NONLINEAR QUARTER-PLANE PROBLEM FOR THE KORTEWEG-DE VRIES EQUATION

Transcription

1 Electronic Journal of Differential Equations, Vol ), No. 113, pp. 1. ISSN: URL: or ftp eje.math.txstate.eu NONLINEAR QUARTER-PLANE PROBLEM FOR THE KORTEWEG-DE VRIES EQUATION NIKOLAI A. LARKIN, EDUARDO TRONCO Abstract. This article concerns an initial-bounary value problem in a quarter-plane for the Korteweg-e Vries KV) equation. For general nonlinear bounary conitions we prove the existence an uniqueness of a global regular solution. 1. Introuction This work concerns the existence an uniqueness of global solutions for the KV equation pose on the first quarter-plane with a general nonlinear bounary conition. Such initial-bounary value problems may serve as moels for waves generate by wavemakers in a channel, or for shallow water waves of the shore, 1,, 3]. There is a number of papers where initial value problems an initial-bounary value problems in a quarter-plane an in a boune omain for ispersive equations were stuie see, 3, 4, 6, 7, 18, 1, 11, 1, 16, 19]). As a rule, simple bounary conitions at x = such as u = for the KV equation or u = u x = for the Kawahara equation were impose. On the other han, general initial-bounary value problems for o-orer evolution equations attracte little attention. We must mention a classical paper of Volevich an Ginikin 14], where general mixe problems for linear b + 1)-hyperbolic equations were stuie by means of functional analysis methos. It is ifficult to apply their metho irectly to nonlinear ispersive equations ue to complexity of this theory. In 4, 5], Bubnov consiere general mixe problems for the KV equation pose on a boune interval an prove solvability results. In 18] also were consiere general mixe problems with linear bounary conitions for the KV equation on a boune interval an for small initial ata the existence an uniqueness of global solutions as well as the exponential ecay of L -norms of solutions while t + were prove. Here we stuy a mixe problem for the KV equation in a quarter plane with a general nonlinear nonhomogeneous conition: x =, xu, t) = ϕt, u, t), x u, t)) an prove the existence an uniqueness of global in t solutions as well as smoothing effect for the initial ata. The presence of a issipative nonlinear term in the Mathematics Subject Classification. 35Q53. Key wors an phrases. KV equation; global solution; semi-iscretization. c 11 Texas State University - San Marcos. Submitte June 18, 11. Publishe August 3, 11. 1

2 N. A. LARKIN, E. TRONCO EJDE-11/113 bounary conition guarantees the existence of global solutions without smallness conitions for the initial ata, whereas posing a general linear bounary conition we i not succee to prove a global existence result because general linear conitions i not imply the first estimate for the KV equation which is crucial for global solvability of a corresponing mixe problem as was pointe out in 3]. To prove our results, we use a linearization technique, semi-iscretization in t to solve the linear problem, the Banach fixe point theorem for local in t existence an uniqueness results an, finally, a priori estimates, inepenent of t, for the nonlinear problem. To prove solvability of a linearize problem with a nonhomogeneous bounary conition, we exploit the metho of semi-iscretization which is transparent an prove its universality, see 6, 7, 16, 17], instea of very popular in the theory of the KV equation with homogeneous bounary conitions semigroups technique, because it is ifficult to aapt this technique for mixe problems with nonhomogeneous an nonlinear bounary conitions. This article has the following structure: Section 1 is Introuction. In Section we formulate the principal problem an some useful known facts. In Section 3 the relate stationary problem is stuie. Section 4 is evote to regular solutions to the linear evolution problem which are obtaine by the metho of semi-iscretization with respect to t. In Section 5, using the contraction mapping arguments, we obtain a local in time regular solution to the nonlinear problem. Finally, in Section 6, necessary a priori estimates are prove which allow us to exten the local solution to the whole interval t, T ) with arbitrary finite T >.. Formulation of the problem an main results Denote R + = {x R : x > } an for a positive number T, Q T = {x, t) R : x R +, t, T )}. In Q T we consier the KV equation subject to initial an bounary conitions u t + Du + D 3 u + udu =.1) ux, ) = u x), x R +,.) D u, t) + αdu, t) + βu, t) + u, t) u, t) + gt) =, t, T );.3) where gt) is a given function, α an β are real coefficients such that β α 1 = a 1 >, 1 α = a >..4) Remark.1. From the technical reasons, we chose the simple nonlinearity in.3). Of course, more general issipative functions may be use. Bounary conitions.3) follow from more general conitions γd u, t) + αdu, t) + βu, t) + u, t) u, t) + gt) =, t, T ).5) when γ. Explicitly, the simple bounary conition u, t) = oes not follow from.3), but it is a singular case of.5): when γ =, in orer to get the first L R + ) estimate, which is crucial for solvability of.1)-.3), see 3], we must put α = an u, t) = that gives exactly the simple bounary conition. Here u : R +, T ) R, or u :, L), T ) R D j = j / x j ; D = D 1. In this article, we aopt the usual notation u, v)t) = + ux, t)vx, t)x, u t) = u, u)t),

3 EJDE-11/113 NONLINEAR QUARTER-PLANE PROBLEM 3 u, v) = u, v) L = + L ux)vx)x, u = u, u), ux)vx)x, u L = u, u) L. Symbols C, C, C i, for i N, mean positive constants appearing uring the text. The main result of this article is the following theorem. Theorem.. Let u H 3 R + ), g H 1, T ), α an β satisfy.4) an for a real k = min{a /, a 1 )/} the following inequality hols: 3 e kx, D i u + u Du ]) <. i= Then for all finite T >, problem.1)-.3) has a unique regular solution: u L, T ; H 3 R + )) L, T ; H 4 R + )), u t L, T ; L R + )) L, T ; H 1 R + )) an the following estimate hols: { 3 sup e kx, D i u ) t) + e kx, u t )t) } t,t ) i= e kx, D i u )t) + e kx, D i u t )t) ] t + i= u, t) + Du, t) ) t + i= CT, k) 3 e kx, D i u ) + e kx, u Du ) + i= 3. Stationary problem u t, t) + Du t, t) ) t g + g t )t) t ]. Our purpose in this section is to solve the stationary bounary-value problem D 3 ux) + ux) = fx), x R +, 3.1) D u) + αdu) + βu) + q 1 =, 3.) where > an q 1 are real constants, α an β satisfy.4) an f is such that e kx/ f L R + ), k >. 3.3) Theorem 3.1. Let > k 3 an f satisfy 3.3). Then 3.1)-3.) amits a unique solution u H 3 R + ) such that 3 e kx, D i u ) Ce kx, f ) + q1]. i= Proof. Consier on an interval, L) the problem D 3 ux) + ux) = fx), x, L), 3.4) D u) + αdu) + βu) + q 1 =, 3.5) ul) = DuL) =, 3.6)

4 4 N. A. LARKIN, E. TRONCO EJDE-11/113 where L is an arbitrary finite positive number, fx) is a restriction on, L) of fx) : e kx/ f L R + ) CR + ). It is known see 8]) that 3.4)-3.6) has a unique classical solution if the corresponing homogeneous problem has only trivial solution. Proposition 3.. Let fx), q 1 = an α an β satisfy.4). Then 3.4)-3.6) has only the trivial solution. Proof. Multiplying 3.4) by u an using 3.5), 3.6), we come to the inequality u L + β α )u ) + 1 α ) Du). Taking into account.4), we obtain u L = which completes the proof. Corollary 3.3. For all finite L > there exists a unique classical solution of 3.4)-3.6). To prove Theorem 3.1, we must exten an interval, L) to R +. To o this, we nee a priori estimates of solutions to the problem 3.4)-3.6) inepenent of L >. These estimates provies the following result. Lemma 3.4. Let > k 3 an fx) : e kx/ f L R + ) CR + ). Then for all finite L > solutions of 3.4)-3.6) satisfy the inequality 3 e kx, D i u ) L C R e kx, f ) + q1], i= where the constant C R oes not epen on L. Proof. Multiplying 3.4) by u an integrating over, L), we obtain D 3 u, u) L + u L = f, u) L, 3.7) an I 1 = D 3 u, u) L 1 1 α ) Du) + β α 1 )u ) q 1 C 4 Du) + u )) q 1, where C 4 = min{ 1 α, β α 1 }. Since D3 u, u) L + q 1, then an u L D3 u, u) L + q 1 + u L 1 f L + u L + q 1 Returning to 3.7), we obtain which implies u L C) f + q 1). 3.8) u L + C 4 Du) + u )) C) f + q 1 Du) + u ) C) f + q 1). 3.9) Multiplying 3.4) by e kx u an integrating over, L), we obtain e kx, u ) L + D 3 u, e kx u) L = f, e kx u) L, 3.1)

5 EJDE-11/113 NONLINEAR QUARTER-PLANE PROBLEM 5 an e kx D 3 u, u) L K 1 u ) + K Du) + 3k ekx, Du ) L k3 ekx, u ) L 1 q 1, where K 1 = β 1 k α + k, K = 1 1 α + k ). With this, 3.1) becomes ekx, u ) L k3 ekx, u ) L + 3k ekx, Du ) L C 5 Du) + u ) ) + C)e kx, f ) + 1 q 1, where C 5 = max{ K 1, K, 1}. Using 3.9), we have e kx, u ) L + e kx, Du ) L C, k) e kx, f ) + q 1). 3.11) Now, multiplying 3.4) by e kx D 3 u an integrating over, L), we obtain e kx, D 3 u ) L C, k) e kx, f ) + q1 ) an multiplying 3.4) by e kx Du, we obtain 3.1) D u)du) + e kx, D u ) L + ke kx D u, Du) L e kx Du, u) L = e kx, fdu) L. Taking into account 3.), 3.9), 3.11), we fin that e kx, D u ) L C, k) e kx, f ) + q 1). Aing to this inequality 3.8), 3.1), we complete the proof. Since estimates of this Lemma o not epen on L, it allows us to exten an interval, L) to R + an by compactness arguments we can eliminate conition f CR + ). Uniqueness of a solution follows from 3.8). This completes the proof of Theorem Linear evolution problem Consier the linear initial-bounary value problem where e kx, u t + D 3 u = fx, t), x, t) Q T ; 4.1) D u, t) + αdu, t) + βu, t) + qt) =, t, T ); 4.) ux, ) = u x), x R + ; 4.3) u H 3 R + ), f, f t C, T ; L R + )), q H 1, T ); 4.4) 3 D i u ) + i= e kx, f )t) + e kx, f t )t)]t + Henceforth we will use the following Lemma see 9]). q t) + q t t))t <. 4.5)

6 6 N. A. LARKIN, E. TRONCO EJDE-11/113 Lemma 4.1 Discrete Gronwall Lemma). Let k n be a sequence of non-negative real numbers. Consier a sequence φ n such that n 1 n 1 φ g, φ n g + p s + k s φ s, n 1 s= s= with g an p s. Then for all n 1 it hols n 1 ) φ n g + p s exp { n 1 } k s. s= To stuy 4.1)-4.3), we use the metho of semi-iscretization with respect to t, 6, 15]. Define h = T >, N N, N u n x) = ux, nh), q n = qnh), f n x) = fx, nh), n = 1,..., N; s= u x) = ux, ) = u x); u n hx) = un x) u n 1 x), qh n = qn q n 1, n = 1,..., N; h h u h u t x, ) = fx, ) D 3 ux, ). We approximate 4.1)-4.3) with the system Lu n un h + D3 u n = un 1 h + f n 1, x R + ; 4.6) D u n ) + αdu n ) + βu n ) + q n 1 =, n = 1,..., N; 4.7) u x) = u x) H 3 R + ), x R ) By Theorem 3.1, given f n 1, q n 1 an u n 1 satisfying e kx, f n 1 x) + u n 1 x) ) + q n 1 C, there exists a unique solution u n x) H 3 R + ) of 4.6)-4.8) such that e kx, u n x) ) C. 4.9) Proposition 4.. Let u an fx, t) be such that for all t, T ) e kx, u x) + fx, t) ) C. Then for all n = 1,..., N an N > k 3 T problem 4.6)-4.8) amits a unique solution u n H 3 R + ) such that e kx, u n x) ) C. Proof. For n = 1 we have f x) = fx, ), u x) = u x), q = q) an 4.6)-4.8) becomes Due to 4.4)-4.5), u 1 x) + D 3 u 1 x) = fx, ) + u x) F 1 x), h h D u 1 ) + αdu 1 ) + βu 1 ) + q =. e kx, F 1 x) ) C.

7 EJDE-11/113 NONLINEAR QUARTER-PLANE PROBLEM 7 Taking 1/h > k 3, by Theorem 3.1, there exists a unique solution u 1 H 3 R + ) of the above problem satisfying e kx, u 1 x) ) C. Repeating this proceure, the result follows. To prove solvability of 4.1)-4.3), it is sufficient to pass to the limit in 4.6)-4.8) as h. For this purpose we nee the following lemma. Lemma 4.3. Assume conition 4.5). Then for all h > sufficiently small an l = 1,..., N the solutions u n x) of 4.6)-4.8) satisfy { sup e kx, u l ) + e kx, u l h ) } + 1 l N C { 3 e kx, D i u ) + i= l { e kx, Du n )h + e kx, Du n h )h } e kx, f + f t )t) t + q t) + q t t)) t }, 4.1) where the constant C > oes not epen on h >. Proof. First we prove a priori estimates inepenent of h > for u n an u n h. Estimate I. Taking 1/h > k 3, multiplying 4.6) by e kx u n an integrating over R +, we obtain We estimate 1 h un u n 1, e kx u n ) + D 3 u n, e kx u n ) = f n 1, e kx u n ). 4.11) I 1 = 1 h un u n 1, e kx u n ) ekx, u n ) h ekx, u n 1 ) ; h I = D 3 u n, e kx u n ) K 1 u n ) + K Du n ) + 3k ekx, Du n ) k3 ekx, u n ) 1 qn 1, where K 1 = β 1 k α+k an K = 1 1 α + k ). I 3 = f n 1, e kx u n ) 1 ekx, u n ) + 1 ekx, f n 1 ). Substituting I 1, I, I 3 in 4.11), multiplying the result by h an summing from n = 1 to n = l N, we obtain l 3k e kx, Du n )h + e kx, u l ) e kx, u ) C 5 + l Du n ) + u n ) ) h ] + k 3 + 1) l e kx, f n 1 )h + l q n 1 h. l e kx, u n )h 4.1)

8 8 N. A. LARKIN, E. TRONCO EJDE-11/113 Making k = in 4.11), using I 1 I 3, multiplying the result by h an summing from n = 1 till n = l N, we obtain where hc 4 h l Du n ) + u n ) ) + u l u l l 1 l 1 u n + h f n + h q n, n= n= C 4 = min{ 1 α, β α 1 }. Consiering < h < 1/, we have u l l 1 l 1 l 1 u + h u n + h f n + h q n. Taking into account 4.4), an N N h f n = h n= n= n= n= f n x) x M l 1 h q n M q t) t, n= where the constants M an M o not epen on h. Using the Discrete Gronwall Lemma, we fin u l l 1 l 1 u + h f n + h q n ) expt ) u + M M 1 u + n= n= f x, t) x t + M f x, t) x t + q t) t with M 1 = max{e T, Me T, M e T }. Now, from 4.13), l Du n ) + u n ) ) h M u + n= f x, t) x t ) q t) t expt ) ), ] f x, t) x t + q t) t an M = 1 + M 1 T + M + M )/C 4 ). Consiering < h < 1 3k l e kx, Du n )h + e kx, u l ) e kx, u ) + CT, k) u ) an using 4.1), ] f x, t) x t + q t) t

9 EJDE-11/113 NONLINEAR QUARTER-PLANE PROBLEM 9 + k 3 + 1) l l 1 l 1 e kx, u n )h + e kx, f n )h + h q n. n= Taking h such that < 1 + k 3 )h < 1/, we obtain l e kx, Du n )h + e kx, u l ) CT, k) e kx, u ) + l 1 + k 3 + 1) e kx, u n )h + h n= n= ] e kx f x, t) x t + q t) t l 1 n= q n l N. Applying Discrete Gronwall Lemma, we obtain e kx, u l ) Ck, T ) e kx, u ) + e kx f x, t) x t + Therefore, l e kx, Du n )h CT, k) e kx, u ) + ] e kx f x, t) x t + q t) t. Estimate II. Writing 4.6) as Lu n Lu n 1 )/h, we have L h u n h = un h un 1 h h ] q t) t. 4.14) 4.15) + D 3 u n h = f n 1 h, x R + ; 4.16) D u n h) + αdu n h) + βu n h) + q n 1 h =, n = 1,..., N; u h u t x, ) = fx, ) D 3 u x); f hx) f t x, ). Multiplying 4.16) by e kx u n h, integrating over R+ an acting as by proving the estimate I, we obtain e kx, u l h ) + l e kx, Du n h )h CT, k) e kx, u t x, )) + 3 CT, k) e kx, D i u ) + i= e kx, ft )t) t + This an 4.14), 4.15) completes the proof. e kx, f t + f )t) t + ] qt t) t ] qt t) t. 4.17) Theorem 4.4. Let u x), qt) an fx, t) satisfy 4.4), 4.5). Then there exists a unique solution of 4.1)-4.3), such that u L, T ; H 3 R + )), u t L, T ; L R + )) L, T ; H 1 R + )).

10 1 N. A. LARKIN, E. TRONCO EJDE-11/113 Proof. Rewriting 4.6)-4.8) as D 3 u n x) + k 3 u n x) = k 3 u n x) u n hx) + f n 1 x) F x), x R + ; D u n ) + αdu n ) + βu n ) + q n 1 =, n = 1,..., N; u x) = u x), x R + ; an taking into account Theorem 3.1, we fin a solution u n H 3 R + ) such that Hence u n H 3 R + ) 3 e kx, D i u n ) C i= 3 e kx, D i u n ) i= e kx, F ) + q n 1 ). C e kx, u n h + u n + f n 1 ) + q n 1 ) C { 3 e kx, D i u ) + i= e kx, f + f t )t) + q t) + q t t)] t }, where the constant C for h > sufficiently small oes not epen on h. Because the estimates 4.14), 4.15), 4.17) an the inequality above are uniform in h >, the stanar arguments see 15]) imply that there exists a function ux, t) such that u n u weakly-* in L, T ; H 3 R + )), u n h u t weakly-* in L, T ; L R + )) L, T ; H 1 R + )). Here u n an u n h are interpolations of un an u n h, respectively, an ux, t) is a solution of 4.1)-4.3). For more etails, see 15]. 5. Nonlinear problem. Local solutions In this section we prove the existence of local regular solutions to the nonlinear problem u t + D 3 u = udu Du, x, t) Q T ; 5.1) D u, t) + αdu, t) + βu, t) + u, t) u, t) + gt) =, t, T ); 5.) ux, ) = u x), x R + ; 5.3) where gt) is a given function, α an β satisfy.4). The main result here is as follows. Theorem 5.1. Let α an β satisfy.4), u x) H 3 R + ), g H 1, T ) an for some k > 3 e kx, D i u ) + e kx, u Du ) <. i= Then there exists a positive number T such that 5.1)-5.3) possesses a unique regular solution in Q T such that u L, T ; H 3 R + )), u t L, T ; L R + )) L, T ; H 1 R + ))

11 EJDE-11/113 NONLINEAR QUARTER-PLANE PROBLEM 11 an the following inequality hols: sup t,t ) + {e kx, u )t) + e kx, u t )t)} + u, t) + u t, t) ) t 3 CT, k) e kx, D i u ) + e kx, u Du ) + i= e kx, Du )t) + e kx, Du t )t)]t ] g t) + gt t)) t. Proof. We prove this theorem using the Banach fixe point theorem. Consier X = L, T ; H 3 R + )), Y = L, T ; L R + )) L, T ; H 1 R + )). Let V be the space with the norm V = {v : R +, T ] R : v X, v t Y, vx, ) = u x)} v V = sup t,t ) + + {e kx, v )t) + e kx, v t )t)} e kx, Dv )t) + e kx, Dv t )t)]t v, t) + v t, t) ) t. Obviously, V, ) is a Banach space. Define B R = {v V : v V R 1C }, C = max{1 + C 5 C 4, C 5δ C 4, δ}, with δ = 1/ min{1, k, C 5 }, an R > 1 is such that 3 e kx, D i u + u Du ) + i= For any vx, t) B R consier the linear problem 5.4) g t) + g t t)) t R. 5.5) u t + D 3 u = vdv Dv, x, t) Q T ; 5.6) D u, t) + αdu, t) + βu, t) + v, t) v, t) + gt) =, t, T ); 5.7) ux, ) = u x), x R + ; 5.8) where gt) H 1, T ), α an β satisfy.4). It is easy to verify that v, t) v, t) H 1, T ); fx, t) = vdv Dv satisfies conitions 4.4) an 4.5). Therefore, by Theorem 4.4, there exists a unique function ux, t) : u L, T ; H 3 R + )), u t L, T ; L R + )) L, T ; H 1 R + )) which solves 5.6)-5.8). Hence, one can efine an operator P relate to 5.6)-5.8) such that u = P v. Lemma 5.. There is a real T = T R) > such that an operator P : u = P v maps B R into itself. Proof. To prove this lemma it suffices to show the necessary a priori estimates:

12 1 N. A. LARKIN, E. TRONCO EJDE-11/113 Estimate I. Multiplying 5.6) by u, integrating over R + an repeating the calculations from Lemma 3.4, we fin t u t) + C 4 u, t) + Du, t) ) 5.9) u t) + vdv t) + Dv t) + C v, t) v, t) + g t). We estimate e kx Dvx, t) x = e kx Dvx, ) x + R + R + τ ekx, Dv )τ) τ e kx, Du ) + e kx Dv + Dv τ ) xτ R + an Moreover, v, t) sup v x, t) v t) Dv t) x R + ) 1/ e kx v x, t) x e kx Dvx, t) x R + R + R 1C ) R + 1R C ) 1/. vdv t) sup v x, t) Dvx, t) x x R + R + Hence 5.9) may be rewritten as ) 1/ R 1C ) R + 1R C ) 1/ R + 1R C ). t u t) + C 4 u, t) + Du, t) ) u t) + CR, C ) + g t). 5.1) Ignoring the secon term on the left sie of this inequality an applying Gronwall s lemma, we obtain u t) e T u + CR, C )T + Taking T > such that e T e CR, C )T R, we have u t) 6R, t, T ). g τ) τ ). Using this inequality an integrating 5.1) over, t), we obtain u, τ) + Du, τ) ]τ 1 C 4 CR, C )T + u + 1 C 4 CR, C )T + u + R ]. g τ) τ ] 5.11) Multiplying 5.6) by e kx u an integrating over R +, we fin t ekx, u )t) + 3ke kx, Du )t) C 5 u, t) + Du, t) ) + C 6 e kx, u )t) + e kx, vdv + Dv )t) + C v, t) v, t) ) + g t) C 5 u, t) + Du, t) ) + C 6 e kx, u )t) + CR, C ) + g t), 5.1)

13 EJDE-11/113 NONLINEAR QUARTER-PLANE PROBLEM 13 where C 6 = 1 + k 3. Ignoring the secon term on the left sie of 5.1), using 5.11) an applying the Gronwall lemma, we obtain e kx, u )t) e C6T e kx, u ) + C 5 e C6T u, τ) + Du, τ) ]τ + e C6T CR, C )T + e C6T g τ) τ e C6T e kx, u ) 1 + C 5 ) + e C 6T C 5 CR, C ) + CR, C )]T C 4 C 4 + e C6T g τ) τ + e C6T C 5 C 4 R. Choosing T > such that e C6T an C5 C 4 CR, C ) + CR, C )]T R C, we obtain e kx, u )t) 6R C + R ; t, T ). Returning to 5.1), we rewrite it as t ekx, u )t) + 3ke kx, Du )t) + C 5 u, t) 3C 5 u, t) + 3C 5 Du, t) + CR, C, k) + CR, C ) + g t). Integrating 5.13) over, t) an using 5.11), we fin e kx, u )t) + e kx, Du )τ) τ + u, τ) τ 5.13) e kx, u ) + δ 3C 5 CR, C )T + e kx, u C ) + R ] + δt CR, C, k) + CR, C )] 4 + δ g τ) τ e kx, u )1 + 3δC 5 C 4 ] + δt 3C 5 C 4 CR, C ) + CR, C, k) + CR, C )] + R C, where δ = 1/ min{1, k, C 5 }. Choosing T > such that we obtain e kx, u )t)+ δt 3C 5 C 4 CR, C ) + CR, C, k) + CR, C )] 3R C, e kx, Du )τ) τ + u, τ) τ 6R C ; t, T ). 5.14) Estimate II. Differentiating 5.6) with respect to t, multiplying by u t, integrating over R + an acting as by proving 5.14), we have t u t t) + C 4 ut, t) + Du t, t) ) Cɛ ) u t t) + ɛ v t Dv t) + ɛ vdv t t) + ɛ Dv t t) + CR, C ) Dv t t) + g t t), where ɛ > will be chosen later. We estimate v t Dv t) sup x R + v t x, t) Dv t) 5.15)

14 14 N. A. LARKIN, E. TRONCO EJDE-11/113 an Then 5.15) becomes 4 v t t) Dv t t) Dv t) CR, C ) Dv t t) vdv t t) CR, C ) Dv t t). t u t t) + C 4 ut, t) + Du t, t) ) Cɛ ) u t t) + ɛ CR, C ) Dv t t) + ɛ CR, C ) Dv t t) + CR, C ) Dv t t) + g t t). By the Gronwall lemma, u t t) e Cɛ)T u t x, ) + ɛ CR, C ) + e Cɛ)T ɛ CR, C ) Due to 5.6), Dv τ τ) τ + Dv τ τ) τ + CR, C ) ) gτ τ) τ. 5.16) ) Dv τ τ)τ u t x, ) 3 u Du + D 3 u + Du ) 3R. 5.17) Since ) 1/ Dv τ τ) τ τ ) 1/ Dv τ 1/ τ) τ T R 1C, we can take T > an ɛ > such that e Cɛ)T an ɛ CR, C )T 1/ R 1C + ɛ CR, C ) + CR, C )T 1/ R 1C CR, C ) in orer to obtain u t t) CR, C ), t, T ). 5.18) Substituting 5.18) into 5.16) an integrating over, t), we obtain which implies u t t) + C 4 uτ, τ) + Du τ, τ) ) τ u t ) + CR, C, ɛ )T + CR, C, ɛ ) + ɛ CR, C ) Dv τ τ) τ + g τ τ) τ Dv τ τ) τ u t ) + CR, C, ɛ )T + CR, C, ɛ )T 1/ + ɛ CR, C ) + R uτ, τ) + Du τ, τ) ) τ 1 C 4 u t ) + CR, C, ɛ )T + CR, C, ɛ )T 1/ ] 5.19) + 1 C 4 ɛ CR, C ) + R ].

15 EJDE-11/113 NONLINEAR QUARTER-PLANE PROBLEM 15 Differentiating 5.6) with respect to t, multiplying by e kx u t, integrating over R + an acting as earlier, we fin t ekx, u t )t) + 3ke kx, Du t )t) C 5 ut, t) + Du t, t) ) + k 3 e kx, u t )t) e kx, vdv) t u t )t) 5.) We estimate e kx, Dv t u t )t) + CR, C ) Dv t t) + g t t). e kx, Dv t u t )t) ɛ 1 e kx, Dv t )t) + Cɛ 1 )e kx, u t )t), where ɛ 1 > will be chosen later, an e kx, vdv) t u t )t) = e kx, vv t ) x u t )t) = v, t)v t, t)u t, t) + vv t, e kx Du t + ku t ])t) v, t)v t, t)u t, t) + ke kx, vv t + u t )t) + ke kx, Du t )t) + 1 k ekx, vv t )t). Since v, t)v t, t)u t, t) CR, C ) Dv t 1/ t) Du t 1/ t), by the Young inequality, v, t)v t, t)u t, t) CR, C, k) Dv t /3 t) + k Du t t). Then 5.) becomes t ekx, u t )t) + ke kx, Du t )t) C 5 ut, t) + Du t, t) ) + Ck, ɛ 1 )e kx, u t )t) + Ck)e kx, vv t )t) + ɛ 1 e kx, Dv t + CR, C ) Dv t /3 t) + CR, C ) Dv t t) + g t t). 5.1) Ignoring the secon term on the left-han sie an applying the Gronwall lemma, we obtain e kx, u t )t) e Ck,ɛ1)T e kx, u t )) + C 5 uτ, τ) + Du τ, τ) ) τ ] + e Ck,ɛ1)T Ck) + e CR, Ck,ɛ1)T C ) + T 1/ e kx, vv τ )τ) τ + ɛ 1 e kx, Dv τ )τ) τ ] T /3 Dv τ τ) τ Dv τ τ) τ ) 1/ ]] + e Ck,ɛ1)T ) 1/3 g τ τ) τ ]. Using 5.17), 5.19), 5.5) an 5.4) together with the choice of B R, we have e kx, u t )t) e Ck,ɛ1)T 3R + C 5 C 4 CR, C, ɛ )T + CR, C, ɛ )T 1/ + ɛ CR, C ) + 4R ] ] + e Ck,ɛ1)T Ck, R)T + ɛ 1 1R C + CR, C )T /3 + T 1/ ) + R ].

16 16 N. A. LARKIN, E. TRONCO EJDE-11/113 Choose T >, ɛ > an ɛ 1 > sufficiently small to obtain Returning to 5.1), it gives e kx, u t )t) CR, C ). 5.) t ekx, u t )t) + ke kx, Du t )t) + C 5 u t, t) 3C 5 ut, t) + Du t, t) ) + Ck, R, C, ɛ 1 ) + Ck)e kx, vv t )t) + ɛ 1 e kx, Dv t )t) + CR, C ) Dv t /3 t) + CR, C ) Dv t t) + g t t). Integration over, t) yiels e kx, u t )t) + e kx, Du τ )τ) τ + u τ, τ) τ e kx, u t )) + δ 3C 5 uτ, τ) + Du τ, τ) ) ] τ + Ck, R, C, ɛ 1 )T + δ Ck) + CR, C ) e kx, vv τ )τ) τ + ɛ 1 e kx, Dv τ )τ) τ ] Dv τ /3 τ) τ + δ CR, C ) Dv τ τ) τ + where δ = 1/ min{1, k, C 5 }. Taking into account 5.17) an 5.19), we fin e kx, u t )t) + e kx, Du τ )τ) τ + u τ, τ) τ ] gτ τ) τ, e kx, u t )) 1 + 3δ C 5 ) C 5 + 3δ CR, C, ɛ )T + CR, C, ɛ )T 1/ C 4 C 4 + ɛ CR, C ) + R ] + δck, R, C, ɛ 1 )T + δ CR, C, k)t + ɛ 1 1R C + CR, C )T /3 + CR, C )T 1/ + R ]. Choose T >, ɛ > an ɛ 1 > sufficiently small in orer to obtain e kx, u t )t) + e kx, Du τ )τ) τ + This inequality an 5.14) completes the proof. u τ, τ) τ 6R C, t, T ). Lemma 5.3. For T > sufficiently small, the operator P is a contraction mapping. Proof. For arbitrary v 1, v B R we enote u i = P v i, i = 1,, s = v 1 v an z = u 1 u. Then zx, t) satisfies the initial bounary value problem z t x, t) + D 3 zx, t) = 1 Dv 1 v ) Ds, x, t) Q T ; 5.3) D z, t) + αdz, t) + βz, t) + v 1, t) v 1, t) v, t) v, t) =, 5.4) Define zx, ) =, x R ) ρ v 1, v ) = ρ s) = supe kx, s )t) + t> e kx, Ds )t) t. 5.6)

17 EJDE-11/113 NONLINEAR QUARTER-PLANE PROBLEM 17 Estimate III. Multiplying 5.3) by z an integrating over R +, we obtain We estimate an t z t) + C 4 z, t) + Dz, t) ) v 1, t) v 1, t) v, t) v, t) z, t) Dv 1 v ), z)t) Ds, z)t). I 1 = Dv 1 v ), z)t) Dv 1 v ) t) z t) CR, C ) s t) + Ds t) ) z t) CR, C ) s t) + ɛ 4 Ds t) + CR, C, ɛ 4 ) z t), I = Ds, z)t) Ds t) z t) ɛ 4 Ds t) + Cɛ 4 ) z t), where ɛ 4 > will be chosen later; v 1, t) v 1, t) v, t) v, t) z, t) z, = v 1, t) s, t) + v, t) v 1, t) v, t) ) t) ) v 1, t) s, t) + v, t) v 1, t) v, t) z, t) ) = v 1, t) s, t) + v, t) s, t) z, t) ) CR, C ) s 1/ t) Ds 1/ t) z, t) CR, C, ɛ) s t) Ds t) ɛ z, t) CR, C, ɛ, ɛ 4 ) s t) ɛ 4 Ds t) ɛ z, t), where ɛ = 1 β α ) 1. Then 5.7) becomes t z t) + C 4 z, t) + Dz, t) ) 3ɛ 4 Ds t) + CR, C, ɛ, ɛ 4 ) s t) + CR, C, ɛ 4 ) z t). 5.7) 5.8) Ignoring the secon term on the left-han sie of 5.8) an applying the Gronwall lemma, we fin ] z t) e CR,C,ɛ 4)T 3ɛ 4 Ds τ) τ + CR, C, ɛ, ɛ 4 ) s τ) τ. Taking T > for a fixe ɛ 4 > such that e CR,C,ɛ 4)T, we have ) z t) 6ɛ 4 + CR, C, ɛ, ɛ 4 )T ρ s). The choice of T > such that CR, C, ɛ, ɛ 4 )T 4ɛ 4 yiels Integrating 5.8) over, t) an using 5.9), we obtain z t) 1ɛ 4 ρ s). 5.9) z, τ) + Dz, τ) ) τ 3ɛ 4 + CR, C, ɛ, ɛ 4 )T ) C 4 ρ s). 5.3)

18 18 N. A. LARKIN, E. TRONCO EJDE-11/113 Multiplying 5.3) by e kx z, we fin t ekx, z )t) + 3ke kx, Dz )t) C 5 z, t) + Dz, t) ) + CR, C, k, ɛ 4 )e kx, z )t) + CR, C, ɛ 4, ɛ)e kx, s )t) + 3ɛ 4 e kx, Ds )t). Due to the Gronwall lemma an 5.3), e kx, z )t) e CR,C,k,ɛ 4)T 3ɛ 4 + CR, C, ɛ, ɛ 4 )T )] C5 ρ s) C 4 + e CR,C,k,ɛ 4)T CR, C, ɛ 4 )T + 3ɛ 4 ]ρ s). Choose T > an ɛ 4 > sufficiently small to obtain 5.31) e kx, z )t) 1 4 ρ s). 5.3) Integrating 5.31) over, t) an using 5.3) an 5.3), we obtain e kx, z )t) + e kx, Dz )τ) τ C 5 δ 3ɛ 4 + CR, C, ɛ, ɛ 4 )T ) ρ s) + δcr, C, k, ɛ 4 )T ρ s) C 4 + δcr, C, ɛ 4, ɛ)t supe kx, s )t) + 3δɛ 4 e kx, Ds )τ) τ. t> Finally, we choose T > an ɛ 4 > sufficiently small to obtain ρ z) γρ s), < γ < 1, with ρ = ρs) efine in 5.6). This competes the proof. Lemmas 5. an 5.3 an the contraction mapping arguments imply the existence an uniqueness of a generalize solution u B R of 5.1)-5.3): u, u t L, T ; L R + )) L, T ; H 1 R + )). It is easy to verify that udu + Du H 1, T ; L R + )). Hence, by Theorem 3.1, u L, T ; H 3 R + )). This completes the proof of Theorem 5.1. Remark 5.4. One can observe that to prove the existence of local solutions, we o not pose any restrictions on k while in conitions of Theorem. the value of k epens on coefficients a 1, a. 6. Global solutions To prove Theorem., we must exten the obtaine local solution to the whole, T ) with an arbitrary fixe T >. For this purpose, we nee a priori estimates of local solutions uniform in t, T ). Estimate I. Multiplying 5.1) by u, we obtain t u t) + D 3 u, u)t) + Du, u )t) + Du, u)t) =. 6.1) We calculate I 1 = D 3 u, u)t)

19 EJDE-11/113 NONLINEAR QUARTER-PLANE PROBLEM 19 β α ɛ ) u, t) + 1 α ) Du, t) + u, t) 3 1 ɛ gt), where ɛ is an arbitrary positive number; I = Du, u)t) = u, t), I 3 = Du, u )t) = 3 u3, t) 3 u, t) 3. Since β α 1 = a 1 > an 1 α = a >, choosing ɛ = a 1 an substituting I 1 I 3 into 6.1), we obtain This implies t u t) + a 1 u, t) + a Du, t) u, t) 3 1 a 1 gt). 6.) u t) + a 1 u, τ) + a Du, τ) u, τ) 3) τ u ) + 1 gτ) τ. a 1 Estimate II. Multiplying 5.1) by e kx u, we obtain t ekx, u )t) + 3ke kx, Du )t) + u, t) 3 + β 1 α k + k a 1 4 ] u, t) + 1 α k) Du, t) k + k 3 )e kx, u )t) e kx u, Du)t) + a 1 gt). 6.3) 6.4) The conitions of Theorem. imply which transforms 6.4) into k + k a 1 4, k a t ekx, u )t) + 3ke kx, Du )t) + u, t) 3 + a 1 u, t) + a Du, t) k + k 3 )e kx, u )t) e kx u, Du)t) + a 1 gt). 6.5) We estimate I 1 = e kx u, Du)t) ke kx, Du )t) + 1 k ekx, u 4 )t) ke kx, Du )t) + 1 k sup x R + e kx u x, t) u t). Taking into account 6.3), we have I 1 ke kx, Du )t) + u ) + 1 ) gτ) τ e kx/ u t) De kx/ u) t) k a 1 ke kx, Du )t) + u ) + 1 ) { gτ) k τ k a 1 ekx/ u t) } + u ) + 1 { e gτ) τ) kx/ u t) e kx/ Du t) }. k a 1

20 N. A. LARKIN, E. TRONCO EJDE-11/113 Using the Young inequality, we obtain I 1 ke kx, Du )t) + C u, g L,T ), k)ekx, u )t). 6.6) Substituting I 1 into 6.5), we obtain t ekx, u )t) + k ekx, Du )t) C u, g L,T ), k)ekx, u )t) + g t)]. Due to the Gronwall lemma an 6.3), ] e kx, u )t) C u, g L,T ) e, k, T ) kx, u ) + g τ) τ. 6.7) Hence e kx, u )t)+ e kx, Du )τ) τ C e kx, u )+ ] g τ) τ, t, T ); 6.8) where C epens on k >, T >, g L,T ) an u. Estimate III. Differentiate 5.1) with respect to t, multiply by e kx u t an acting as by proving Estimate II, we obtain We efine t ekx, u t )t) + 3ke kx, Du t )t) + e kx u ) xt, u t )t) + 4 u, t) u t, t) + a 1 u t, t) + a Du t, t) k + k 3 )e kx, u t )t) + a 1 g t t). I 1 = e kx u ) xt, u t )t) = u, t)u t, t) ke kx u, u t )t) + e kx Du, u t )t) which we estimate as follows: an I 11 = ke kx u, u t )t) k sup x R + ux, t) e kx, u t )t) k 1 + u t) + e kx, Du )t) ) e kx, u t )t), I 1 = e kx Du, u t )t) sup x R + u t x, t) e kx Du, u t )t) u t 1/ t) Du t 1/ t) e kx/ Du t) e kx/ u t t) ke kx, Du t )t) + 1 k 1 + ekx, Du )t)]e kx, u t )t). Substituting I 1 into 6.9) an using 6.3), we come to the inequality t ekx, u t )t) + ke kx, Du t )t) + a 1 u t, t) + a Du t, t) + 3 u, t) u t, t) Ck)1 + u t) + Du t)]e kx, u t )t) + a 1 g t t). Taking into account 6.8), it is easy to see that 1 + u t) + Du t) L 1, T ). 6.9)

21 EJDE-11/113 NONLINEAR QUARTER-PLANE PROBLEM 1 Hence, by the Gronwall lemma, we obtain e kx, u t )t) + e kx, Du τ )τ) τ 3 ] C e kx, D i u ) + e kx, u Du ) i= + C g H 1,T ), t, T ); where C epens on k >, T >, g L,T ) an u. Returning to 5.1), we can write it as D 3 u + u = u u t udu Du. 6.1) By Theorem 3.1, 3 e kx, D i u )t) C e kx, D i u ) + e kx, u Du ) + g H 1,T )] ). i= On the other han, i= D 4 u = Du t D u Du ud u L, T ; L R + )) 6.11) an e kx, D 4 u )τ) τ 3 This gives 3 e kx, D 4 u )τ) τ e kx, D 4 u )τ) τ + 3 e kx, D 4 u )τ) τ + 3 e kx, D u )τ) τ + 3 e kx, ud u )τ) τ. e kx, Du τ )τ) τ e kx, D u )τ) τ + 3 Du t) D u t) + 3 u t) Du t) This an 6.11) imply e kx, D 4 u )τ) τ e kx, D u )τ) τ. e kx, Du τ )τ) τ e kx, Du 4 )τ) τ e kx, Du )τ) τ C e kx, D i u ) + e kx, u Du ) + g H 1,T )] ). i= Hence, u L, T ; H 4 R + )). Estimates I, II, III complete Section 6. Uniqueness of the obtaine regular solution follows from the contraction mapping principle. The proof of Theorem. is complete.

22 N. A. LARKIN, E. TRONCO EJDE-11/113 Acknowlegements. The authors appreciate very much the etaile an constructive remarks mae by the anonymous referee. References 1] T. B. Benjamin; Lectures on Nonlinear Wave Motion, Lecture Notes in Applie Mathematics ), ] J. L. Bona, V. A. Dougalis; An Initial an Bounary-Value Problem for a Moel Equation for Propagation of Long Waves, J. Math. Anal. Appl ), ] J. L. Bona, S. M. Sun, B. Y. Zhang; A Non-Homogeneous Bounary-Value Problem for the Korteweg-e Vries Equation in a Quarter Plane, Trans. Amer. Math. Soc. 354 ), no., electronic). 4] B. A. Bubnov; General Bounary-Value Problems for the Korteweg-e Vries Equation in a Boune Domain, Differentsial nye Uravneniya 151) 1979), Translation in: Differ. Equ ), ] B. A. Bubnov; Solvability in the large of nonlinear bounary value problems for the Kortewege Vries equation in a boune omain, Differential Equations ), no 1, ] G. G. Doronin, N. A. Larkin; Kawahara equation in a boune omain, Discrete an Continuous Dynamic Systems, Serie B, 1 8), ] G. G. Doronin, N. A. Larkin; KV Equation in Domains with Moving Bounaries, J. Math. Anal. Appl. 38 7), ] C. N. Dorny; A vector space approach to moels an applications, New York, Wiley 1975). 9] E. Emmrich; Discrete versions of Gronwall s lemma an their applications to the numerical analysis of parabolic problems, Prepint No 637, July 1999, Prepint Reihe Mathematik Techniche Universitat Berlin, Fachbereich Mathematik. 1] A. V. Faminskii; An initial bounary-value problem in a half-strip for the Korteweg-e Vries equation in fractional orer Sobolev spaces, Comm. Partial Differential Equations ) 4) ] A. V. Faminskii, N. A. Larkin; Initial-bounary value problems for quasilinear ispersive equations pose on a boune interval, Electron. J. Differential Equations, v.1 1), No. 1, pp ] A.v. Faminskii, N. A. Larkin; O-orer quasilinear evolution equations pose on a boune interval, Bol. Soc. Paranaense e Mat., 35) v. 8, 1 1), ] A. V. Faminskii; On an Initial Bounary Value Problem in a Boune Domain for the Generalize Korteweg-De Vries Equation, Functional Differential Equations, v. 8 1), ] S. Ginikin, L. Volevich; Mixe problem for partial ifferential with quasihomogeneous principal part, Translations of Mathematical Monographs, 147, Amer. Math. Soc., Provience, RI, ] O. A. Layzhenskaya; The Bounary Value Problems of Mathematical Physics, Applie mathematical sciences, New York, ] N. A. Larkin; Correct initial bounary value problems for ispersive equations, J. Math. Anal. Appl., 344 8), ] N. A. Larkin; Korteweg-e Vries an Kuramoto-Sivashinsky Equations in Boune Domains, J. Math. Anal. Appl. 971) 4), ] N. A. Larkin, J. Luchesi; General mixe problems for the KV equations on boune intervals, Electron. J. of Differential Equations, vol. 1 1), No. 168, ] F. Linares, G. Ponce; Introuction to nonlinear Dispersive Equations, Eitora IMPA, 6. Nikolai A. Larkin Departamento e Matemática, Universiae Estaual e Maringá, Av. Colombo 579: Agência UEM, 87-9, Maringá, PR, Brazil aress: nlarkine@uem.br Euaro Tronco Departamento e Matemática, Universiae Estaual e Maringá, Av. Colombo 579: Agência UEM, 87-9, Maringá, PR, Brazil aress: etronco@uem.br