EXISTENCE OF SOLUTIONS TO THE ROSENAU AND BENJAMIN-BONA-MAHONY EQUATION IN DOMAINS WITH MOVING BOUNDARY

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1 Electronic Journl of Differentil Equtions, Vol. 24(24), No. 35, pp. 2. ISSN: URL: or ftp ejde.mth.txstte.edu (login: ftp) EXISTENCE OF SOLUTIONS TO THE ROSENAU AND BENJAMIN-BONA-MAHONY EQUATION IN DOMAINS WITH MOVING BOUNDARY RIOCO K. BARRETO, CRUZ S. Q. DE CALDAS, PEDRO GAMBOA, & JUAN LIMACO Abstrct. In this rticle, we prove the existence of solutions for hyperbolic eqution known s the the Rosenu nd Benjmin-Bon-Mhony equtions. We study incresing, decresing, nd mixed non-cylindricl domins. Our min tools re the Glerkin method, multiplier techniques, nd energy estimtes.. Introduction To investigte the dynmics of certin discrete systems, Philip Rosenu obtined the eqution u t + (u + u 2 ) x + u xxxxt =. The study of this eqution in cylindricl domins ws done by Mi Ai Prk [3], who proved the existence nd uniqueness of locl nd globl solutions. The Rosenu eqution could be seen s vrint of Benjmin-Bon-Mhony (BBM) eqution, u t + (u + u 2 ) x u xxt =, which models long wves in non liner dispersive system. In [3], Benjmin-Bon-Mhony proved the existence nd uniqueness of globl solutions for the BBM eqution in cylindricl domins. In this work, we study the existence of solutions for the Rosenu nd BBM equtions for incresing, decresing, nd mixed noncylindricl domins. We introduce the following nottion: Let α, β, = β α, be C 2 -functions of rel vrible, such tht α(t) < β(t), for ll t. We represent the noncylindricl domin by = {(x, t) R 2 : α(t) < x < β(t), t }, nd its lterl boundry by Σ = t T {α(t), β(t)} {t}. In the present work we investigte the following two equtions: u t + (u + u 2 ) x + u xxxxt = u(x, t) = u x (x, t) = u(x, ) = u (x) for (x, t) Σ for (x, t) Σ in for α() < x < β() (.) 99 Mthemtics Subject Clssifiction. 35M, 35B3. Key words nd phrses. Benjmin-Bon-Mhony eqution, Rosenu eqution, noncylindricl domins. c 24 Texs Stte University - Sn Mrcos. Submitted April 28, 23. Published Mrch, 24.

2 2 R. K. BARRETO, C. DE CALDAS, P. GAMBOA, & J. LIMACO EJDE-24/35 nd u t + (u + u 2 ) x u xxt = u(x, t) = for (x, t) Σ in (.2) u(x, ) = u (x) in Ω. This pper is orgnized s follows: The next section is devoted to the existence nd uniqueness of solution for (.) nd (.2), stisfying the hypothesis (H) α (t) nd β (t) for t [, T ]. Note tht this hypothesis implies decreses in the sense tht if t 2 > t, then the projection of [α(t 2 ), β(t 2 )] in the subspce t = is contined in the projection of [α(t ), β(t )] in the sme subspce. In the third section of this rticle, we study the existence of solutions for (.) nd (.2) stisfying the hypothesis (H2) α (t) nd β (t) for t [, T ]. Anlogously hypothesis (H2) implies tht increses In the lst section of this rticle, we study the (.) nd (.2), stisfying the hypothesis: (H3) = 2 where is incresing nd 2 is decresing. In the following, by Ω we represent the intervl ], [, Ω t nd Ω denote the intervls ]α(t), β(t)[ nd ]α(), β()[ respectively. We denote, s usul, by (.,.), respectively the sclr product nd norm in L 2 (Ω). In the sequel, w m,x denotes w m x, nlogously w m,xx = 2 w m x, w 2 m,xt = 2 w m t x, etc. 2. Solutions on decresing domins In this section we study the existence nd uniqueness for (.) nd (.2) stisfying the hypothesis (H). Let (t) = β(t) α(t) >, for ll t. Then < x α(t) (t) <, for ll t [, T ]. With the chnge of vrible u(x, t) = v(y, t) where y = x α(t) (t), for ll t [, T ], problem (.) is trnsformed into v t + (v + v2 ) y + 4 v yyyyt (α + y) v y 4 5 v yyyy (α + y) 5 v yyyyy = in Ω ], T [ v(, t) = v(, t) = in ], T [ Also problem (.2) is trnsformed into v y (, t) = v y (, t) = in ], T [ v(y, ) = v (y) in Ω. v t + (v + v2 ) y 2 v yyt (α + y) v y v yy + (α + y) 3 v yyy = in Ω ], T [ v(, t) = v(, t) = in ], T [ v(y, ) = v (y) in Ω. Under these conditions, we estblish the following existence results. (2.) (2.2)

3 EJDE-24/35 ROSENAU AND BENJAMIN-BONA-MAHONY EQUATIONS 3 Theorem 2.. For ech u H 2 (Ω ) H 4 (Ω ), there exists unique function u : R, stisfying u C ([, T ]; H 2 (Ω t )) C(, T ; H 3 (Ω t ) H 2 (Ω t )) nd u t φ dx dt + (u + u 2 ) x φ dx dt + u xxt φ xx dx dt =, for ll φ L 2 (, T ; H 2 (Ω t )), u(x, ) = u (x), for ll x Ω. Theorem 2.2. For ech u H (Ω ) H 2 (Ω ), there exists unique function u : R, stisfying u L (, T ; H (Ω t )), u t L (, T ; H (Ω t )) nd u t φ dx dt + (u + u 2 ) x φ dx dt + u xt φ x dx dt =, for ll φ L 2 (, T ; H (Ω t )), u(x, ) = u (x), for ll x Ω. To prove Theorem 2., we need the following lemms. Lemm 2.3. For ech v H 2 (Ω) H 4 (Ω), there exists unique function v : Ω ], T [ R, stisfying v L (, T ; H 2 (Ω) H 4 (Ω)), v t L (, T ; H 2 (Ω)), nd [v t ψ + (v + v2 ) y ψ + 4 v yytψ yy Ω ],T [ (α + y) v y ψ 4 5 v yyψ yy + ( (α + y) 5 ψ) y v yyyy ]dy dt =, for ll ψ L 2 (, T ; H 2 (Ω)), v(y, ) = v (y), for ll y Ω. Lemm 2.4. For ech f C([, T ]; H 2 (Ω)), there exists unique function z : Ω ], T [ R, stisfying z C([, T ]; H 2 (Ω)) nd z z = f Lemm 2.5. For ech v H 2 (Ω) H 4 (Ω), there exists unique function v : Ω ], T [ R, stisfying v C ([, T ]; H 2 (Ω)) C([, T ]; H 3 (Ω) H 2 (Ω)) nd [v t ψ + (v + v2 ) y ψ + 4 v yytψ yy Ω ],T [ (α + y) v y ψ 4 5 v yyψ yy + ( (α + y)ψ 5 ) y v yyyy ] dy dt =, for ll ψ L 2 (, T ; H 2 (Ω)); v(y, ) = v (y), for ll y Ω. Lemm 2.6. For ech v H (Ω) H 2 (Ω), there exists unique function v : Ω ], T [ R, stisfying v L (, T ; H (Ω) H 2 (Ω)), v t L (, T ; H (Ω)) nd [v t ψ + (v + v2 ) y ψ 2 v yytψ Ω ],T [ (α + y) v y ψ v yyψ ( (α + y)ψ 3 ) y v yy ]dy dt =, for ll ψ L 2 (, T ; H (Ω)); v(y, ) = v (y), for ll y Ω. In this rticle, we prove Theorem 2. nd Lemms 2.3, 2.4, 2.5 which correspond to Rosenu Eqution. However, we omit the proofs of Theorem 2.2 nd Lemm 2.6 which correspond to Benjmin Bon-Mhony Eqution; becuse the proofs re mde in similr wy.

4 4 R. K. BARRETO, C. DE CALDAS, P. GAMBOA, & J. LIMACO EJDE-24/35 Proof of Lemm 2.3. Let (w i ) i N be the specil bsis of H 2 (Ω), such tht w i,yyyy = λ i w i, in Ω w i () = w i () = w i,y () = w i,y () =, i N. We denote by V m the subspce generted by w,..., w m. Our strting point is to construct the Glerkin pproximtion of the solution v m V m, which is given by the solution of the pproximte eqution (v m,t, w) + ( (v m + v 2 m) y, w) + 4 (v m,yyyyt, w) 4 5 (v m,yyyy, w) +( (α + y) 5 v m,yyyyy, w) + ( (α + y) v m,y, w) = v m () = vm v in H 4 (Ω) for ll w V m (2.3) First Estimte. Tking w = v m (t) in (V), we hve: d dt ( v m(t) v m,yy(t) 2 ) + 5 v m,yy(t) 2 + v m(t) 2 + α 5 v2 m,yy() β 5 v2 m,yy() = Integrting this eqution over [, t] nd using hypothesis (H), we obtin v m (t) v m,yy(t) 2 v () v yy t 4 v m,yy(s) 2 ds t v m (s) 2 ds where,, 2 re positive constnts, such tht (t), nd (2.4) (2.5) 2 = mx t T (t), = mx t T α (t). This implies v m (t) v m,yy(t) 2 C + C t [ v m (s) v m,yy(s) 2 ] ds (2.6) where C, C,... denote positive constnts. Applying Gronwll inequlity, we hve the first estimte v m (t) v m,yy(t) 2 C 2 (2.7) Second Estimte Tking w = v m,yyyy (t) in the first eqution of (2.3), we obtin d 2 dt [ v m,yy(t) v m,yyyy(t) 2 ] v m,yyyy(t) 2 + v m,y(t) v m,yyyy (t) + (α + 2 ) v m,y (t) 2 v m,yyyy (t). (2.8)

5 EJDE-24/35 ROSENAU AND BENJAMIN-BONA-MAHONY EQUATIONS 5 From (2.7) nd using Schwrtz s inequlity nd Poincre s inequlity, we obtin d 2 dt [ v m,yy(t) v m,yyyy(t) 2 ] v m,yyyy(t) v m,yyyy(t) C 3 v m,yy (t) 2 + C 3 v m,yy (t) 2 v m,yyyy (t) 2 ( ) 4 v m,yyyy(t) C 3C 2 + (α + 2 ) C 3 C 2 v m,yyyy (t) 2 ( ) 4 v m,yyyy(t) C 3C C v m,yyyy(t) 2. Let c 4 = (α+2) C 3 C 2 nd let c 5 = 2 2 C 3 C C4. 2 Then d dt [ v m,yy(t) v m,yyyy(t) 2 ] C 5 + ( ) 4 v m,yyyy(t) 2. (2.9) Integrting this inequlity over [, t] nd pplying Gronwll inequlity, we obtin v m,yy (t) v m,yyyy(t) 2 C 6 (2.) Third Estimte Tking w = v m,t (t) in (V), we hve v m,t (t) v m,yyt(t) 2 = ((v m(t) + v 2 m(t)) y, v m,t (t)) (v m,yyyy(t), v m,t (t)) ( (α + y) 5 v m,yyyy (t), v m,yt (t)) + ( (α + y) 5 v m,y (t), v m,t (t)). From (2.7), (2.), nd (2.), we obtin (2.) v m,t (t) v m,yyt(t) 2 C 7. (2.2) These three estimtes permit to pss to the limit in the pproximte eqution nd we obtin wek solution v in the sense of Lemm 2.3. The uniqueness of solution nd the verifiction of initil dt re showed by the stndrd rguments. Proof of Lemm 2.4. To prove the existence we consider two stges: First stge f C([, T ]; H 2 (Ω)). Let (w i ) i N be the specil bsis of H 2 (Ω) used in the proof of Lemm 2.3. Consider the sequence (f n ), such tht f n (t) = n i= (f(t), w i)w i. It is cler tht f n f strongly in C([, T ]; H 2 (Ω)). The pproximted solution z m (t) to z z = f is z m (t) = m i= g im(t)w i, where g im re solutions of the pproximted system (z m (t), w i ) + 4 ( z m(t), w i ) = (f m (t), w i ), i =,... m (2.3) A priori estimte. Let us prove tht (z m ) is Cuchy sequence in C([, T ]; H 2 (Ω)). In fct, let m nd n be positive integer such tht m > n nd g in (t) = for

6 6 R. K. BARRETO, C. DE CALDAS, P. GAMBOA, & J. LIMACO EJDE-24/35 n i m. Then z m nd z n re solutions of (2.3) in V m. Consider the Cuchy difference z m z n. We hve from (2.3) tht, for i =,..., m, (z m (t) z n (t), w i ) + 4 ( (z m(t) z n (t)), w i ) = (f m (t) f n (t), w i ), (2.4) Tking w i = z m (t) z n (t) in (2.4), using the Cuchy-Schwrz inequlity nd the equivlent norms, we obtin z m z n C([,T ];H 2 (Ω)) c f m f n C([,T ];H 2 (Ω)) Then z n z strongly in C([, T ]; H 2 (Ω)). Therefore, tking limit in (2.3), we obtin z + 2 z = f in C([, T ]; H 2 (Ω)). 4 Second stge f C([, T ]; H 2 (Ω)). By density, there exists sequence (f n ), f n C([, T ]; H 2 (Ω)), such tht f n f strongly in C([, T ]; H 2 (Ω)). Using the first stge we hve tht there exist sequence (z n ), z n C([, T ]; H 2 (Ω)) such tht z n z n = f n in C([, T ]; H 2 (Ω)). (2.5) Consider the Cuchy difference z m z n, m > n. We obtin z m z n (z m z n ) = f m f n in C([, T ]; H 2 (Ω)). (2.6) Composing (2.6) with z m z n C([, T ]; H 2 (Ω)). nd integrting in Ω, we hve z m z n C([,T ];H 2 (Ω)) c f m f n C([,T ];H 2 (Ω)); therefore, z n z strongly in C([, T ]; H 2 (Ω)) nd tking limit in (2.5) we hve z + 2 z = f in C([, T ]; H 2 (Ω)). The uniqueness of the solutions is showed by 4 the stndrd rguments. Proof of Lemm 2.5. From Lemm 2.4 we cn define the opertor B(t) = (I + 2 ) from C([, T ]; H 2 (Ω)) to C([, T ]; H 2 4 (Ω)) by B(t)f = z with f C([, T ]; H 2 (Ω)) where z is solution of z + 2 z = f. Note tht B(t) is 4 liner nd continuous. By Lemm 2.3, v L 2 (, T ; H 4 (Ω) H 2 (Ω)) nd v t L 2 (, T ; H 2 (Ω)). From Lions-Mgenes, Theorems 3. nd 9.6, chpter I [], we conclude tht v C([, T ]; H 2 (Ω) H 3 (Ω)) (2.7) On the other hnd, from the trnsformed problem (2.), we obtin (I )v t = f, (2.8) where, f = (v + v2 ) y + (α + y) v y v yyyy + ( (α + y) 5 )v yyyyy From (2.7), we conclude tht f C([, T ]; H 2 (Ω)). Then from (2.8) we hve tht v t = B(t)f, where v t C([, T ]; H 2 (Ω)) nd we get the required result. The proof of Theorem 2. follows immeditely from Lemm 2.5 nd the Chnge of Vrible Theorem. Therefore, we omit it. Observe tht Theorem 2. in cylindricl domin, hs the regulrity u C ([, T ]; H 2 (Ω)) C([, T ]; H 4 (Ω) H 2 (Ω)). In fct, s we consider n dditionl estimte with w i = u m,txxxx, in the Glerkin pproximtion, tht llows us

7 EJDE-24/35 ROSENAU AND BENJAMIN-BONA-MAHONY EQUATIONS 7 to obtin the regulrity u t L 2 (, T ; H 4 (Ω) H 2 (Ω)). However, in our noncylindricl domin, this is not possible since the trnsformed problem (III), contins term v yyyyy tht does not llow us to use the estimte with w i = v m,tyyyy in the Glerkin pproximtion. 3. Solutions on incresing domins In this section we study the existence of solution for the systems (.) nd (.2) stisfying the hypothesis (H2). We use the Penliztion Method given by Lions []. Let Q =], b[ ], T [ be the cylinder such tht Q. We define the function M : Q R, by { in Q \ M(x, t) = in To show the existence result we will use the following Lemm. Lemm 3.. If u, u t L 2 (, T ; L 2 (, b)), then t Proof. We hve (Mu(s), u t (s))ds 2 M(t)u(t) 2 L 2 (,b) 2 M()u() 2 L 2 (,b). t (Mu(s), u t (s))ds = 2 = 2 t b [,t] [,b] M(u 2 (s)) t dξ ds M(u 2 (s)) t dξ ds. From Fubini s Theorem nd reclling the definition of M, it follows tht t = 2 = 2 (Mu(s), u t (s))ds α(t) t β(t) β (x) β() α(t) β(t) β() [u 2 (s)] t ds dξ + 2 α() α (x) α(t) [u 2 (s)] t ds dξ + 2 [u 2 (t, ξ) u 2 (, ξ)] dξ + 2 α(t) 2 [ u 2 (t, ξ) dξ + + = 2 α() α(t) b b β(t) α() α(t) [u 2 (β (x), ) u 2 (, ξ)] dξ + 2 u 2 (, ξ) dξ + b β(t) β(t) β() M(t, ξ) u 2 (t, ξ) dξ 2 u 2 (t, ξ) dξ ( u 2 (, ξ) dξ + b t [u 2 (s)] t ds dξ [u 2 (s)] t ds dξ [u 2 (α (x), ) u 2 (, ξ)] dξ b β(t) α(t) b β(t) M(, ξ)u 2 (, ξ) dξ = 2 M(t)u(t) 2 L 2 (,b) 2 M()u() 2 L 2 (,b) [u 2 (t, ξ) u 2 (, ξ)] dξ u 2 (, ξ) dξ u 2 (, ξ) dξ)]

8 8 R. K. BARRETO, C. DE CALDAS, P. GAMBOA, & J. LIMACO EJDE-24/35 which completes the proof. The existence of solution for (.) nd (.2), stisfying the hypothesis (H2), is estblished in the next theorems. Theorem 3.2. For ech u H 2 (Ω ), there exists function u : R, stisfying u L (, T ; H 2 (Ω t )), u t L (, T ; H 2 (Ω t )) nd u t φ dx dt + (u + u 2 ) x φ dx dt + u xxt φ xx dx dt = (3.) for ll φ L 2 (, T ; H 2 (Ω t )); u(x, ) = u (x) Theorem 3.3. For ech u H (Ω ), there exists function u : R, stisfying u L (, T ; H (Ω t )), u t L (, T ; H (Ω t )) nd u t φ dx dt + (u + u 2 ) x φ dx dt + u xt φ x dx dt =, for ll φ L 2 (, T ; H (Ω t )); u(x, ) = u (x) Proof of Theorem 3.2. To prove this result we use the penliztion method. For ech ɛ > we consider the problem u ɛ,t + (u ɛ + u 2 ɛ) x + u ɛ,xxxxt + ɛ Mu ɛ,t ɛ (Mu ɛ,xt) x = u ɛ (, t) = u ɛ (b, t) = u ɛ,x (, t) = u ɛ,x (b, t) = in ], T [ u ɛ (x, ) = ũ (x) in ], b[ in Q (3.2) Let {w i } i N be bsis of H 2 (, b), such tht w = ũ. We denote by V m = [w,..., w m ] the subspce of H 2 (, b), generted by ũ, w 2,..., w m. We seek u ɛm (t) in V m solution to the pproximte problem (u ɛm,t, w) + ((u ɛm + u 2 ɛm) x, w) + (u ɛm,xxxxt, w) + ɛ (Mu ɛm,t, w) ɛ ((Mu ɛm,xt) x, w) = for ll w V m (3.3) u ɛm () = u (x) ũ in H 2 (, b) First Estimte. Tking w = u ɛm in (3.3) nd pplying Lemm 3., we obtin u ɛm (t) 2 + u ɛm,xx (t) 2 + ɛ M(t)u ɛm(t) 2 + ɛ M(t)u ɛm,x(t) 2 c 8, (3.4) where denotes the norm in L 2 (, b). Second Estimte. Tking w = u ɛm,t (t) in (3.3) nd using (3.4) we hve u ɛm,t (t) 2 + u ɛm,xxt (t) 2 + ɛ (M(t)u ɛm,t(t), u ɛm,t (t)) ɛ (M(t)u ɛm,xt(t), u ɛm,xt (t)) c u ɛm,t(t) 2 (3.5) From where we obtin u ɛm,t (s) 2 + u ɛm,xxt (s) 2 + ɛ M(t)u ɛm,t(t) 2 + ɛ M(t)u ɛm,xt(t) 2 c 9

9 EJDE-24/35 ROSENAU AND BENJAMIN-BONA-MAHONY EQUATIONS 9 From the estimtes bove, we pss to the limit in the pproximte eqution, nd we obtin tht u ɛ is solution of the penlized problem T b u ɛ,t v dx dt + + ɛ T b T b (u ɛ + u 2 ɛ) x v dx dt + Mu ɛ,t v dx dt + ɛ T b T b u ɛ,xxt v xx dx dt Mu ɛ,xt v x dx dt = (3.6) for ll v L 2 (, T ; H 2 (, b)). From (3.4), (3.5) nd the Bnch-Steinhuss Theorem, we pss to the limit s ɛ in (3.6) nd we obtin (3.). Regulrity. From the first estimte, we hve ɛ t (Mu ɛm,t (s), u ɛm (s)) ds c On the other hnd, from Lemm 3. we obtin t (Mu ɛm,t (s), u ɛm (s)) ds ɛ 2ɛ M(t)u ɛm(t) 2. Then M(t)u ɛm (t) 2 2cɛ. Thus T b M(t) u2 ɛm(t) dx dt 2cɛT or T b M(t)u ɛ (t) 2 dx dt lim inf T b M(t)u ɛ (t) 2 dx dt 2cɛT Then Mu ɛ in L 2 (, T ; L 2 (, b)). On the other hnd, Mu ɛ Mu in L 2 (, T ; L 2 (, b)) nd Mu ɛm Mu ɛ in L 2 (, T ; L 2 (, b)). So, we conclude tht Mu =.e. in Q or u = in Q \. Anlogously, pplying the Lemm 3. to u ɛm,xt insted of u ɛm,t, we obtin: u x = in Q\. Since u L (, T ; H (, b)) nd u t L (, T ; H (, b)), then u C([, T ]; H (, b)). Therefore, u(t) H (, b) for ll t nd u = in ], b[\]α(t), β(t)[. From where u(t) H (α(t), β(t)), for ll t. Thus u L (, T ; H (Ω t )). Anlogously, u x L (, T ; H (Ω t )). From these two sttements, we hve tht u L (, T ; H 2 (Ω t )). From the second estimte, T b M(t)u ɛm,t (t) 2 dx dt + T b M(t)u ɛm,xt (t) 2 dx dt 2cɛT. By similr rguments, we obtin tht u t L (, T ; H 2 (Ω t )), which prove the regulrity of the solution. The proof of Theorem 3.3 is similr to the proof of Theorem 3.2 nd is ommitted. Remrk 3.4. Theorems 2., 2.2, 3.2 nd 3.3 re invrible by trnsltion. In fct, the prticulr problem u t + (u + u 2 ) x + u xxxxt = in Ω ]T, T [ u(x, t) = u x (x, t) = in Σ in Σ u(x, T ) = u (x) in Ω T, with the chnge of vrible u(x, t) = u(x, t T ), cn be trnsformed into problem of type (.).

10 R. K. BARRETO, C. DE CALDAS, P. GAMBOA, & J. LIMACO EJDE-24/35 4. Solutions on mixed domins Here we nlyze the cse when is mixed domin; i.e., = B B 2 where B = {(x, t) : < t T } nd B 2 = {(x, t) : T < t < T }, where B is decresing stisfying (H), nd B 2 is incresing stisfying (H2). We define i by i = int(b i ), i =, 2. i.e., = {(x, t) : < t < T } nd 2 = {(x, t) ; T < t < T }. To find solution to (.) in = 2, we consider the following two cses: () Solution on : For ech u H 2 (Ω ) H 4 (Ω ), by Theorem 2., there exist u solution of u,t + (u + u 2 ) x + u,xxxxt = in u (x, t) = in Σ u,x (x, t) = in Σ (4.) u (x, ) = u o (x) in Ω stisfying u L (, T ; H 2 (Ω t )), u t L (, T ; H 2 (Ω t )); therefore u is in C([, T ]; H 2 (Ω t )). (2) Solution on 2 : For ech u = u (T ) H 2 (Ω T ), by Theorem 3.2, there exist u 2 solution of u 2,t + (u 2 + u 2 2) x + u 2,xxxxt = in 2 u 2 (x, t) = in Σ 2 u 2,x (x, t) = in Σ 2 u 2 (x, T ) = u (x) in Ω T stisfying u 2 in L (T, T ; H 2 (Ω t )), u 2t in L (T, T ; H 2 (Ω t )), nd u 2 in C([T, T ]; H 2 (Ω t )). (3) Solution on : We define u : R by { u (x, t), (x, t) u(x, t) = u 2 (x, t), (x, t) 2, (4.2) where u nd u 2 re solutions of (4.) nd (4.2), respectively. Given tht u H 2 (Ω ), by Remrk 3.4, we deduce tht the function u defined bove is solution of u t + (u + u 2 ) x + u xxxxt = in Ω ], T [ u(x, t) = in Σ u x (x, t) = in Σ (4.3) u(x, ) = u (x) in Ω stisfying u L (, T ; H 2 (Ω t )), u t L (, T ; H 2 (Ω t )) nd u t φ dx dt + (u + u 2 ) x φ dx dt + u xxt φ xx dx dt =, for ll φ L 2 (, T ; H 2 (Ω t )); u(x, ) = u (x). following theorem. This result is summrized in the

11 EJDE-24/35 ROSENAU AND BENJAMIN-BONA-MAHONY EQUATIONS Theorem 4.. For ech u H 2 (Ω ) H 4 (Ω ), nd mixed domin defined s bove, there exists function u : R stisfying u L (, T ; H 2 (Ω t )), u t L (, T ; H 2 (Ω t )) nd u t φ dx dt + (u + u 2 ) x φ dx dt + u xxt φ xx dx dt =, for ll φ L 2 (, T ; H 2 (Ω t )); u(x, ) = u (x) for ll x Ω. In nlogous wy we hve the following result Theorem 4.2. For ech u H (Ω ) H 2 (Ω ), nd mixed domin defined s bove, there exists unique function u : R, stisfying u L (, T ; H (Ω t )), u t L (, T ; H (Ω t )) nd u t φ dx dt + (u + u 2 ) x φ dx dt + u xt φ x dx dt =, for ll φ L 2 (, T ; H (Ω t )); u(x, ) = u (x), for ll x Ω. Acknowledgement. We wish to cknowledge the nonymous referee t the Electronic Journl of Differentil Equtions, for his constructive remrks nd corrections on the originl mnuscript. References [] Adms, R. A.; Sobolev Spces, Acdemic, New York, 975. [2] Avrin, J. nd Goldstein, J. A.; Globl existence for the Bejmin-Bon-Mhony eqution in rbitrry dimensions, Nonliner Anlysis TMA, 9 (985), [3] Bon, J. L. nd Douglis, V.A.; An initil boundry-vlue problem for model eqution for propgtion of long wves, J. Mth. Anlysis Appl. 75 (98), [4] Brill, H.; A semiliner Sobolev eqution in Bnch spce, J. Diff. Eqns., (977) [5] Clds, C. S. Q., Limco, J. nd Brreto, K.; Liner thermoelstic system in noncylindricl domins, Funckcilj Ekvcioj, (999), [6] Friedmn, A.; Prtil Differentil Equtions, Holt, Rinehrt nd Winston, New York, 969. [7] Goldstein, J. A. Semigrups of Liner Opertors nd Applictions, Oxford University, New York, 985. [8] Goldstein, J. A.; Mixed problems for the generlized Benjmin-Bon-Mhony eqution, Nonliner Anlysis TMA, 4 (98), [9] Lions, J. L.; Une remrque sur les problèmes d èvolution linéires dns des domines non cylindriques, Revue Roumine de Mth. Pures et Appliquèes, V. 9, (964) -8. [] Lions, J. L. nd Mgenes E.; Problèmes ux Limites non Homogènes et Applictions V., Dunod, Pris 968. [] Medeiros, L. A., Limco J. nd Bezerr S.; Vibrtions of Elstic Strings, J. of Computtionl Anlysis nd Appliction 4(22), No. 2, 9-27, No. 3, [2] Prk, M. A.; On the Rosenu Eqution, Mt. Aplic. Comp. V. 9, (99), [3] Rosenu, P. H.; Dynmics of dense discrete systems, Prog. Theoreticl Phys., 79 (988), Rioco K. Brreto Instituto de Mtemátic - UFF, Ru Mário Sntos Brg s/n o, CEP: 242-4, Niterói, RJ, Brsil E-mil ddress: rikb@vm.uff.br Cruz S. Q. de Clds, Instituto de Mtemátic - UFF, Ru Mário Sntos Brg s/n o, CEP: 242-4, Niterói, RJ, Brsil E-mil ddress: gmcruz@vm.uff.br

12 2 R. K. BARRETO, C. DE CALDAS, P. GAMBOA, & J. LIMACO EJDE-24/35 Pedro Gmbo Instituto de Mtemátic - UFRJ, Cix Postl 6853, CEP , Rio de Jneiro, RJ, Brsil E-mil ddress: pgmbo@dmm.im.ufrj.br Jun Limco Instituto de Mtemátic - UFF, Ru Mário Sntos Brg s/n o, CEP: 242-4, Niterói, RJ, Brsil E-mil ddress: junbrj@hotmil.com