Kaikina, I. and uiz-paredes, F. Osaka J. Math. 4 (5), 47 4 SECOND TEM OF ASYMPTOTICS FO KdVB EQUATION WITH LAGE INITIAL DATA ELENA IGOEVNA KAIKINA and HECTO FANCISCO UIZ-PAEDES (eceived December, 3) Abstract We consider the Cauchy problem for the Korteweg-de Vries-Burgers equation. Global existence and smoothing effect for the Cauchy problem to the Kortewegde Vries-Burgers equation was proved previously for the case of large initial data. ecently the first term of the large time asymptotics was obtained without restriction on the size of the initial data. The aim of the present paper is to obtain the second term of the large time asymptotic behavior of solutions to the Cauchy problem for the Korteweg-de Vries-Burgers equation in the case of the initial data of arbitrary size.. Introduction We consider the Korteweg-de Vries-Burgers equation (.) (here the subscripts denote the differentiation with respect to the spatial and time coordinates, respectively). Korteweg-de Vries-Burgers equation (.) is a nonlinear model taking into account the simplest dispersive and dissipative processes, thus it is applicable in many fields of Physics and Technoy (see, e.g. [7], [], and references cited therein). Global existence and smoothing effect for the Cauchy problem to the Korteweg-de Vries-Burgers equation was proved in [7], [3]: there exists a unique solution C (( ); H ()) to the Cauchy problem for the Kortewegde Vries-Burgers equation (.) with initial data H (),. Here and below we denote the Sobolev space by H () L (): L,, L () is the usual Lebesgue space, L () L (): L is the weighted Lebesgue space. We now mention some results on the large time asymptotic behavior of solutions to (.). Everywhere below we suppose that the total mass of the initial data. (For the case, see paper [3]). In paper [6] it was proved that for small initial data L () H 7 (), the solutions 99 Mathematics Subject Classification : 35Q35.
48 I. KAIKINA AND F. UIZ-PAEDES of (.) have the asymptotics (.) ( ) ( ( ) ) as, where ( ) and cosh 4 sinh 4 Erf is the self-similar solution for the Burgers equation [] (.3) defined by the total mass of the initial data. Here Erf is the error function. The conditions on the initial data were generalized in paper [5], where it was proved that the solution of (.) with small initial data L () L () have asymptotics (.4) ( ) ( ) as. ecently in paper [3] it was proved that if the initial data H () L (), where, then there exists a unique solution C (( ); H ()) to the Cauchy problem for the Korteweg-de Vries-Burgers equation (.), which have asymptotics (.4). Moreover if additionally the initial data L (), then the asymptotics (.) is true. In paper [8], it was proved that for small initial data L () H 5 () such that L () the solutions of (.) have the following two terms of the large time asymptotics (.5) ( ) ( ) as uniformly with respect to where 4 ( ) 3
KDVB EQUATION 49 with cosh 4 sinh 4 Erf In the present paper we will extend this result for the case, when the initial data have an arbitrary size. Some other results for dissipative equations with critical nonlinearities were shown in papers [], [4], [5], [6], [7], [8], [], [], []. The aim of the present paper is to obtain the second term of the large time asymptotic behavior of solutions to the Cauchy problem for equation (.) in the case of the initial data of arbitrary size. Theorem. Let H () L (), where, and. Then the solution to the Cauchy problem for the Korteweg-de Vries- Burgers equation (.) with the initial condition has asymptotics (.5) as uniformly with respect to. EMAK. In the case of zero total mass the main term of the asymptotics is the same as that for the linear heat equation, and the second term of the asymptotics can be found by a similar approach. EMAK. The term in the estimate of the remainder in formula (.5) comes from estimate (.8) of Lemma 3 below. It could be removed by a more delicate consideration. EMAK 3. similar method. We believe that the third term of the asymptotics can be found by a Below we denote the Sobolev spaces H () L (); L where, the usual Lebesgue space is L (), and is the weighted Lebesgue space.. Proof of Theorem L () L (): L ecently in paper [3] it was proved that if the initial data H () L (), where, then there exists a unique solution C (( ); H ()) to the Cauchy problem for the Korteweg-de Vries-Burgers equation (.), which have the
4 I. KAIKINA AND F. UIZ-PAEDES following optimal time decay estimates (.) L ( )( ) for, where, 3. From now on denotes various positive constants. Denote the Green operator where the Green function G 3 We write the following integral equation associated with the Cauchy problem for the Korteweg-de Vries-Burgers equation (.) (.) G G We use the following notation,. First we give estimates for the Green operator G (see [3]). Lemma. The estimates are true ( )( ) ( )( ) L for all,, and 4 ( )( ) L for all,,. Proof. We have by the Plancherel theorem L 3 L ( ) ( )( ) ( )( ) L ( ) for all,,. Therefore by the Cauchy inequality we get L
KDVB EQUATION 4 4 L 4 4 L The second estimate of the lemma follows now by the Hölder inequality. Lemma is proved. Now let us prove the estimate of the L -norm of solutions of the Cauchy problem (.) (.3) L We multiply equation (.) by sign( ) and integrate with respect to over to get sign( ) sign( ) sign( ) sign( ) We have sign( ) L sign( ) ( ) sign L and sign( ) sign L ( ) Therefore we get (.4) L sign( )
4 I. KAIKINA AND F. UIZ-PAEDES L Next we give estimates for the norm L L. Multiplying equation (.) by and integrating with respect to we get (.5) 3 3 Since 3 3 4 3 L L L L L L and 3 3 L L 3 4 L we get L 3 4 L 3 4 3 4 L L L hence integrating we see that (.6) L 4 In the next lemma we obtain the estimates for the norm L. Denote W 3 () L (); 3 L. Lemma. Let the initial data H () W 3 () and estimate (.) be valid. Then the estimate is true (.7) L
KDVB EQUATION 43 for all. Proof. First we need to estimate the norm L. By the integral equation (.) we have L G L L L L L L ( L L L L L L ) whence by estimate L 4 we get L 3 4 In the same manner by the integral equation (.) we have L 3 G L 4 L L 4 L L L ( L ( L L L L ) L ( L L L L )) hence we obtain L 3 3 4 3 Thus the estimate of the lemma is true. Lemma is proved.
44 I. KAIKINA AND F. UIZ-PAEDES Integration of inequality (.4) yields L L Therefore estimate (.3) is true for all. Now we obtain the second term of the large time asymptotics as of solutions to the Cauchy problem for the Korteweg-de Vries-Burgers equation (.). We take the initial time to be sufficiently large and define as a solution to the Cauchy problem for the Burgers equation with as the initial data (.8) By the Hopf-Cole [4] transformation equation (.8) is converted to the heat equation. Therefore we obtain (.9) exp where (4 ) 4 is the Green function for the heat equation. Note that the following estimates are true (.) L ( )( ) for all,,. Consider now the difference for. By (.) and (.8) we get the Cauchy problem (.) We have the estimates (see [3]) (.) L L for all and (.3) L ( )( ) for all,,, where ( ). Following the heuristic considerations of paper [8] we easily see that the main term of the asymptotic expansion of as is determined by the linear
KDVB EQUATION 45 Cauchy problem (.4) To eliminate the second term from (.4), let us integrate (.4) with respect to make the substitution where is defined by (.9). We obtain and (.5) It is easy to integrate (.5) to get (.6) G In the following lemma we evaluate the large time asymptotics of the solution linear problem (.4) of (.7) ( G G ) Lemma 3. Let H () L (). Then the asymptotics (.8) is valid as uniformly with respect to, where 4 4 3 cosh 4 sinh 4 Erf Proof. Let us represent the integral with respect to in (.7) as the sum of three parts ( ) (.9) 3
46 I. KAIKINA AND F. UIZ-PAEDES For all and we have and for each (.) the following inequalities hold: L L L for all 3,. By using these inequalities, we readily estimate the first two integrals in representation (.9) (.) L ( ) as, and L ( L L L ) (.) 3 since changing variables of integration we have 3 3 3 3 4 3 as. In the third integral 3 we integrate by parts with respect to to obtain 3 4 5 6 ( )
where and KDVB EQUATION 47 ( ) Since sup sup sup L () 3 3 and therefore L () we obtain (.3) 4 3 3 3 3 ( ) The integral 5 can be estimated similarly. Since ( ) for, we derive the estimate (.4) 6 ( ) from the estimate (.5) ( ) 3 Since integration by parts yields, by virtue of (.5), 3
48 I. KAIKINA AND F. UIZ-PAEDES 3 ( 3 ) Then from (.) (.4) we obtain (.8). Lemma 3 is proved. It follows from (.) and (.4) that the remainder is the solution to the Cauchy problem (.6) To eliminate the second term from (.6) as above we integrate this equation with respect to and introduce the new unknown function Then we obtain (.7) where In view of (.3) and (.) we find (.8) L L L L 3 for all,. Using the integral equation associated with (.7) we obtain L ( ) L L ( ) L L hence in view of estimate (.8) we find L ( ) ( ) 3 4
KDVB EQUATION 49 for all,, if we take ( 4 ). In the same manner we have L ( ) L L ( ) ( ) L L ( ) 3 4 for all,. Using the identity we obtain the estimate for all. When L, then L ( ) ( ) for and in view of Lemma 3, the asymptotics of the theorem follows. Theorem is proved. ACKNOWLEDGEMENTS. The work is partially supported by CONACYT. We are grateful to an unknown referee for many useful suggestions and comments. eferences [] C.J. Amick, J.L. Bona and M.E. Schonbek: Decay of solutions of some nonlinear wave equations, J. Differential Equations 8 (989), 49. [] J.M. Burgers: A mathematical model illustrating the theory of turbulence: in Advances in Applied Mechanics, Academic Press, Inc., New York (948), 7 99. [3].E. Cardiel and P.I. Naumkin: Asymptotics for nonlinear dissipative equations in the super critical case, Contemp. Math. 37 (), 47 67. [4] D.B. Dix: Large-Time Behavior of Solutions of Linear Dispersive Equations, Lecture Notes in Mathematics 668, Springer-Verlag, Berlin, 997. [5] M. Escobedo, O. Kavian and H. Matano: Large time behavior of solutions of a dissipative semilinear heat equation, Comm. Partial Differential Equations (995), 47 45.
4 I. KAIKINA AND F. UIZ-PAEDES [6] M. Escobedo, J.L. Vázquez and E. Zuazua: Asymptotic behavior and source-type solutions for a diffusion-convestion equation, Arch. ational Mech. Anal. 4 (993), 43 65. [7] V.A. Galaktionov, S.P. Kurdyumov and A.A. Samarskii: On asymptotic eigenfunctions of the Cauchy problem for a nonlinear parabolic equation, Math. USS Sbornik 54 (986), 4 455. [8] A. Gmira and L. Véron: Large time behavior of the solutions of a semilinear parabolic equation in, J. Differential Equations 53 (984), 58 76. [9] N. Hayashi and E.I. Kaikina: Nonlinear Theory of Pseudodifferential Equations on a Half-Line, Elsevier Science B.V. Amsterdam, 4. [] N. Hayashi, E.I. Kaikina and P.I. Naumkin: Large-time behavior of solutions to the dissipative nonlinear Schrödinger equation, Proc. oy. Soc. Edinburgh Sect. A 3 (), 9 43. [] N. Hayashi, E.I. Kaikina and F.. uiz-paredez: Korteweg-de Vries-Burgers equation on a halfline with large initial data, J. Evol. Equ. (), 39 347. [] N. Hayashi and P.I. Naumkin: Asymptotics for the Burgers equation with pumping, Commun. Math. Phys. 39 (3), 87 37. [3] N. Hayashi and P.I. Naumkin: Asymptotics for the Korteweg-de Vries-Burgers equation, Preprint 3. [4] E. Hopf: The partial differential equation, Comm. Pure Appl. Math. 3 (95), 3. [5] G. Karch: Self-similar large time behavior of solutions to the Korteweg-de Vries-Burgers equation, Nonlinear Anal. 35 (999), 99 9. [6] P.I. Naumkin: On the asymptotic behavior for large time values of the solutions of nonlinear equations in the case of maximal order, Differential Equations 9 (993), 7 74. [7] P.I. Naumkin and I.A. Shishmarev: Nonlinear Nonlocal Equations in the Theory of Waves, American Mathematical Society, Providence, I, 994. [8] P.I. Naumkin and I.A. Shishmarëv: An asymptotic relationship between solutions of different nonlinear equations for large time values. I, Differential Equations 3 (994), 86 84. [9] P.I. Naumkin and I.A. Shishmarëv: An asymptotic relationship between solutions of different nonlinear equations for large time values. II, Differential Equations 3 (994), 39 34. [] H.F. uiz-paredez: Localización de Fallas en Sistemas Electricos de Potencia, Analysis of Electric Circuits, Zeveke, Ionkin, Netushil, Strakhov, MI, 999. [] E. Zuazua: A dynamical system approach to the self-similar large time behavior in scalar convection-diffusion equations, J. Differential Equations 8 (994), 35. Elena Igorevna Kaikina Departamento de Ciencias Básicas Instituto Tecnológico de Morelia CP 58, Morelia, Michoacán, México e-mail: ekaikina@matmor.unam.mx Hector Francisco uiz-paredes Posgrado Electrica Instituto Tecnológico de Morelia CP 58, Morelia, Michoacán, México e-mail: hruiz@sirio.tsemor.mx