DISSIPATIVE INITIAL BOUNDARY VALUE PROBLEM FOR THE BBM-EQUATION

Transcription

1 Electronic Journal of Differential Equations, Vol. 8(8), No. 149, pp ISSN: URL: or ftp ejde.math.txstate.edu (login: ftp) DISSIPATIVE INITIAL BOUNDARY VALUE PROBLEM FOR THE BBM-EQUATION NIKOLAI A. LARKIN, MIKHAIL P. VISHNEVSKII Abstract. This paper concerns a dissipative initial boundary value problem for the Benjamin-Bona-Mahony (BBM) equation. We prove the existence and uniqueness of global solutions and the decay of the energy as time tends to infinity. 1. Introduction This paper concerns the dissipative initial boundary value problems for the Benjamin-Bona-Mahoney (BBM) equation u t u txx + uu x = (1.1) which was derived by Benjamin-Bona-Mahony, [, 3], and usually is called the alternative Korteweg-de Vries (KdV) equation. In spite of the fact that both (1.1) and the KdV equation, u t + au xxx + uu x = (1.) are dispersive equations and have almost the same names, formulations of initial boundary value problems for them are completely different. Considering (1.1) and (1.) in a rectangle Q = (, 1) (, T ), T >, one must put for (1.1) one condition at x = and one condition at x = 1. On the other hand, for (1.) one must put three conditions at the ends of the interval (, 1). A number of conditions at x = and x = 1 depends on a sign of the coefficient a: if a >, then we pose one condition at x = and two conditions at x = 1. If a <, then we pose two conditions at x = and one condition at x = 1. Historically, interest in dispersive-type evolution equations dates from the 19th century when Russel [], Airy [1], Boussinesq [9] and later Korteweg and de Vries [15] studied propagation of waves in dispersive media. Due to physical reasons, these and posterior studies mostly dealt with one-dimensional problems posed on the entire real line, see [, 4, 7, 8, 14, 1] and references therein. Moreover, the emphasis in these works was mainly focused on the existence and qualitative structure of the solitary, cnoidal and other specific types of waves, whereas correctness of the corresponding mathematical problems attracted minor interest. Mathematics Subject Classification. 35Q53, 37K45. Key words and phrases. BBM equation; regular solution; asymptotic behavior. c 8 Texas State University - San Marcos. Submitted June 5, 8. Published October 9, 8. 1

2 N. A. LARKIN, M. P. VISHNEVSKII EJDE-8/149 Initial boundary value problems for (1.1) with Dirichlet boundary conditions were considered in [8, 11, 13, 19, 18, 6]. Bubnov in [1] studied general boundary conditions and proved existence of local solutions to a corresponding mixed problem. Mixed problems for multi-dimensional versions of (1.1) were considered in [13, 19, 18]. It is easy to see that mixed problems for (1.1) with Dirichlet boundary conditions imply conservation of the energy: d dt E(t) = d dt {u(x, t) + u x (x, t) }dx =. It means that the energy can not decay with time. Differently, the KdV equation itself has dissipative properties and solutions of initial boundary value problems for it decay with time see [1, 16, 17]. The goal of our paper is to find such boundary conditions which guarantee existence of global regular solutions and decay of the energy for the BBM equation. For this purpose we pose dissipative nonlinear boundary conditions (.). From the physical point of view, if to consider dynamics of a fluid in a cylinder, the Dirichlet boundary conditions mean that the walls of a cylinder are impermeable: a fluid cannot enter or exit the cylinder. On the other hand, nonlinear boundary conditions (.) allow a fluid to exit, for a example, when a cylinder has porous walls. This effect stabilizes the system and dissipates the energy. This paper has the following structure: in Chapter we formulate a nonlinear problem and consider decay properties of linearized problems. In Chapter 3, first we prove local existence of regular solutions to the nonlinear problem, using the theory of elliptic equations with a parameter t, then global existence and uniqueness of regular solutions. In Chapter 4, decay properties of the energy, as t, are proved.. Formulation of the problem In Q = (, 1) (, T ) we consider the following initial boundary value problem: u t u txx + uu x =, x (, 1), t (, T ), (.1) u(, t) =, u tx (1, t) = 1 3 u (1, t) u(1, t), t >, (.) u(x, ) = u (x), x (, 1). (.3).1. Linear problem. First we study the linearized version of (.1)-(.3): u t u txx =, (x, t) Q, (.4) u(, t) =, u tx (1, t) = u(1, t), t >, (.5) u(x, ) = u (x), x (, 1). (.6) We also assume that the initial data admits the compatibility condition u () =. It is easy to see that problem (.4)-(.6) has a unique solution. Considering solutions of the form u(x, t) = v(x)w(t), we obtain w t (t)(v(x) v(x) xx ) =, x (, 1), t R +, (.7) This problem has two type of solutions: v()w(t) =, w t (t)v x (1) = w(t)v(1). (.8) λ 1 =, w 1 (t) = C 1 exp(λ 1 t), v 1 (x) C [, 1], v 1 () = v 1 (1) =

3 EJDE-8/149 BBM-EQUATION 3 and λ = e 1 e + 1, w (t) = C exp(λ t), v (x) = ex e x. Since u () =, φ(x) = u (x) C ex e x, where C = eu (1) e 1, is a stationary solution of (.7),(.8) corresponding to λ 1 =. This implies that u(x, t) = φ(x) + eu (1) e x e 1 [ex ] exp( e 1 e + 1 t) is a unique solution of (.4)-(.6) and u(x, t) φ(x) u (1) exp( e 1 e + 1 t). These results can be summarized as follows. Theorem.1. Problem (.4)-(.6) has a continuum of stationary solutions and any nonstationary solution converges to a stationary one exponentially as t. Remark.. Consider the linearized problem with the Dirichlet boundary conditions, u t u txx =, (x, t) Q, (.9) u(, t) = u(1, t) =,, t >, (.1) u(x, ) = u (x), (.11) it is easy to show that this problem has only stationary solutions. 3. Nonlinear Problem 3.1. Local Solutions. We start with the linear problem u t u txx = f(x, t), (x, t) Q, (3.1) u(, t) =, u tx (1, t) = g(t), t >, (3.) Denote w(x, t) = u t (x, t), then the problem becomes u(x, ) = u (x), x (, 1). (3.3) w w xx = f(x, t), x (, 1), t >, (3.4) w(, t) =, w x (1, t) = g(t), t > (3.5) which is an elliptic problem with a parameter t. Lemma 3.1. Regular solutions of (3.4) (3.5) satisfy the inequality w(t) H (,1) C( f(t) L (,1) + g(t) ). (3.6) Here and in the sequel the constants C do not depend on g(t), f(x, t).

4 4 N. A. LARKIN, M. P. VISHNEVSKII EJDE-8/149 Proof. Considering w(x, t) = z(x, t) g(t)(1 x)x, (3.7) we rewrite (3.4)-(3.5) as the following elliptic problem with a parameter t: z z xx = f 1 (x, t) f(x, t) + g(t)x(1 x) g(t), x (, 1), (3.8) Standard elliptic estimates [5] give z(, t) = z x (1, t) =, t >. (3.9) z(., t) H (,1) C f 1 (t) L (,1) C( f(., t) L (,1) + g(t) ). This and (3.7) imply (3.6). Remark 3.. Let u(x, t) be a solution to the problem u t u txx = f(x, t), (x, t) Q, u(, t) = u t (, t) =, u tx (1, t) = 1 3 u (1, t) u(1, t), t >, Then (3.4)-(3.6) imply u(x, ) = u (x). u t (., t) H (,1) C( f(., t) L (,1) + u(1, t) + u(1, t) ). (3.1) Lemma 3.3. Regular solutions of (3.1)-(3.3) in the cylinder Q T = (, 1) (, T ), T >, satisfy the inequality u C([,T ];H1 (,1)) u H1 (,1) + CT ( ) f C([,T ];L (,1)) + u(1, t) C[,T ] + u(1, t) C[,T ]. (3.11) Proof. Because we have u(x, t) = u (x) + u(t) H1 (,1) u H1 (,1) + t( u s (x, s)ds, u s (., s) H 1 (,1) ds)1/, u(x, t) C([,T ];H 1 (,1)) u (x) H 1 (,1) + T u t (x, t) C([,1];H 1 (,1)). Using (3.1), we obtain and max (x,t) Q T ( u t (x, t) ) u t (x, t) C([,T ];H 1 (,1)) C( f(x, t) C([,T ];L (,1)) + u(1, t) C[,T ] + u(1, t) C[,T ] ) u C(,t;H1 (,1)) u H1 (,1)+T C{ f C(,t;L (,1))+ u(1, t) C(,t) + u(1, t) C(,t) }. (3.1) This completes the proof. Using the estimates of Lemmas 3.1 and 3.3, we can solve locally in t the nonlinear problem (.1)-(.3). Theorem 3.4. Let u H 1 (, 1). Then there is T > such that for all t (, T ) there exists u(x, t) such that u C(, T ; H 1 (, 1)), u t C(, T ; H (, 1)), u tt C(, T ; H (, 1)), which is a unique regular solution of (.1)-(.3).

5 EJDE-8/149 BBM-EQUATION 5 Remark 3.5. If u H (, 1), then u C(, T ; H (, 1)). Proof of Theorem 3.4. We use the contraction mapping theorem. Let u H1 (,1) < R, R > 1 and B R be a ball of functions w(x, t) such that w C(, T ; H 1 (, 1)), T >, w C(,T;H 1 (,1)) < R, w(x, ) = u (x), w(, t) =, t (, T ), where the constant T will be defined later. For w B R consider the linear problem v t v txx = ww x, (x, t) (, 1) (, T ), (3.13) v(, t) =, v tx = 1 3 w (1, t) w(1, t), t (, T ), (3.14) v(x, ) = u (x), x (, 1). (3.15) Since (3.13)-(3.15) is a linear, elliptic problem for v t, solvability of this problem follows from Lemmas 3.1 and 3.3. Therefore, we can define the operator P : v(x, t) = P (w(x, t)) in B R. The proof will be completed after proving the following two propositions. Proposition 3.6. The operator P maps B R into B R for T > sufficiently small. Proof. Fixing 1 < R < and taking into account (3.6),(3.1),(3.1) and the obvious inequality w(1, t) C[,T) w C[,T;H 1 (,1)), we find ww x C(,T;L (,1)) max [,T ] ( Using Lemma 3.3, we obtain w (x, t)w x(x, t)dx) 1/ w C[,T;H 1 (,1)) w x C[,T;L (,1)) w C[,T ;H 1 (,1)). v C[,T;H 1 (,1)) u H 1 (,1) + C T {1 + w C[,T;H 1 (,1))} Taking < T < 1/(4C R ), we get which completes the proof. u H1 (,1) + C T w C[,T ;H 1 (,1)) u H1 (,1) + C T R. v C[,T;H 1 (,1)) R + R 4 < R Proposition 3.7. For T > sufficiently small the operator P is a contraction mapping in B R. Proof. For any w 1, w B R denote v i = P (w i ), i = 1, ; s = w 1 w, z = v 1 v. From (3.13)-(3.15), we obtain z t z txx = (w s x + w 1x s), (x, t) (, 1) (, T ), z(, t) =, z tx (1, t) = 1 3 (w 1(1, t) + w (1, t)) 1)s(1, t), t (, T ), z(x, ) =, x (, 1).

6 6 N. A. LARKIN, M. P. VISHNEVSKII EJDE-8/149 By Lemma 3.3, z C(,T;H 1 (,1)) C T R s C(,T;H 1 (,1)), where the constant C does not depend on s. Taking < T < 1/(C R), we obtain z C(,T;H 1 (,1)) γ s C(,T;H 1 (,1)) with < γ < 1. This completes the proof. Propositions 3.6 and 3.7 imply that the operator P : B R B R is a contraction mapping provided T > sufficiently small. Hence, there exists a unique function u(x, t) : u C(, T ; H 1 (, 1)) such that u = P u. More regularity follows directly from (.1)-(.3) and estimates of elliptic problems for u t, u tt, see Lemmas 3.1 and 3.3. This proves Theorem Global Solutions. Theorem 3.8. Let u H 1 (, 1). Then there exists a function u(x, t) such that u L (, ; H 1 (, 1)), u t L (, ; H (, 1)), u tt L (, ; H (, 1)) which is a unique solution of (.1)-(.3). Proof. Due to Theorem.1, it is sufficient to extend local solutions to any finite interval (, T ). For this purpose we need a priori estimate independent of t. Multiplying (.1) by u and integrating over (, 1) (, t), t (, T ), we get E(t) = 1 (u (x, t) + u x(x, t))dx = E() u (1, s)ds E(). (3.16) This estimate guarantees prolongation of local solutions, provided by Theorem.1, for any finite interval (, T ). Moreover, since it does not depend on T, the interval of the existence is (, ) : u L (, ; H 1 (, 1)). Returning to (.1)-(.4), we rewrite it as an elliptic problem for u t : (I xx)u t = uu x L (, ; L (, 1)), (3.17) u(, t) =, u tx = 1 3 u (1, t) u(1, t) L (, ; L (, 1)). (3.18) By Lemmas 3.1 and 3.3, Differentiating (3.17), (3.18) with respect to t, we get Hence, u t L (, ; H (, 1)). (3.19) (I xx)u tt = uu tx u t u x L (, ; L (, 1)), u tt (, t) =, u ttx (1, t) = 3 u(1, t)u t(1, t) u t (1, t) L (, ; L (, 1)). This proves the existence part of Theorem 3.8. u tt L (, ; H (, 1)). (3.)

7 EJDE-8/149 BBM-EQUATION 7 To prove uniqueness of solutions, assume that there exist two different solutions u 1, u of (.1)-(.3). For z = u 1 u we have the following problem: Lz = z t z txx = 1 (u 1x + u x )z 1 (u 1 + u )z x, (x, t) (, 1) (, ), z(, t) =, z tx (1, t) = 1 3 (u 1(1, t) + u (1, t))z(1, t) z(1, t), t >, z(x, ) =, x (, 1). Multiplying Lz by z and integrating over (, 1) (, t), we obtain (z (x, t) + z x(x, t))dx = {(u 1x (x, s) + u x (x, s))z (x, s) 1 (u 1(x, s) + u (x, s))z(x, s)z x (x, s)}dxds { 1 3 [u 1(1, s) + u (1, s)]z (1, s) + z (1, s)}ds. Since z(1, s) z x (s) L (,1), we arrive to the inequality C (z (x, t) + z x(x, t))dx (1 + u 1x (s) L (,1) + u x (s) L (,1)) z x (s) L (,1) ds. Because u i L (, ; H 1 (, 1)), i = 1,, (z (x, t) + z x(x, t))dx C By the Gronwall lemma, (z (x, s) + z x(x, s))dxds. (z (x, t) + z x(x, t))dx =, t >. Then z(x, t) =, (x, t) (, 1) (, ) that completes the proof. 4. Uniform Decay of Solutions as t Lemma 4.1. For regular solutions of (.1)-(.3), lim t + u(1, t) =. Proof. From (3.16), Due to (3.19), u (1, s)ds E(), and E(t) E() for all t >. (4.1) Assume that sup u t (t) H (,1) C 1, t> lim t + u (x, t). sup d t> dt u (x, t) C. (4.)

8 8 N. A. LARKIN, M. P. VISHNEVSKII EJDE-8/149 This implies that there exist a positive number ε 1 > and a sequence of t n + such that u (1, t n ) ε 1 for all n N. Since it follows that sup d t> dt u (x, t) C, u (1, t) > 1 ε 1 for t [t n ε 1 C, t n + ε 1 C 1 ] and all n N. We may assume that t n ε1 C n+ ε 1 C 1 u (1, s)ds >. Therefore, n i=1 i+ ε 1 C 1 u nε 1 (1, s)ds > ε 1 t i ε 1 C C +. Thus we have a contradiction with (4.1) which completest the proof. Theorem 4.. For regular solutions of (.1)-(.3), lim t + E(t) =. Proof. Let τ > and t n +. Consider a sequence u n (x, t) = u(x, t n + t), (x, t) Q τ = [, 1] [, τ]. It follows from (3.16), (3.19), (3.) that from the sequence u n (x, t) we can extract a subsequence, which we again denote by u n (x, t), such that u n (x, t) w(x, t) in C α1 ( Q τ ), α 1 (, 1 ); (4.3) u n t (x, t) w t (x, t) in C α1 ( Q τ ); (4.4) u n x(x, t) w x (x, t) weakly in L (, τ; L (, 1)), (4.5) u n txx(x, t) w txx (x, t) weakly in L (, τ; L (, 1)). (4.6) We will return to this proof after the following proposition. Proposition 4.3. It holds Proof. Writing u n (x, t)u n x(x, t) w(x, t)w x (x, t) weakly in L (, τ; L (, 1)). u n u n x ww x = u n x(u n w) + w(u n x w x ), from (4.3), we have lim n un x(u n w) L (Q τ ) =. A function w(x, t) is bounded in C α1 ( Q τ ), whence by (4.5), w(u n x w x ) weakly in L (, τ; L (, 1)). This completes the proof of Proposition 4.3 Due to (4.4), (4.6), Proposition 4.3 implies w t w txx + ww x =, (x, t) Q τ and w(, t) =. (4.7) By Lemma 4.1 and (4.3), w(1, t) =, but since w tx (1, t) = 1 3 w (1, t) w(1, t), we have w(1, t) = w tx (1, t) =.

9 EJDE-8/149 BBM-EQUATION 9 Denoting v(x, t) = w t (x, t), from (4.7), we get v v xx + ww x =, x (, 1), v() = v(1) = v x (1) =. Let g(x, y) be a Green function of the problem It is known that where Simple calculations give and From here, v(x, t) = z xx z =, x (, 1), z() = z(1) =. { g(x, y) = 1 v 1 (x)v (y), x y; D() v 1 (y)v (x), y x 1, v 1xx v 1 =, v 1 () =, v 1x () = 1; v xx v =, v (1) =, v x (1) = 1; v 1(x) v (x) v 1x (x) v x (x) = D(x). v 1 (x) = ex e x, v (x) = e x e x e g(x, y)w(y, t)w y (y, t)dy = 1 g y (x, y)w (y, t)dy. v x (x, t) = 1 g xy (x, y)w (y, t)dy. The function g xy (x, y) is negative for < x, y < 1. On the other hand, Hence, v x (1, t) = 1 g xy (1, y)w (y, t)dy = 1 ( e y e y )w (y, t)dy =. ( e y + e y )w (y, t)dy = and, consequently, w (y, t) =. It implies that E(t) tends to zero when t [t n, t n + τ] and n. Due to monotonicity of E(t), we have lim t + E(t) =. This completes the proof of Theorem 4.. References [1] A. Chapman, George Biddell Airy; F. R. S. ( ): a centenary commemoration, Notes and Records Roy. Soc. London 46(1) (199) [] T. B. Benjamin; Lectures on nonlinear wave motion, Lecture notes in applied mathematics 15 (1974) [3] T. B. Benjamin, J. L. Bona, J. J. Mahony; Model equations for long waves in nonlinear dispersive system, Philos. Trans. Soc., London S. R., A. 7(1) (197) [4] D. J. Benney, Long waves on liquid films, J. Math. and Phys. 45 (1966) [5] J. M. Berezanskii; Expansions in eigenfunctions of selfadjoint operators, Translations of Mathematical Monographs, 17. AMS, Providence, Rhode Island, [6] G. Boling; Initial boundary value problem for one class of system of multi-dimensional inhomogeneous GBBM equations, China Ann. Math. Ser B (1987) 6-38.

10 1 N. A. LARKIN, M. P. VISHNEVSKII EJDE-8/149 [7] J. L. Bona, R. Smith; The initial value problem for the Korteweg-de Vries equation. Philos. Trans. Royal Soc. London A 78 (1975) [8] J. L. Bona, P. L. Bryant; A mathematical model for long waves generated by wavemakers in non-linear dispersive systems, Proc. Cambridge Philos. Soc. 73 (1973) [9] J. Boussinesq; Théorie des ondes et des remous qui se propagent le long d un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fund, J. Math. Pures Appl. 17 (187) [1] B. A. Bubnov; Boundary value problems for the alternative Korteweg-de Vries equation in a bounded domain. (Russian) Dokl. Akad. Nauk SSSR. 47, N (1979) 7-75 [11] C. S. Caldas, J. Limaco, P. Gamboa, R. Bareto; On the Roseneau and Benjamin-Bona- Mahony equations with moving boundaries, Electr. J. Different. Equats. 4 (4), no. 35, 1-1. [1] G. G. Doronin, N. A. Larkin; KdV equation in domains with moving boundaries, J. Math Anal. Appl. 38 (7) [13] J. Goldstein, B. Wichnoski; On the Benjamin-Bona-Mahony equation in higher dimensions, Nonlinear Analysis, TMA 4(4) (198) [14] T. Kawahara; Oscillatory solitary waves in dispersive media, J. Phys. Soc. Japan. 33 (197) [15] D. J. Korteweg, G. de Vries; On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag. 39 (1895), [16] N. A. Larkin; Korteweg-de Vries and Kuramoto-Sivashinsky equations in bounded domains, J. Math. Anal and Appl. 97 (4) [17] N. A. Larkin; Modified KdV equation with a source term in a bounded domain, Math. Meth. in the Appl. Sci. 9 (6) [18] J. L. Limaco, H. R. Clark, L. A. Medeiros; On equations of Benjamin-Bona-Mahony type, Nonlinear Analysis, TMA 59 (4) [19] L. A. Medeiros, M. M. Miranda; Weak solutions for a nonlinear dispersive equation. J. Math. Anal. Appl. 59(3) (1977) [] J. S. Russel; Report on waves, Rep. 14th Meet. Brit. Assoc. Adv. Sci. (1844) [1] J. Topper, T. Kawahara; Approximate equations for long nonlinear waves on a viscous fluid. J. Phys. Soc. Japan 44 (1978), no., Nikolai A. Larkin Departamento de Matemática, Universidade Estadual de Maringá, Agência UEM, CEP: 87-9, Maringá, PR, Brazil address: nlarkine@uem.br Mikhail P. Vishnevskii Laboratorio de Matematika, Universaidade Estadual de Norte Flumenense, Alberto Lamego, CEP 815-6, Campos dos Goytacazes, RJ, Brasil address: mikhail@uenf.br