MARKO NEDELJKOV, DANIJELA RAJTER-ĆIRIĆ

Transcription

1 GENERALIZED UNIFORMLY CONTINUOUS SEMIGROUPS AND SEMILINEAR HYPERBOLIC SYSTEMS WITH REGULARIZED DERIVATIVES MARKO NEDELJKOV, DANIJELA RAJTER-ĆIRIĆ Abstract. We aopt the theory of uniformly continuous operator semigroups for use in Colombeau generalize function spaces. The main objective is to fin a unique solution to a class of semilinear hyperbolic systems with singularities. The iea of regularize erivatives is to transform unboune ifferential operators into boune, integral ones. This iea is use here to permit working with uniformly continuous operators. 1. Introuction Generalize uniformly continuous semigroups are introuce in [5] in orer to use the classical theory of semigroups in solving linear partial ifferential equations with singularities in Colombeau generalize function spaces. This paper is a continuation of that effort. Our aim is to solve some semilinear hyperbolic systems. Singularities in initial ata or in coefficients are represente by generalize functions, so the theory of semigroups can be use efficiently. Regularize erivatives (see [3]) are use for transforming ifferential into integral (boune) operators. This was the main tool permitting a use of uniformly continuous semigroups for ifferential operators. Generalize functions use here are base on Sobolev spaces instea of istributions, an the use of Banach space methos is available. The main results of the paper are assertions concerning existence an uniqueness of solutions to a class of semilinear hyperbolic systems. Cauchy problem for semilinear first-orer hyperbolic system with smooth coefficients an Lipschitz nonlinearity is solve in the usual Colombeau generalize function space in [6]. In this paper, besies giving a solution to the problem above, we also investigate its relation with the solution from [6] in the case when the system has smooth coefficients with appropriate type of initial ata. The efinitions of basic generalize function spaces suitable for work with generalize uniformly continuous semigroups are given in the secon section. Let us note that similar proceure can be performe for generalize C -semigroups (see [4], for example). In the thir section, the efinitions given previously by the authors in [5] are repeate, an followe with some assertions about uniformly continuous semigroups an their infinitesimal generators. The existence an uniqueness theorem is given an prove in the fourth section. The last, fifth, section is evote to examination of the connection between the results from the previous section an the result in [6] for a class of systems escribe above an the L 2 -association of the solutions is establishe Mathematics Subject Classification. 35L45,47F6. Key wors an phrases. generalize functions, semigroups of operators, regularize erivatives. 1

2 2 M. NEDELJKOV, D. RAJTER-ĆIRIĆ 2. Basic spaces Let (E, ) be a Banach space an L(E) the space of all linear continuous mappings from E into E. SE M ([, ) : L(E)) is the space of nets S ε : [, ) L(E), ε (, 1) ifferentiable with respect to t [, ), with the property that for every T > there exist N N, M > an ε (, 1) such that (1) sup α t αs ε(t) Mε N, ε < ε, α {, 1}. L(E) t [,T) It is an algebra with respect to composition of operators. SN ([, ) : L(E)) is the space of nets N ε : [, ) L(E), ε (, 1) ifferentiable with respect to t [, ), with the property that for every T > an a R there exist M > an ε (, 1) such that (2) sup α t α N ε(t) Mε a, ε < ε, α {, 1}. L(E) t [,T) It is an ieal of SE M. Thus, we efine Colombeau space as SG([, ) : L(E)) = SE M ([, ) : L(E)) SN ([, ) : L(E)). Elements of SG([, ) : L(E)) will be enote by S = [S ε ] where S ε is a representative of the class. Similarly, one can efine following spaces: SE M (E) is the space of nets of linear continuous mappings A ε : E E, ε (, 1) with the property that there exist constants N N, M > an ε (, 1) such that (3) A ε L(E) Mε N, ε < ε. SN(E) is the space of nets of linear continuous mappings A ε : E E, ε (, 1) with the property that for every a R there exist M > an ε > such that (4) A ε L(E) Mε a, ε < ε. Now, Colombeau space is efine by SG(E) = SE M(E) SN(E). Elements of SG(E) will be enote by A = [A ε ] where A ε is a representative of the class. Let H m (R n ) be the usual Sobolev space H m,2 (R n ). In the paper we shall use the following spaces: E M ([, ) : H m (R n )) is the space of nets G ε : [, ) R n C, G ε (t, ) H m (R n ), for every t [, ),

3 GENERALIZED UNIFORMLY CONTINUOUS SEMIGROUPS... 3 with the property that for every T > there exist C >, N N an ε > such that sup t α G ε (t, ) H m Cε N, α {, 1}, ε < ε. t [,T) N ([, ) : H m (R n )) is the space of nets G ε E M ([, ) : H m (R n )) with the property that for every T > an a R there exist C > an ε > such that Define the quotient space sup t α G ε(t, ) H m Cε a, α {, 1}, ε < ε. t [,T) G ([, ) : H m (R n )) = E M ([, ) : H m (R n )) N ([, ) : H m (R n )). In similar way, by omitting t-variable, one can efine spaces E M (H m (R n )), N (H m (R n )) an G (H m (R n )). Note that the above spaces are not algebras with respect to multiplication (which is the case for the original efinition of generalize function spaces). 3. Generalize semigroups Definition 1. S SG([, ) : L(E)) is calle a uniformly continuous Colombeau semigroup if it has a representative S ε which is a uniformly continuous semigroup for every ε small enough, i.e. (i) S ε () = I. (ii) S ε (t 1 t 2 ) = S ε (t 1 )S ε (t 2 ),for every t 1, t 2. (iii) lim t S ε (t) I =. The following proposition is prove in [5]. Proposition 1. Let S ε an S ε be representatives of a uniformly continuous Colombeau semigroup S, with infinitesimal generators A ε, an Ãε, respectively, for ε small enough. Then A ε Ãε SN(E). Definition 2. A SG(E) is calle the infinitesimal generator of a uniformly continuous Colombeau semigroup S SG([, ) : L(E)) if A ε is the infinitesimal generator of the representative S ε, for every ε small enough. Definition 3. Let h ε be a positive net satisfying h ε ε 1. It is sai that A SG(E) is of h ε -type if it has a representative A ε such that (5) A ε L(E) = O(h ε ), ε. G G ([, ) : H m (R n )) is sai to be of h ε -type if it has a representative G ε such that G ε H m = O(h ε ), ε. In the classical theory semigroups of boune linear operators the following theorem hols. Theorem 1. ([7], Theorem 1.2) A linear operator A is the infinitesimal generator of a uniformly continuous semigroup if an only if A is a boune linear operator. The following lemma hols for generalize operators. Lemma 1. Every A SG(E) of h ε -type, where h ε C log 1 ε, is the infinitesimal generator of some T SG([, ) : L(E)).

4 4 M. NEDELJKOV, D. RAJTER-ĆIRIĆ Proof. Accoring to Theorem 1 every boune operator A ε is the infinitesimal generator of the uniformly continuous semigroup (ta T ε (t) = e taε ε ) n =. n! Let us show that T ε SE M ([, ) : L(E)). We have that ta ε n 1 T ε (t) M n! n! (th ε) n = Me thε. n= Since h ε C log 1 ε we have sup t [,T) T ε (t) Mε TC, for ε small enough. Since t T ε(t) = A ε, for every ε small enough, we have t T ε(t) = A ε C log 1 ε Cε 1, for every such ε, i.e., T ε SE M ([, ) : L(E)). Thus, the proof is complete. Proposition 2. Let A be the infinitesimal generator of a uniformly continuous Colombeau semigroup S, an B be the infinitesimal generator of a uniformly continuous Colombeau semigroup T. If A = B, then S = T. Proof. Let N ε = A ε B ε SN(E). We have t (S ε(t) T ε (t))x = A ε (S ε (t) T ε (t))x N ε T ε (t)x. Duhamel principle an S ε () = T ε () = I imply (S ε (t) T ε (t))x = n= n= S ε (t s)n ε T ε (s)xs. One can easily show that S ε (t) T ε (t) Cε a, for every real a, because N ε SN(E). The same bouns for t-erivative of S ε (t) T ε (t) can be obtaine by a successive ifferentiation of the above term. 4. Semilinear hyperbolic systems with regularize erivatives Definition 4. Let α N n. Regularize α-th erivative of a generalize function G is efine by the representative (6) α hε G ε = G ε α φ hε, where φ hε (x) = h n εφ(xh ε ), φ(y) = φ 1 (y 1 )... φ 1 (y n ), φ 1 C (R), φ 1 (ξ), φ is symmetric function with φ 1 (ξ)ξ = 1. For efinition an some basic properties of regularize erivatives we refer to [3] an [8]. In the sequel, we shall use the symbol hε for the first orer erivative in one-imensional case. Theorem 2. Let u ( G ( H 1 (R) )) ( n an hε be a net satisfying h ε = O (log 1/ε) 1/2), as ε. Let function f(x, t, u(t)) = (f 1 (x, t, u(t)),..., f n (x, t, u(t))) be globally Lipschitz with respect to x an u with boune secon orer erivative with respect to u an f(x, t, ) =. Also, suppose that x f(x, t, u(t)) is globally Lipschitz function with respect to u.

5 GENERALIZED UNIFORMLY CONTINUOUS SEMIGROUPS... 5 (7) Let operator A SG (( H 1 (R) ) n) be represente by the nets of operators A ε : ( H 1 (R) ) n ( H 1 (R) ) n, A ε u = λ ε hε u, where λ ε = iag(λ 1 ε,..., λn ε ) ( H 1, (R) ) n n an λε L (Ê) = O ( (log 1/ε) 1/2). Then there exists the generalize function u ( G([, ) : (H 1 (R)) ) n, which uniquely solves the Cauchy problem (8) t u(t) = Au(t) f(, t, u(t)), u() = u. That solution is represente by (9) u i ε (t) = S ε(t)u i ε S ε (t s)f i (, s, u i (s))s, i = 1,...,n, where S SG ( [, ) : ( H 1 (R) ) n) is the uniformly continuous Colombeau semigroup generate by A. Proof. Take u (H 1 (R)) n. Then A ε u L 2 λ i ε hε u i L 2 an λ i ε L ui hε φ hε L 2 λ i ε L ui L 2 hε φ hε L 1 C log 1/ε x (A ε u) L 2 u i L 2 x (λ i ε ) hε u i L 2 C 1 (log 1/ε) 1/2 C 2 (log 1/ε) 1/2 u i L 2 λ i ε x( hε u i ) L 2 x λ i ε L hε u i L 2 λ i ε L hε ( x u i ) L 2 x λ i ε L hε L 1 u i L 2 C log 1 ε λ i ε L hε L 1 ( x u i ) L 2 ( u i L 2 x u i ) L 2. Since u ( H 1 (R) ) n it follows that operator A is of log 1 ε-type. Therefore, it is the infinitesimal generator of a uniformly continuous Colombeau semigroup S ( SG([, ) : (H 1 (R)) ) n. By well known classical result (see [7]) we know that (9) represents a solution to (8). Let us show that this solution is an element of ( G([, ) : (H 1 (R)) ) n. We have u ε (t) L 2 S ε (t)u ε L 2 S ε (t s)f(, s, u ε (s)) L 2s

6 6 M. NEDELJKOV, D. RAJTER-ĆIRIĆ an t u ε(t) A ε (t)u ε L 2 f(x, t, u ε (t)) L 2. L 2 Since f(x, t, u) is globally Lipschitz with respect to u an f(x, t, ) =, one gets f(, t, u(t)) (H 1 (R)) n if u(t) (H 1 (R)) n. Thus, the moerate bouns for u ε (t) L 2 an t u ε(t) L 2 immeiately follow. After a ifferentiation of (9) with respect to x we have x u ε (t) L 2 S ε (t) x u ε L 2 S ε (t s)( u f(x, s, u ε (s)) x u ε (s)) L 2s S ε (t s) x f(x, s, u ε (s))u ε (s) L 2s S ε (t) x u ε L 2 S ε (t s) u f L x u ε (s) L 2s S ε (t s) x f L u ε (s) L 2s. Since f is Lipschitz with respect to u an x the moerate boun for x u ε L 2 follows. Thus, u ( G([, ) : (H 1 (R)) ) n. To show that the solution is unique in ( G([, ) : (H 1 (R n )) ) n, suppose that there exist two solutions u an v to equation (8) an set w ε = u ε v ε. This ifference satisfies (1) t w ε(t) = A ε (t)w ε (t) f(, t, u ε (t)) f(, t, v ε (t)) N ε (t), w ε () = w ε, where N ε (t) N([, ) : (H 1 (R)) n ) an w ε N((H 1 (R) n ). Then (11) an wε i (t) =S ε(t)wε i S ε (t s)(f i (, s, u i ε (s)) f i(, s, vε i (s)))s w ε (t) L 2 S ε (t)w ε L 2 S ε (t s)n i ε (s)s, S ε (t s)n ε (s) L 2s. Using the fact that uε f L C < an S ε (t s)(f(, s, u ε (s)) f(, s, v ε (s))) L 2s f(x, s, u ε (s)) f(x, s, v ε (s)) L 2 u f L u ε (s) v ε (s) L 2 we obtain N-boun for w ε (t) L 2. Equation (1) implies t w ε(t) A ε w ε (t) L 2 f(x, s, u ε (s)) f(x, s, v ε (s)) L 2 N ε (t) L 2. L 2 Since, as we showe in previous step, w ε (t) H 1 has the N-boun an N ε (t) N([, T) : (H 1 (R)) n ) we obtain that t w ε(t) L 2 has the N-boun, too.

7 GENERALIZED UNIFORMLY CONTINUOUS SEMIGROUPS... 7 Finally, But x w ε L 2 S ε (t) x w ε L 2 x N ε L 2 S ε (t s)(( u f(, s, u ε (s)) x u ε (s) u f(, s, v ε (s)) x v ε (s))) L 2s S ε (t s)(( x f(, s, u ε (s))u ε (s) x f(, s, v ε (s))v ε (s))) L 2s. ( u f(, s, u ε (s)) x u ε (s) u f(, s, v ε (s)) x v ε (s) L 2 ( u f(, s, u ε (s)) x u ε (s) u f(, s, v ε (s)) x u ε (s) L 2 u f(, s, v ε (s)) x u ε (s) u f(, s, v ε (s)) x v ε (s) L 2 x u ε (s) L 2 f uu (, s, ỹ(s)) L u ε (s) v ε (s) L an u f(, s, v ε (s)) L x u ε (s) x v ε (s) L 2 C 1 x u ε (s) L 2 w ε (s) H 1 C 2 x w ε (s) L 2, x f(, s, u ε (s))u ε (s) x f(, s, v ε (s))v ε (s) L 2 x f(, s, u ε (s))u ε (s) x f(, s, u ε (s))v ε (s) L 2 x f(, s, u ε (s))v ε (s) x f(, s, v ε (s))v ε (s) L 2 x f(, s, u s ) L u ε (s) v ε (s) L 2 v ε (s) L 2 uf(, s, ỹ(s)) L u ε (s) v ε (s) L 2. for some functions ỹ(s) (H 1 (R)) n. Since w ε ( N((H 1 (R)) ) n an Nε ( N([, T) : (H 1 (R)) ) n, Gronwall inequality gives the N-boun for x w ε (t) L 2. Thus, w ε := u ε v ε ( N([, ) : (H 1 (R)) ) n, i.e. the solution is unique. Definition 5. The solution u of the problem (8) introuce in Theorem 2 is calle generalize solution of the equation t u(t) = λ xu(t) f(, t, u(t)) with regularize erivatives. 5. Relations with the previous results In [6] Oberguggenberger gives the following theorem: Theorem 3. ([6],Theorem 16.1) Let λ(x, t) be a smooth, real value iagonal matrix. Let f be smooth an u f(x, t, u) be polynomially boune together with all erivatives, uniformly for (x, t) varying in compact subsets of R 2. Let u u f(x, t, u) be globally boune, uniformly with respect to (x, t) varying in compact subsets of R 2. Then, for given u G(R), the problem (12) ( t λ(x, t) x )u = f(x, t, u), u t= = u has a unique solution u G(R 2 ). Let K T = K {(x, t) : t [, T]}, K is a region boune by the external characteristics emerging from the en points of K = [A, B] suppu ε. The following Lemma will be useful.

8 8 M. NEDELJKOV, D. RAJTER-ĆIRIĆ Lemma 2. ([6], Lemma 16.2) Let v be a smooth solution to the linear problem ( t λ(x, t) x )v(x, t) = f(x, t)v(x, t) g(x, t), v(x, ) = b(x), where f is a smooth matrix an g, b are smooth. Then v L (K T) b L (K ) T g L (K T ) exp(nt f L (K T)). In orer to compare solutions to regularize an non-regularize equations we nee some aitional a priori bouns for solution to (12). Lemma 3. Let u be a solution to (12), where all erivatives of f with respect to u i, i = 1,..., n, an x of orer less or equal to three are boune when (x, t) belongs to some compact set. Suppose that i u L (K ) b ε, i =, 1, 2,where b ε 1. Then u L (K T ) constb ε, x u L (K T ) constb ε, an 2 xu L (K T) constb 2 ε. Proof. We ifferentiate system in (12) three times with respect to x an use the Lemma 2. The bouns for u an x u irectly follows from the proof of Theorem 3 given in [6], so we shall give the estimate for the secon erivative. After ifferentiating the system two times with respect to x we have ( t λ x )w =( u f 2 x λ)w 2 xf ( 2 ufv u x f 2 xλ)v, where v = x u an w = x 2 u. Now, the above lemma gives the estimate w L (K T ) w L (K ) 2 xf L (K T) exp(nt u f 2 x λ L (K T)) T( 2 ufv 2 L (K T) u x fv L (K T ) 2 xλv L (K T )). Using the assumptions an the bouns for u an v one can see that there exists a constant C such that xu 2 L (K T) Cb 2 ε. Theorem 4. Let u be a solution of (12), with λ = λ(x) from Theorem 3. Let the initial ata be compactly supporte an satisfy (13) i x u ε L g M/3 ε, i =, 1, 2, for some M > an g ε = log h ε, h ε < log 1 ε. Then, u an the solution v to (8) with the initial ata v() = u given in Theorem 2 are L 2 - associate. Proof. In Lemma 3 is prove that (14) i xu ε L g M ε, as ε, i =, 1, 2, where L -norm can taken over all Ê [, T], since supp(u ε ) K T. Also, the assumption on compact supports for the initial ata an finite propagation spee ensures that u(, t) L 2 (Ê), an (14) hols for L 2 -norm, too for every t T. The following equations are satisfie t u ε (t) λ x u ε (t) = f(, t, u ε (t)) t v ε (t) λ x v ε φ hε (t) = f(, t, v ε (t)).

9 GENERALIZED UNIFORMLY CONTINUOUS SEMIGROUPS... 9 (15) Difference of these two equalities gives t (u ε (t) v ε (t)) λ x (u ε (t) v ε (t)) φ hε =λ( x u ε (t) x u ε φ hε (t)) f(, t, u ε (t)) f(, t, v ε (t)), for every t < T. Let us fix t for a moment. Then x u ε (, t) φ hε x u ε (, t) L (Ê) = sup φ hε (y)( x u ε (x, t) x u ε (x y, t))y x Ê y h ε ( 1 ) = sup φ hε (y) x 2 x Ê u ε(x σy, t)σ yy y h ε Cg M ε h 2 ε, ε, since g ε = log h ε, an (14) hols. Since K T {(x, τ) : τ = t} is a boune set with a finite Lebesgue measure, say l(t), one can immeiately see that (16) λ(t)( i xu ε (, t) φ hε i xu ε (, t)) L 2 (Ê) const(λ) l(t) i xu ε (, t) φ hε i xu ε (, t) L (Ê), ε, i =, 1 by the previous estimate on L -norm. Put w ε = u ε v ε. One can write (15) in the form (17) t w ε(t) = A ε (t)w ε (t) f(, t, u ε (t)) f(, t, v ε (t)) N ε (t), w ε () =, where A ε u = λ hε u an N ε (t) = λ(t)( x u ε (, t) φ hε x u ε (, t)). Inequality (16) means that N ε H 1 as ε. Then, each component of w ε satisfies (18) an wε i (t) = S ε (t s)(f i (, s, u i ε (s)) fi (, s, vε i (s)))s w ε (t) L 2 S ε (t s)nε i (s)s, i = 1,...,n, S ε (t s)(f(, s, u ε (s)) f(, s, v ε (s))) L 2s S ε (t s)n ε (s) L 2s. If u H 1, then A ε u L 2 M u H 1, where M oes not epen on ε. Since S ε (t) = e taε = (ta ε) n n= n!, one can see that S ε u L 2 e tm u H 1. Particularly, S ε (t s)n ε (s) L 2, as ε. Using the fact that u f L C < an the estimate f(x, s, u ε (s)) f(x, s, v ε (s)) L 2 u f L u ε (s) v ε (s) L 2 C w ε (s), use of Gronwall lemma gives sup t (,T) w ε (t) L 2, as ε. Immeiate consequence of the above theorem is the following corollary. Corollary 1. If there exists a L classical solution to problem (12) with L -initial ata then it can be regularize so that it is L -associate with the solution to system (8), when A given by (7).

10 1 M. NEDELJKOV, D. RAJTER-ĆIRIĆ Acknowlegment The authors are grateful to Michael Oberguggenberger for his help uring the preparation of the paper. References [1] Biagioni W A an Oberguggenberger M (1992), Generalize solutions to the Korteweg-e Vries an the regularize long-wave equations, SIAM J. Math. Anal. 23, [2] Colombeau J F (1985), Elementary Introuction in New Generalize Functions, North Hollan, Amsteram. [3] Colombeau JF, Heibig A an Oberguggenberger M (1996), Generalize solutions to partial ifferential equations of evolution type, Acta Appl. Math. 45,2, [4] Neeljkov M, Pilipović S an Rajter-Ćirić D (25), Heat equation with singular potential an singular ata, Proc. Roy. Soc. Einburgh Sect. A 135 no. 4, [5] Neeljkov M, an Rajter D (24), Semigroups an PDEs with perturbations, in: Delcroix A, Hasler M, Marti J-A, Valmorin V, Nonlinear Algebraic Analysis an Applications, Proc. of ICGF 2, [6] Oberguggenberger M (1992), Multiplication of Distributions an Applications to Partial Differential Equations, Pitman Res. Not. Math. 259, Longman Sci. Techn., Essex. [7] Pazy A (1983), Semigroups of Linear Operators an Applications to Partial Differential Equations, Springer- Verlag, New York Inc. [8] Rosinger E (1978), Distributions an Nonlinear Partial Differential Equations, Springer Lectures Notes in Mathematics, vol.684, Springer-Verlag, New York Inc. Department for Mathematics an Informatics, University of Novi Sa, Trg D. Obraovića 4, 21 Novi Sa, Serbia