharacteristic of the hyperbolic operator L = t n a ij t, x x i x j passing through t 0,x 0 ; τ 0,ξ 0 T IR n+ \0. For any fixed n n+ + s r s, ifu H s and θ H s / are solutions of. satisfying u H r t 0,x 0 ; τ 0,ξ 0, then we have.4 u HΓ r and θ H r / Γ. Theorem.. Consider the following Cauchy problem for.: t a ij x ix j u = fu, θ, t u, u, θ,.5 t b ij x ix j θ + c j tx j u + d ij x ix j u = gu, θ, t u, u, j= u0,x = u 0 x, u t 0,x = u x, θ0,x = θ 0 x, for any fixed n +<s r<s n, ifu 0,θ 0 H s IR n H r x 0,ξ 0, u H s IR n H r x 0,ξ 0,andγt =t, xt,τt,ξt T IR n+ \ 00 t<t 0 is a null bicharacteristic for the hyperbolic operator appeared in.5 with x0 = x 0 and ξ0 = ξ 0, then the problem.5 has unique solutions: u C [0,T], H s H r xt,ξt C [0,T], H s H r xt,ξt,.6 θ C [0,T],H s H r xt,ξt. These two theorems indicated that the rule of singularity propagation for semilinear hyperbolic parabolic coupled systems is quite similar to that in pure hyperbolic case. Moreover, from Theorem. we can derive the following assertion, which improves the corresponding conclusion in [0, 3, 4]. Theorem.3. For the Cauchy problem.5, let C = t = gx} be the forward characteristic cone issuing from the origin, and denote by C + = t >gx}. If u 0,θ 0 H s IR n C IR n \ 0 and u H s IR n C IR n \ 0 for s>n/+, then the local solution u, θ of.5 obtained in Theorem. satisfies: s n/ ɛ u C [0,T], H loc C + C [0,T], H s n/ ɛ loc C +,.7 θ C [0,T], H s n/ ɛ loc C +, for any ɛ>0. The remainder of this paper is arranged as follows: In, we shall first prove Theorem. by using the hypoellipticity of heat equations to decouple the hyperbolic parabolic coupled system. and using the paralinearization to deal with nonlinear

4 Chen and Wang, Singularities in Hyperbolic Parabolic Coupled Systems 49 terms. The proof of Theorem. will be given in 3 by using a decoupling technique initiated in [5]. The method reduces the system to a weakly coupled one, then the corresponding hyperbolicity and hypoellipticity yield the required conclusion. Finally, at the end of 3, Theorem.3 will be derived as a consequence of Theorem... Propagation of singularities in interior domains In this section we are going to prove Theorem.. For readers convenience, let us first recall some basic facts in microlocal analysis. Let X be an open set in IR n and x, ξ be the coordinate of points on T X. For any m IR and 0 δ<ρ, a function ax, ξ defined on T X is called a symbol in the class Sρ,δ m, if for any compact subset K of X, and multi indices α, β IN n, the following estimate holds:. sup ξ α xax, β ξ C α,β,k + ξ m ρ α +δ β. x K Correspondingly, the associated operator. Ax, Du = e ixξ ax, ξûξ dξ, for all u C π n 0 X, is called pseudodifferential operator in the class Ψ m ρ,δ. Direct computation implies.3 t n b ij x i x j + j= c j x j Ψ,0 IR n+. Suppose that ψθ, η C IR n IR n \ 0 is nonnegative, homogeneous of order zero, and there are small 0 <ɛ <ɛ such that, when θ ɛ η,.4 ψθ, η = 0, when θ ɛ η, moreover, let Sη C IR n satisfy: 0, when η R,.5 Sη =, when η R. Then for any a, u S IR n, and χθ, η =ψθ, ηsη the operator T a defined by.6 T a ux = F ξ x χξ η, ηâξ ηûη dη is called a paraproduct operator. Correspondingly, assume that lx, ξ is homogeneous of order m for ξ IR n,and for any α IN n, Dξ αlx, ξ isinhs s>n/ with respect to x, then the following operator T l Op Σs n/ m is called a paradifferential operator of order m with symbol lx, ξ:.7 T l ux = F ξ x χξ η, ηˆlξ η, ηûη dη

5 50 Math. Nachr where ˆlθ, η represents the Fourier transform of lx, η with respect to x IR n. For paraproduct and paradifferential operators we have see [5]: Lemma.. For any t>n/, ifa H t IR n, then the operator T a : H s H s is bounded for any s IR. For any t>n/ and m Z, the operator T l Op Σ m t n/ : H s H s m is bounded for any s IR. 3 For any two paradifferential operators T l Op Σ m s n/ and T l Op Σ m s n/ with m,m IR and s> n +, we have [ ].8 Tl,T l Op Σ m +m s n/. The following result on paralinearization will be used later. Lemma.. Suppose that F y,...,y N is smooth with respect to its arguments, and each derivative of F is bounded on any compact set K IR n. Then, for any u i H s IR n s>n/, i =,...,N, we have.9 F u x,...,u N x = N j= T F y j u x,...,u N x uj x+rx where R H s n/ IR n. Return to the hyperbolic parabolic coupled system.. Denote by A the pseudodifferential operator with symbol iτ + n b jkξ j ξ k outside a given compact domain in τ, ξ space, then Ah satisfies n t b jk Ah = h + smooth x j x k for an arbitrary function h. Let n.0 w = θ + A we have. and t n k= c k u t x k + k,l= d kl u x k x l a ij x ix j u = F u, w, t u, u, w, Pu, Qu

6 Chen and Wang, Singularities in Hyperbolic Parabolic Coupled Systems 5. t b ij x ix j w = Gu, w, t u, u, P u where F and G are smooth with respect to their arguments, P = P k,p kl } n k,l=, Q = Q kl,q klm } n k,l,m= with P k = Ac k tx k, P kl = Ad kl x kx l, Q kl = xl Ac k tx k, Q klm = xl Ad km x kx m. Lemma.3. We have the following results:.3 xl t x lx m t b ij x ix j Ψ,0 IR n+, b ij x ix j Ψ 0,0 IR n+, P k,p kl Ψ IR n+, Q,0 kl,q klm Ψ IR n+,0. Proof. Since b jk is definitely positive, we have ξ j c which implies ξ j ξ k c iτ + n b jk ξ j ξ k b jk ξ j ξ k, Hence, direct calculation yields and ξ j iτ + ξ j ξ k iτ + τξ j iτ + which lead to the conclusion of the lemma. c iτ + τξ j c τ iτ + b jk ξ j ξ k b jk ξ j ξ k n S,0 IR n+, b jk ξ j ξ k S 0,0 IR n+ b jk ξ j ξ k S,0 IR n+, b jk ξ j ξ k. Proof of Theorem.. Paralinearizing the right side of., we obtain t a ij x ix j u T u T 3 u t T 4 u T 6 Pu T 7 Qu.4 = T w + T 5 w + R

7 5 Math. Nachr where T k k 7 are paraproduct operators derived from F with respect to corresponding arguments. For instance, T u = T F u,etc.. From assumptions u H s and θ H s /,weobtainw H s / for the function w given in.0 by using Lemma.3. Hence, when s n +,wehave.5 Then,. leads to.6 Meanwhile,.7 [ w = A t Gu, w, u t, u, P u H s. w H s. ] b ij xix j w + smooth. In view of., Lemma.3 and.5 we have w H s. Hence on the right side of.4, T w H s, T 5 w H s. Besides, due to the property of paralinearization the remainder R in.4 is in n+ s H H s when s n +. Therefore, by using the result on propagation of singularities for hyperbolic equations, we have.7 u H minr,s+/ Γ, where Γ is the bicharacteristics passing through t 0,x 0,τ 0,ξ 0. Together with.6, it follows.8 θ H s / H minr /,s Γ. If s<r s +/, we have proved what we want. If r>s+/,.7 and.8 mean that.9 u H s H s+/ Γ, θ H s / H s Γ which give rise to.0 Gu, w, u t, u, P u H s H s / Γ. Comparing.0 with.5 and the assumptions of Theorem., we can improve the microlocal smoothness of u, θ oforder along Γ. Continuously doing in the same way we conclude that u H s H r Γ, θ H s / H r / Γ if r s n+ holds.

8 Chen and Wang, Singularities in Hyperbolic Parabolic Coupled Systems Propagation of singularities for Cauchy problems This section is devoted to the proof of Theorem.. As an application of this microlocal regularity result, we will consider the propagation of local regularity for Cauchy problems of semilinear hyperbolic parabolic coupled systems to conclude Theorem.3 at the end of this section. At first, let us give two results for Cauchy problems of linear parabolic equations and hyperbolic equations. Lemma 3.. Consider the following Cauchy problem: Lw = gt, x, 3. w0,x = w 0 x, where L = t n b ij x ix j. Suppose that γt = xt,ξt T IR n \ 0:0 t<t} is a smooth curve, g L 0,T; H s IR n H r γt and w 0 H s IR n x0,ξ0. Then, the solution wt, x to 3. has the following regularity: H r 3. w L 0,T; H s+ IR n H r+ γt H 0,T; H s IR n H r γt. Proof.Let Ktbeaconicneighborhoodofγt inir n x IR n ξ, such that for 0 t<t, we have γt Kt, and w 0 H r K0, g L 0,T; H r Kt. Let χt, x, ξ,χ t, x, ξ S,0 0 IR n x IR n ξ be smooth in t [0,Twithχt, x, ξ γt, supp x,ξ χ t, x, ξ Kt andχ t, x, ξ on the support of χt, x, ξ. From 3. we know that χwt, x satisfies: Lχw = G, 3.3 where 3.4 with 3.5 χw t=0 H r, G = χt, x, D x g χ [χ, L]w + G G = χ [χ, L]w H 0,T IR n. From assumptions w 0 H s IR n andg L 0, T; H s IR n, we immediately have w L 0,T; H s+ IR n H 0,T; H s IR n which implies χ [χ, L]w L 0,T; H s IR n H 0,T; H s IR n, because [χ, L] only involves pseudodifferential operators of first order. Thus, we have 3.6 G L 0,T; H minr,s IR n.

9 54 Math. Nachr According to the standard theory of parabolic equations, we have χw L 0,T; H minr+,s+ IR n H 0,T; H minr,s IR n which implies 3.7 w L 0,T; H minr+,s+ γt H 0,T; H minr,s γt. If s r s +, we have proved 3., otherwise, we proved 3.8 w L 0,T; H s+ H s+ γt H 0,T; H s H s γt. Applying 3.8 to 3.4 and repeating the same argument, we can improve the regularity of w once again, and eventually attain the conclusion in the lemma. In the following discussion, we will always use the small and the capital letters to denote symbols and the associated pseudodifferential operators respectively. Lemma 3.. Let s>n/+, a C[0,T], H s IR n, T a be the paraproduct operator associated with a, p t, x, ξ S,0 IR n x IR n ξ be real symbol and smooth in t [0,T, P 0 t, x; D t,d x, Q 0 t, x; D t,d x be two pseudodifferential operators of order zero. Consider the following Cauchy problem: 3.9 Dt P t, x, D x u + P 0 T a Q 0 u = ft, x, u0,x = u 0 x. Assume that γt =t, xt; p t, xt,ξt,ξt is a null bicharacteristic for L = D t P t, x, D x, f L 0,T; H s H r xt,ξt, u 0 H s H r x0,ξ0 with n/+<s r s n/, then we have 3.0 u C [0,T], H s H r xt,ξt. Proof. Let K t Kt T IR n \0 be conic neighbourhoods of xt,ξt such that f L 0,T; H s H r Kt,u 0 H r K0. Similar to the proof of Theorem.. in [4], we can find bt, x, ξ S,0 0 IR n IR n ξ such that suppx,ξ bt, x, ξ Kt, bt, x, ξ onk t, and [D t P t, x, D x,bt, x, D x ] Ψ,0 IRn. From 3.9, we know that Bt, x, D x u satisfies 3. Dt P t, x, D x Bu = Bf +[D t P t, x, D x,b]u BP 0 T a Q 0 u, Bu0,x H r IR n. First, let us study the term BP 0 T a Q 0 u. Obviously, we have 3. BP 0 T a Q 0 = [B, P 0 ]T a Q 0 + P 0 BT a Q 0. Since the difference of Bt, x, D x with the paradifferential operator T b is a smoothing operator see [5], then we have 3.3 BT a Q 0 = T a [B, Q 0 ]+T a Q 0 B +[T b,t a ]Q 0 + R

10 Chen and Wang, Singularities in Hyperbolic Parabolic Coupled Systems 55 and 3.4 BP 0 T a Q 0 = [B, P 0 ]T a Q 0 + P 0 T a [B, Q 0 ]+P 0 T a Q 0 B + P 0 [T b,t a ]Q 0 + R where R and R are smoothing operators. Thus, from 3. we know that Bu satisfies the following problem: Dt P t, x, D x +P 0 T a Q 0 Bu = Bf + P u + R u, 3.5 where 3.6 Bu0,x H r, P = [D t P t, x, D x,b] [B, P 0 ]T a Q 0 P 0 T a [B, Q 0 ] P 0 [T b,t a ]Q 0, which maps H s IR n toh s+ IR n byvirtueof[t b,t a ] Op Σ s. By using n assumptions f L 0,T; H s IR n and u 0 H s IR n in the problem 3.9, we immediately obtain 3.7 and then 3.8 u C[0,T], H s IR n, P u C [0,T], H s+ IR n. Noting Bf L 0,T; H r IR n and applying the theory of hyperbolic equations in 3.5, we obtain: Bu C [0,T], H minr,s+ IR n which implies 3.9 u C [0,T], H minr,s+ xt,ξt. When r s +, the proof has been done, otherwise, 3.9 means 3.0 u C [0,T], H s H s+ xt,ξt. Choose b t, x, ξ S,0 0 IR n x IR n ξ, which has the same property as bt, x, ξ given above. Moreover, b t, x, ξ on a neighborhood K t ofγt and supp x,ξ b K t. Then, by using 3.0 we have [D t P t, x, D x,b ]u = [D t P t, x, D x,b ] Bu +[D t P t, x, D x,b ]Bu C [0,T], H s+ IR n. Similar to the analysis in we can derive 3. Dt P t, x, D x +P 0 T a Q 0 B u = B f + P u + R u, B u0,x H r, where P u C [0,T], H s+ IR n,and R is a smoothing operator. Therefore, by applying the theory of hyperbolic equations in 3. again, we obtain which implies B u C [0,T], H mins+,r IR n,

11 56 Math. Nachr u C [0,T], H mins+,r xt,ξt. Continuing this process we conclude the assertion 3.0 eventually. The above result can be generalized to the system case as follows: Lemma 3.3. Assume N N matrices P t, x, D x Ψ,0IR n, P 0 t, x; D t,d x, Q 0 t, x; D t,d x Ψ,0 0 IR n+, L = D t P t, x, D x is strictly hyperbolic with respect to t, andγt =t, xt; τt,ξt 0 t<t is a null bicharacteristic of L. At, x = aij with aij C[0,T], H s IR n is an N N matrix and T A is the corresponding matrix of paraproduct operators T aij. Consider the Cauchy problem 3.3 Dt P t, x, D x u + P 0 T A Q 0 u = F t, x, u0,x = u 0 x. If F L 0,T; H s H r xt,ξt and u 0 H s H r x0,ξ0 with n +<s r s n, then we have 3.4 u C [0,T], H s Hxt,ξt r. Proof. By applying a suitable transformation we can assume 3.5 P t, x, ξ = diag[λ t, x, ξ,...,λ N t, x, ξ] without loss of generality. Meanwhile, we assume γt is the null bicharacteristic of the operator D t λ t, x, D x. Denote by P II = diag[λ t, x, ξ,...,λ N t, x, ξ], and u = u,u II t with u II = u,...,u N t for any vector u. From assumptions F L 0,T; H s IR n and u 0 H s IR n, we immediately obtain 3.6 u C[0,T], H s IR n. By noting that D t P II t, x, D x is elliptic near γt, we have 3.7 u II C [0,T], H s+ xt,ξt. Substituting 3.7 into the problem of u in 3.3, and using Lemma 3., it follows: 3.8 u C [0,T], H s H minr,s+ xt,ξt. Continuing this process, we can obtain the conclusion of the lemma eventually. Pro o f of Theorem.. For convenience, let us first briefly recall the decoupling process introduced in [6] for the hyperbolic parabolic coupled system.. Denote by / / Λ a = a jk x jx k, Λ b = b jk x j x k and 3.9 u ± = t ± iλ a u.

12 Chen and Wang, Singularities in Hyperbolic Parabolic Coupled Systems 57 Then, from. we know that U =u +,u,θ t satisfies the following system: 3.30 t U + A t, x, D x U + A t, x, D x U = F U where F U = F U,F U,F 3 U t with F k U = F k Ψ0 U, Ψ 0 U, Ψ U 3 k =, and F 3 U =F 3 Ψ0 U, Ψ 0 U, Ψ 0 U 3 being smooth in their arguments, Ψ j = Ψ j,0 IRn, A t, x, D x = , 0 0 Λ b A t, x, D x = iλ a iλ a 0 C + i t D Λ a 0 C i t D Λ a with C =c,...,c n t, = x,..., xn andd =d jk. Similar to [5, 6], we choose K t, x, D x Ψ,0 IRn withsymbol 3.3 K t, x, ξ = ξ b p t, x, ξ p t, x, ξ 0 outside a compact set of ξ in IR n,where ξ b = + n b / jkξ j ξ k and p t, x, ξ = i n c j ξ j + ξ a d jk ξ j ξ k, j= p t, x, ξ = i n c j ξ j ξ a d jk ξ j ξ k. j= Then, direct computation leads to that 3.3 V = I + K t, x, D x u +,u,θ t satisfies the following Cauchy problem [ ] iλ a 0 t + 0 iλ a 3.33 t b ij x ix j V V V 3 = P 0 G V k 0,x = V k 0 x H s IR n H r x 0,ξ 0 k =,, V 3 0,x = V 3 0 x H s IR n H r x 0,ξ 0, = P 0 F Q 0 V,Q 0 V,Q V 3, Q 0 V,Q 0 V,Q 3 0 V 3 where P k 0 t, x, D x,q l 0 t, x, D x Ψ 0,0IR n andq t, x, D x Ψ,0IR n aresmooth in t.,

13 58 Math. Nachr By using the classical theory of parabolic equations and hyperbolic equations, we immediately obtain that 3.33 admits unique local solutions: 3.34 V,V C [0,T], H s IR n, V 3 L 0,T; H s+ IR n H 0,T; H s IR n. Employing Lemma. for the right hand side of the first two equations in 3.33, we know that V,V satisfies [ ] iλa 0 V t + P 0 T 0 iλ a V F Q 0 V + T F Q 0 V =P 0 T F 3 Q V 3 + R, 3.35 V k0,x = V k 0 x H s IR n H r x 0,ξ 0 k =,, where 3.36 R C [0,T], H s n IR n. By using 3.34 and Lemma 3.3 in the problem 3.35, we obtain 3.37 V,V C [0,T], H s H minr,s xt,ξt when n/+<s r<s n/. If s r s +,wehaveproved 3.38 V,V C [0,T], H s H r xt,ξt which implies 3.39 u C [0,T], H s Hxt,ξt r C [0,T], H s H r xt,ξt. From 3.38 and 3.34, we have P 0 G Q 0 V,Q 0 V,Q 3 0 V 3 C [0,T], H s H r xt,ξt. Using Lemma 3. for the problem of V 3 in 3.33, it follows 3.40 V 3 L 0,T; H s+ H r+ xt,ξt H 0,T; H s H r xt,ξt which implies 3.4 θ C [0,T], H s H r xt,ξt. When r>s+, we can continue this process and obtain the conclusion of the theorem eventually. Proof of Theorem.3. For any t 0,xt 0 C +,andξt 0 IR n \ 0, let τt 0 IR such that P 0 =t 0,xt 0,τt 0,ξt 0 is a characteristic point for the hyperbolic operator L = t n a ij x ix j appeared in.5. Denote by Γt =t, xt,τt,ξt} T IR n+ \ 0 the null bicharacteristic of L passing through P 0. Obviously, the projection in t, x space of Γt intersects with t =0} at x 0,whereu, θ is smooth by using Theorem. of [6].

14 Chen and Wang, Singularities in Hyperbolic Parabolic Coupled Systems 59 Thus, by applying Theorem. we obtain s n/ ɛ u C [0,T], H xt,ξt C [0,T], H s n/ ɛ xt,ξt θ C [0,T], H s n/ ɛ xt,ξt, for any ɛ>0, which is equivalent to the assertion.7 by using the arbitrariness of xt 0,ξt 0, because the above result holds obviously for the case that Γ = t, xt,τt,ξt} is a bicharacteristic of L when t 0,xt 0,τt 0,ξt 0 is not a characteristic point of L. It is not difficult to extend the result of Theorem.3 to the case when the initial data of u, θ have singularities on submanifolds of IR n. This result shows that for the Cauchy problem of hyperbolic parabolic coupled system., if the initial data have singularities at the origin for instance, then in the forward characteristic cone issuing from the origin the solutions u and θ will have the same order regularity in the x variable as in hyperbolic problems []. This conclusion improves the formal observations in [0, 3, 4] for semilinear systems in thermoelasticity. Acknowledgements The first author was partially supported by the National Science Foundation and Doctoral Programme Foundation of Institution High Education of China. The second author was partially supported by the National Science Foundation, Educational Ministry of China and Shanghai Qimingxing Foundation. References [] Beals, M., and Reed, M.: Propagation of Singularities for Hyperbolic Pseudodifferential Operators with Non Smooth Coefficients, Comm. Pure Appl. Math , [] Bony, J. M.: Calcul Symbolique et Propagation des Singularités pour les Équations aux Dérivées Partielles Non Linéaires, Ann. Scient. E. N. S. 4 98, [3] Chazarain, J., and Piriou, A.: Introduction to the Theory of Linear Partial Differential Equations, North Holland Publishing Company, 98 [4] Chen, Shuxing: Analysis of Singularities for Partial Differential Equations in Chinese, Shanghai Scientific & Technical Publishers, 998 [5] Chen, Shuxing, Qiu, Qingjiu and Li, Chengzheng: Introduction to Paradifferential Operators in Chinese, Sci. Publishers, 990 [6] Egorov, Y. V., and Schulze, B. W.: Pseudo Differential Operators, Singularities, Applications, Birkhäuser Verlag, 997 [7] Hörmander, L.: The Analysis of Partial Differential Equations, Vol. I IV, Springer Verlag, 985 [8] Nirenberg, L.: Lectures on Linear Partial Differential Equations, AMS, 973 [9] Racke, R.: Lectures on Nonlinear Evolution Equations. Initial Value Problems, Vieweg & Sohn, Braunschweig/Wiesbaden, 99 [0] Racke, R., and Wang, Y. G.: Propagation of Singularities in One Dimensional Thermoelasticity, J. Math. Anal. Appl , 6 47 [] Rauch, J.: Singularities of Solutions to Semilinear Wave Equations, J. Math. Pures et Appl ,

15 60 Math. Nachr [] Rauch, J., and Reed, M.: Propagation of Singularities for Semilinear Hyperbolic Equations in One Space Variable, Ann. of Math. 980, [3] Reissig, M., and Wang, Y. G.: Analysis of Weak Singularities of Solutions to d Thermoelasticity. In: Partial Differential Equations and Their Applications Chen Hua and L. Rodino eds., pp. 7 8, World Scientific Publishing Comp., Singapore, 999 [4] Reissig, M., and Wang, Y. G.: Propagation of Mild Singularities in Higher Dimensional Thermoelasticity, J. Math. Anal. Appl , [5] Taylor, M.: Reflection of Singularities of Solutions to Systems of Differential Equations, Comm. Pure Appl. Math , [6] Wang, Y.-G.: Microlocal Analysis in Nonlinear Thermoelasticity, preprint Center of Mathematical Sciences at Zhejiang University Hangzhou 3007 China E mail: sxchen@mail.hz.zj.cn Department of Mathematics Shanghai Jiao Tong University Shanghai China E mail: ygwang@mail.sjtu.edu.cn and Institute of Mathematics Fudan University Shanghai China E mail: sxchen@fudan.ac.cn