Scientiae Mathematicae Japonicae Online, Vol. 5, (2001), Ryo Ikehata Λ and Tokio Matsuyama y

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1 Scientiae Mathematicae Japonicae Online, Vol. 5, (2), L 2 -BEHAVIOUR OF SOLUTIONS TO THE LINEAR HEAT AND WAVE EQUATIONS IN EXTERIOR DOMAINS Ryo Ikehata Λ and Tokio Matsuyama y Received November 2, 2; revised April 9, 2 Abstract. Uniform L 2 -decay of solutions for the linear heat equations will be given. In order to derive the L 2 -decay of solutions, the modified method of Morawetz [6] will be used and we shall show thatthe L 2 norm of solutions decays like O(t )as t! + for some kinds of weighted initial data. Furthermore, by the same argument, one can also derive the L 2 -bound and L 2 -decay for weak solutions of the linear free and dissipative wave equations, respectively.. Introduction. Let Ω be an exterior domain in R N (N 2) with a compact C 2 Without loss of generality, wemay assume =2 Ω. In this paper, first we are concerned with the initial-boundary value problem (.) u t (t; x) u(t; x) =; (t; x) 2 (; ) Ω; (.2) u(;x)=u (x); x 2 Ω; (.3) =; t 2 (; ): Throughout this paper, k kmeans the usual L 2 (Ω)-norm. Furthermore, we set Z (f;g)= f(x)g(x) dx: Ω First of all, we shall state the well-posedness to the problem (.)-(.3) (c.f. Cazenave and Haraux [2, Proposition 3.5.3]). Proposition.. For each u 2 H (Ω), there exists a unique solution u(t; x) in the class C([; ); H (Ω)) C ((; ); L 2 (Ω)) C((; ); H 2 (Ω)) to the problem (.)-(.3) satisfying (.4) 2 kru(t; )k2 + ku t (s; )k 2 ds = 2 kru k 2 on [; ), (.5) (u t (t; );u(t; )) + kru(t; )k 2 = on [; ). 2 Mathematics Subject Classification. 35L5, 35L2. Key words and phrases. Heat equation, wave equation, L 2 decay, unbounded domains. Λ The research of the first author was in part supported by Grant-in-Aid for Scientific Research (No.7468), Ministry of Education, Science and Culture. y The research of the second author was in part supported by Grant-in-Aid for Scientific Research (C)(2)(No.6423), Japan Society for the Promotion of Science.

2 8 RYO IKEHATA AND TOKIO MATSUYAMA For the equation (.), as will be seen in the proof of Corollary.3 below, we can easily derive the decay of energy kru(t; )k 2. But then, it seems unknown at least for the exterior problem whether L 2 -norm of the solution u(t; x) to the problem (.)-(.3) decays or not. Since we treat the continuous orbit fu(t; )g t in the phase space H (Ω), it is quite natural to investigate L 2 -decay of solutions. On the other hand, in the case when Ω = R N, although we can use the explicit formula through the fundamental solution for the heat (see Racke [, Lemma.6]), our result will make sense (at least) of alternative proof. The first purpose of this paper is to derive a certain decay rate of L 2 -norm of the solution to the problem (.)-(.3) with the weighted initial data in an exterior domain". Our argument is based on the (modified) method of Morawetz [6] (for another use of Morawetz' method, see also Ikehata and Matsuyama [4] and Nakao [8]), and the so called Hardy inequality (see Dan and Shibata [3]). Before introducing our main theorem, we must define a function d(x) as follows: (.6) d(x) = ( jxj; N 3; jxj log(bjxj); N =2; where B> is a constant such that inf jxj 2 x2ω B main result reads as follows. >. Then, based on Proposition. our Theorem.2. Let N 2 and assume that the initial data u belongs to H (Ω) and further satisfies kd( )u k < +. Then, the solution u(t; x) to the problem (.)-(.3) satisfies ( + t)ku(t; )k 2» C Λ kd( )u k 2 for all t, where C Λ > is a certain constant. Corollary.3. Under the same assumptions as in Theorem.2, one has ( + t)ku(t; )k 2 H» C(ku k 2 H + kd( )u k 2 ) for all t with some constant C>, where k k H denotes the usual H (Ω)-norm. The second purpose of this paper is to derive L 2 -bound for the free wave equation: (.7) u tt (t; x) u(t; x) =; (t; x) 2 (; ) Ω; (.8) u(;x)=u (x); u t (;x)=u (x); x 2 Ω; (.9) =; t 2 (; ): Generally speaking, in studying the local energy decay of solutions to the problem (.7)- (.9) it seems essential to derive the L 2 -bound for solutions (see [6] and [8]). Further, since kru(t; )k is bounded for all t, from the point of view of the dynamical system, it is important toknow whether the L 2 -norm of the solution to the problem (.7)-(.9) is bounded or not for the initial data fu ;u g2h (Ω) L 2 (Ω). In this occasion, we shall proceed our argument based on the energy identity.

3 L 2 -BEHAVIOUR OF SOLUTIONS TO THE LINEAR HEAT 9 Proposition.4. For each fu ;u g2h (Ω) L 2 (Ω), there exists a unique solution u 2 C([; ); H (Ω)) C ([; ); L 2 (Ω)) to the problem (.7)-(.9) such that 2 kru(t; )k2 + 2 ku t(t; )k 2 = 2 kru k ku k 2 : As in the results of Morawetz [6] and Nakao [8], we can see that the boundedness of ku(t; )k plays an essential role in deriving the local energy decay of solutions to the wave equations with the compactly supported initial data fu ;u g.however, the following result implies that the L 2 -boundedness holds true without any compactness of the support of the initial data. If, in particular, we impose the initial data as u =, then we can obtain the L 2 -bound for the solution without any condition of compact support for the initial data u. Generalizing these observations, we have the following result. Theorem.5. Let N 2 and assume the initial data fu ;u g belongs to H (Ω) L 2 (Ω) and further satisfies kd( )u k < +. Then, the solution u(t; x) to the problem (.7)-(.9) satisfies for all t with a certain constant C>. ku(t; )k 2»ku k 2 + Ckd( )u k 2 In [3] Dan and Shibata have investigated the asymptotic behaviour of the dissipative wave equations in an exterior domain and proved that if the initial data fu ;u g has a compact support, then the local energy E loc (t) and L 2 -norm of solutions decay like C(ku k 2 H + ku k 2 )( + t) N as t! +. The proof in [3] is based on a spectral analysis and the Poincaré type inequality (see [3, Lemma 2.3]), and the dissipative term u t plays an essential role. Our final result is concerned with the L 2 -decay of solutions for the following dissipative wave equation: (.) (.) u tt (t; x) u(t; x)+u t (t; x) =; (t; x) 2 (; ) Ω; u(;x)=u (x); u t (;x)=u (x); x 2 Ω; (.2) =; t 2 (; ): In [4], we have derived the uniform L 2 -decay of a solution to the problem (.)-(.2) for an appropriately selected initial data. Compared with (especially) the Cauchy problem in R N for the dissipative wave equation (.) treated by Kawashima et al. [5], our result is much stronger than [5] in the sense that if we apply the result of [5] to our problem, we can only derive thel 2 -bound for the solution with the same initial data as in [4]. On the contrary, if the initial data fu ;u g has the weight d(x), we can derive thel 2 -decay of the solution to the problem (.)-(.2). To state the result, we need the well-posedness of the problem (.)-(.2). Proposition.6. For each fu ;u g2h (Ω) L 2 (Ω), there exists a unique solution u 2 C([; ); H (Ω)) C ([; ); L 2 (Ω)) to the problem (.)-(.2) such that (.3) 2 kru(t; )k2 + 2 ku t(t; )k 2 + ku t (s; )k 2 ds = 2 kru k ku k 2 ; (.4) d dt (u t(t; );u(t; )) + kru(t; )k 2 +(u t (t; );u(t; )) = ku t (t; )k 2 :

4 2 RYO IKEHATA AND TOKIO MATSUYAMA Our final result reads as follows. Theorem.7. Let N 2 and assume the initial data fu ;u g2h (Ω) L2 (Ω) further satisfies kd( )(u + u )k < +. Then, the solution u(t; x) to the problem (.)-(.2) satisfies (.5) ( + t)ku(t; )k 2» Cfku k 2 H + ku k 2 + kd( )(u + u )k 2 g for all t with a constant C> independent of t 2 [; ). Remark.8. If u + u =in L 2 (Ω), then this result coincides with that of [4]. On the other hand, if supp u [ supp u is compact, then kd( )(u + u )k < + obviously follows. Thus, in the framework of compactly supported initial data, the decay estimate (.5) is also valid. 2. Proof of Theorem.2. In this section, we shall prove our main Theorem.2. First, we shall prepare the Hardy inequality (see [3]). Lemma 2.. For each u 2 H (Ω), it holds that Z ju(x)j 2 d(x) dx» 2 Ckruk2 ; where d(x) is the function defined in(.6). Ω Lemma 2.2. For each u 2 H (Ω), the solution u(t; x) to the problem (.)-(.3) satisfies for all t>. ( + t)ku(t; )k 2»ku k 2 + ku(s; )k 2 ds Proof. Multiplying the both sides of the identity (.5) by + t and integrating it over [;t], one has ( + s) d 2 ds ku(s; )k2 ds + Then integrating by parts, we see ( + s) 2 ( + s)kru(s; )k 2 ds =: d ds ku(s; )k2 ds = +t 2 ku(t; )k2 2 ku k 2 2 which completes the proof of Lemma 2.2. ku(s; )k 2 ds; The following lemma plays an essential role in our argument and Theorem.2 is an immediate consequence of Lemmas 2.2 and 2.3 below. Lemma 2.3. Under the assumptions as in Theorem.2, one has for all t> with some constant C Λ >. ku(s; )k 2 ds» C Λ kd( )u k 2

5 L 2 -BEHAVIOUR OF SOLUTIONS TO THE LINEAR HEAT 2 Proof. As in [4], set w(t; x) = u(s; x) ds: Then, since k u(t; )k = ku t (t; )k 2L loc (; ) holds (c.f. [2, Proposition 3.5.3]), it follows from the identity (.4) that w 2 C ([; ); H (Ω)) C((; ); H 2 (Ω)) satisfies w t (t; ) w(t; ) =u on (; ); w(;x)=; x 2 Ω; =; t 2 (; ); so that kw t (t; )k 2 + d (2.) 2 dt krw(t; )k2 =(u ;w t (t; )): Hence, noting w t = u and integrating the both sides of the identity (2.), one has (2.2) ku(s; )k 2 ds + 2 krw(t; )k2 = (u ;w t (s; )) ds = (u ;w(t; )) (u ;w(; )) = (u ;w(t; )): Here we note that (2.3) j(u ;w(t; ))j» ψ Z Ω ju (x)j 2 d(x) 2 dx! =2 ψ Z Then, combining Lemma 2. with the inequality (2.3), we have (2.4) j(u ;w(t; ))j» Ckd( )u kkrw(t; )k Ω jw(t; x)j 2 d(x) 2» C 2" kd( )u k 2 + C" 2 krw(t; )k2 ; dx! =2 where C > is a constant appearing in Lemma 2. and "> is an arbitrarily fixed real number. Thus, it follows from (2.2) and (2.4) that ku(s; )k 2 ds + 2 ( C")krw(t; )k2» C 2" kd( )u k 2 : Taking "> so small that <"<, one obtains the desired inequality. C Secondly, let us prove Corollary.3. The essential part of the proof of Corollary.3 lies in deriving the decay of kru(t; )k. Proof of Corollary :3. Since the function t 7! kru(t; )k 2 is monotone decreasing because of the energy identity (.4), we have : (2.5) d dt t)kru(t; )k2 g = kru(t; )k 2 +(+t) d kru(t; )k2 dt» kru(t; )k 2 :

6 22 RYO IKEHATA AND TOKIO MATSUYAMA Integrating the inequality (2.5) over [;t], we see (2.6) ( + t)kru(t; )k 2»kru k 2 + On the other hand, it follows from the identity (.5) that (2.7) Hence, we see from (2.6) and (2.7) that kru(s; )k 2 ds: kru(s; )k 2 ds + 2 ku(t; )k2 = 2 ku k 2 : (2.8) ( + t)kru(t; )k 2»kru k ku k 2»ku k 2 H : Combining (2.8) with the estimate in Theorem.2, we have the desired result. Finally, let us remark the Cauchy problem in R N (N 3) for the heat equation (.): (2.9) u t (t; x) u(t; x) =; (t; x) 2 (; ) R N ; (2.) u(;x)=u (x); x 2 R N : Then we can obtain the similar result to the problem (.)-(.3) by using the Hardy inequality replaced by the following one. Lemma 2.4. (Benci and Cerami []) Let N 3. For each u 2 H (R N ), it holds that Z ju(x)j 2 dx» Ckruk2 2 ( + jxj) with a certain constant C>. RN Our result concerning the Cauchy problem reads as follows. Theorem 2.5. Let N 3 and assume that the initial data u 2 H (R N ) further satisfies k( + jxj)u k < +. Then, the solution u(t; x) to the problem (2.9)-(2.) in the sense of Proposition. replaced byω=r N satisfies ( + t)ku(t; )k 2» C Λ k( + jxj)u k 2 for all t>, where C Λ > is a certain constant. 3. Proof of Theorems.5 and.7. In this section, we shall prove Theorems.5 and.7. This is done by modifying the arguments in the previous section. To begin with, let us prove Theorem.5. Proof of Theorem :5. Set w(t; x) = u(s; x) ds: Then, w 2 C ([; ); H (Ω)) C 2 ([; ); L 2 (Ω)) satisfies (3.) w tt (t; x) w(t; x) =u ; (t; x) 2 (; ) Ω;

7 L 2 -BEHAVIOUR OF SOLUTIONS TO THE LINEAR HEAT 23 (3.2) w(;x)=; w t (;x)=u (x); x 2 Ω; (3.3) =; t 2 (; ) and (3.4) Since 2 kw t(t; )k krw(t; )k2 = 2 ku k 2 + (u ;w t (s; )) ds = it follows from the identity (3.4) that (3.5) d (u ;w t (s; )) ds: dt (u ;w(s; )) ds =(w(t; );u ); 2 kw t(t; )k krw(t; )k2 = 2 ku k 2 +(u ;w(t; )): On the other hand, by the similar argument to the proof of the inequality (2.4) we see that (3.6) j(u ;w(t; ))j» C " kd( )u k 2 + C 2" krw(t; )k2 2 with some constants C i > (i =; 2), where we have just used the Hardy inequality in Lemma 2.. Thus, (3.4)-(3.6) imply 2 kw t(t; )k ( C 2")krw(t; )k 2» 2 ku k 2 + C " kd( )u k 2 : Taking "> so small above, one has ku(t; )k 2 = kw t (t; )k 2»ku k 2 +2C " kd( )u k 2 ; which completes the proof of Theorem.5. Finally, we shall prove Theorem.7. Indeed, setting w(t; x) = u(s; x) ds; we see that w 2 C ([; ); H (Ω)) C 2 ([; ); L 2 (Ω)) satisfies w tt (t; x) w(t; x)+w t (t; x) =u + u ; (t; x) 2 (; ) Ω; w(;x)=; w t (;x)=u (x); x 2 Ω; and =; t 2 (; ) (3.7) 2 kw t(t; )k krw(t; )k2 + = 2 ku k 2 + (u + u ;w t (s; )) ds: kw t (s; )k 2 ds

8 24 RYO IKEHATA AND TOKIO MATSUYAMA Since (u + u ;w t (s; )) ds = d ds (u + u ;w(s; )) ds =(w(t; );u + u ); in a completely analogous way to the proof of Theorem.5 we see from the identity (3.7) that which implies 2 kw t(t; )k krw(t; )k2 + kw t (s; )k 2 ds» 2 ku k 2 + C 3 kd( )(u + u )k 2 + C 4" 2 krw(t; )k2 ; 2 ku(t; )k2 + 2 ( C 4")krw(t; )k 2 + ku(s; )k 2 ds» 2 ku k 2 + C 3 kd( )(u + u )k 2 for "> with some constants C i > (i =3; 4). Taking "> so small, we arrive atthe following estimate. Lemma 3.. Under the same assumptions as in Theorem.7, it holds that for all t>. 2 ku(t; )k2 + ku(s; )k 2 ds» 2 ku k 2 + C 3 kd( )(u + u )k 2 On the other hand, we have the following estimate which has been proved in [4, Lemma 3.]. For the convenience of the readers we shall prove it briefly. Lemma 3.2. Under the assumptions as in Theorem.7, one has (3.8) ( + t)ku(t; )k 2» C 5 fku k 2 H + ku k 2 g + ku(s; )k 2 ds for all t>, where C 5 > is a constant. Proof. Multiplying the both sides of (.4) by t and integrating it over [;t], one has Z s d t ds (u s(s; );u(s; )) ds sku s (s; )k 2 ds + Integrating by parts, we see that skru(s; )k 2 ds + 2 s d ds ku(s; )k2 ds =: and s d ds (u s(s; );u(s; )) ds = t(u t (t; );u(t; )) 2 ku(t; )k2 + 2 ku k 2 s d ds ku(s; )k2 ds = tku(t; )k 2 ku(s; )k 2 ds:

9 L 2 -BEHAVIOUR OF SOLUTIONS TO THE LINEAR HEAT 25 Thus, we get (3.9) t(u t (t; );u(t; )) + 2 ku k 2 + = 2 ku(t; )k2 + sku s (s; )k 2 ds + 2 skru(s; )k 2 ds + t ku(t; )k2 2 ku(s; )k 2 ds: On the other hand, it is easy to prove (cf. [4] or [5]) (3.) ku(t; )k 2» I ; ( + t)e(t)» E() + fi; E(s) ds» fi; where we setfi = 2 E() + 8 I with I =2ku k 2 +(u ;u )+8E() and E(t) = 2 fkru(t; )k2 + ku t (t; )k 2 g: Therefore, integrating by parts with respect to t, one gets fi E(s) ds = te(t) se (s) ds = te(t)+ sku s (s; )k 2 ds; from which it follows that (3.) sku s (s; )k 2 ds» fi; where we have just used the relation E (t) = ku t (t; )k 2 (see (.3)). Note that (3.2) (u t (t; );u(t; ))»ku t (t; )kku(t; )k» 4 ku(t; )k2 + ku t (t; )k 2 : From the estimates (3.9), (3.) and (3.2) it follows that 2 ku(t; )k2 + which implies that tku t (t; )k sku s (s; )k 2 ds + 2 ku(s; )k 2 ds + ku(s; )k 2 ds t(u t (t; );u(t; )) + t ku(t; )k2 2 t 4 ku(t; )k2 tku t (t; )k 2 ; sku s (s; )k 2 ds + 2 ku(t; )k2 t 4 ku(t; )k2 : Noting that ( + t)e(t)» fi + E() implies tku t (t; )k 2» 2(fi + E()) and combining it with the estimates (3.), we arrive at the desired inequality (3.8). Therefore, Theorem.7 is an immediate consequence of Lemmas 3. and 3.2. Remark 3.3. Using Lemma 2.4 instead of Lemma 2., we can obtain the similar result associated with Theorem 2.5 for the Cauchy problem in R N to the equation (.7) or (.). So we shall omit the details.

10 26 RYO IKEHATA AND TOKIO MATSUYAMA References N+2 [] V. Benci and G. Cerami, Existence ofpositive solutions of the equation u + a(x)u = un 2 in R N,J.Func. Anal. 88 (99), 9 7. [2] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford Lecture Series in Mathematics and its Applications 3, Oxford Science Publ., 998. [3] W. Dan and Y. Shibata, On a local energy decay of solutions of a dissipative wave equation, Funkcial. Ekvac. 38 (995), [4] R. Ikehata and T. Matsuyama, Remarks on the behaviour of solutions to the linear wave equations in unbounded domains, Proc. School of Science, Tokai Univ. 36 (2), 3. [5] S. Kawashima, M. Nakao and K. Ono, On the decay property of solutions to the Cauchy problem of the semilinear wave equation with a dissipative term, J. Math. Soc. Japan 47 (995), [6] C. Morawetz, The limiting amplitude principle, Comm. Pure Appl. Math. 5 (962), [7] C. Morawetz, Exponential decay of solutions of the wave equations, Comm. Pure Appl. Math. 9 (966), [8] M. Nakao, Stabilization of local energy in an exterior domain for the wave equation with a localized dissipation, J. Diff. Eq. 48 (998), [9] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences 44, Springer, Berlin, 983. [] R. Racke, Lectures on Nonlinear Evolution Equations: initial value problems, Vieweg, 992. [] H. Tanabe, Equations of Evolution, Pitman, London, 979. Λ Department of Mathematics, Graduate School of Education, Hiroshima University Higashi-Hiroshima , Japan y Department of Mathematics, Faculty of Science, Tokai University Hiratsuka , Japan

Two dimensional exterior mixed problem for semilinear damped wave equations

J. Math. Anal. Appl. 31 (25) 366 377 www.elsevier.com/locate/jmaa Two dimensional exterior mixed problem for semilinear damped wave equations Ryo Ikehata 1 Department of Mathematics, Graduate School of