THE CAUCHY PROBLEM FOR GENERALIZED ABSTRACT BOUSSINESQ EQUATIONS

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1 Dynamic Systems and Alications 25 ( THE CAUCHY PROBLEM FOR GENERALIZED ABSTRACT BOUSSINESQ EQUATIONS VELI B. SHAKHMUROV Deartment of Mechanical Engineering, Okan University, Akfirat, Tuzla Istanbul, Turkey Khazar University, Baku, Azerbaijan ABSTRACT. In this aer, the existence and uniqueness of solution of the Cauchy roblem for abstract Boussinesq equations is obtained. By alying this result, the Wentzell-Robin tye mixed roblem for Boussinesq equations and the Cauchy roblem for finite or infinite systems of Boussinesq equations are studied. AMS 2: 35Lxx, 35Qxx, 43Axx, 47Axx, 47Hxx, 39A.. Introduction, Definitions and Background The subject of this aer is to study the local existence and uniqueness of solution of the Cauchy roblem for the following Boussinesq-oerator equation (. u tt u tt + Au = f (u, x R n, t (, T, (.2 u (x, = ϕ (x, u t (x, = ψ (x, where A is a linear oerator in a Banach sace E, u(x, t denotes the E-valued unknown function, f(u is the given nonlinear function, ϕ (x and ψ (x are the given initial value functions, subscrit t indicates the artial derivative with resect to t, n is the dimension of sace variable x and denotes the Lalace oerator in R n. This is a first aer that devoted to initial value roblem for abstract Boussinesq equations. Since the equation contain generally, unbounded oerator in a abstract Banach sace E and the roblem (. is considered in UMD-valed function sace, the methods alied for Boussinesq equations in scalar case does not ass here. By this reason, in this aer, the methods of roofs naturally differs to those used in scalar case. Since the Banach sace E is arbitrary and A is a ossible linear oerator, by choosing E and A we can obtain numerous classis of Boussinesq tye equations which have a different alications (see [ 7] and the references therein. Received March 3, $5. c Dynamic Publishers, Inc.

2 V. B. SHAKHMUROV Here, by insiring [8] and [9] in this aer, we obtain the local existence and uniqueness of small-amlitude solution of the roblem (. (.2. Note that, differential oerator equations were studied e.g. in [ 3]. The Cauchy roblem for abstract hyerbolic equations were treated e.g. in [ 2]. The strategy is to exress the abstract Boussinesq equation as an integral equation with oerator coefficient, to treat in the nonlinearity as a small erturbation of the linear art of the equation, then use the contraction maing theorem and utilize an estimate for solutions of the linearized version to obtain a riori estimates on E-valued L norms of solutions. The key ste is the derivation of the uniform estimate for the solutions of the linearized Boussinesg-oerator equation. If we choose the UMD sace E as a abstract Hilbert sace H, then we obtain the existence and uniqueness results of the Cauchy roblem for abstract Boussinesg equation defined in Hilbert sace valued classes. For examle, if we choose E a concrete Hilbert sace, for examle E = L 2 (R n and A =, we obtain the scalar Cauchy roblem for generalized Boussinesq tye equation (.3 u tt u tt u = f (u, x R n, t (, T, (.4 u (x, = ϕ (x, u t (x, = ψ (x. The equation (.3 arise in different situations (see [, 2]. For examle, equation (.3 for n = describes a limit of a one-dimensional nonlinear lattice [3], shallowwater waves [4,5] and the roagation of longitudinal deformation waves in an elastic rod [6]. In [8] and [9] the existence of the global classical solutions and the blow-u of the solution for the initial boundary value roblem and the Cauchy roblem (.3 (.4 are obtained. Moreover, let we choose E = L (, and A to be differential oerator with generalized Wentzell-Robin boundary condition defined by { } (.5 D (A = u W 2 (,, B j u = Au (j + α ji u (i (j, j =,, i= Au = au (2 + bu ( + cu, in (. (.2, where α ji are comlex numbers, a, b, c are comlex-valued functions. Then, we get the following Wentzell-Robin tye mixed roblem for Boussinesq equation (.6 u tt x u tt + a 2 u y 2 + b u y + cu = xf (u, x R n, y (,, t (, T, (.7 B j u = Au (x, t, j + α ji u (i (x, t, j =, j =,, i= (.8 u (x, y, = ϕ (x, y, u t (x, y, = ψ (x, y,

3 ABSTRACT BOUSSINESQ EQUATIONS Note that, the regularity roerties of Wentzell-Robin tye BVP for ellitic equations were studied e.g. in [3,32] and the references therein. Here Ω = R n (,, =(, and L denotes the sace of all -summable comlex-valued functions with mixed norm i.e., the sace of all measurable functions f defined on Ω, for which f L = ( R n ( f (x, y dy dx <. By alying the general Theorem 2.5 obtained here, we established the local existence and uniqueness of small-amlitude solution of the roblem (.6 (.8 in mixed L sace. In order to state our results recisely, we introduce some notations and some function saces. Let E be a Banach sace. L (Ω; E denotes the sace of strongly measurable E-valued functions that are defined on the measurable subset Ω R n with the norm ( f L (Ω;E = f (x E dx, <, f L = ess su f (x E. Ω Let E and E be two Banach saces and E is continuously and densely embeds into E. Let H s, (R n ; E, < s < denotes the E-valued Sobolev sace of order s which is defined as: H s, = H s, (R n ; E = (I s 2 L (R n ; E, u H s, = H s, (R n ; E, E denotes the Sobolev-Lions tye sace, i.e., { H s, (R n ; E, E = u H s, (R n ; E L q (R n ; E, u H s, (R n ;E,E = u L (R n ;E + u H s, (R n ;E < }. x Ω (I s 2 L u. (R n ;E Let B s,q (Rn ; E denote E-valued Besov sace (see e.g. [34, 5]. Let B s,q (Rn ; E, E denote the sace L (R n ; E B s,q (R n ; E with the norm u B s,q (R n ;E,E = u L (R n ;E + u B s,q (Rn ;E <. For estimating lower order derivatives we use following embedding theorem that is obtained from [22, Theorem ]: Theorem.. Suose the following conditions are satisfied: ( E is a UMD sace and A is an R-ositive oerator in E (see e.g. [29] for definitions; (2 α = (α, α 2,...,α n is a n-tules of nonnegative integer number and s is a ositive number such that κ = [ ( α + n s ], µ κ, < q < ; < h h, q

4 2 V. B. SHAKHMUROV where h is a fixed ositive number. Then the embedding D α H s, (R n ; E (A,E L q (R n ; E (A κ µ is continuous and for u H s, (R n ; E (A,E the following uniform estimate holds D α u L q (R n ;E(A κ µ hµ u H s, (R n ;E(A,E + h ( µ u L (R n ;E. In a similar way as [9, Theorem A ] we obtain: Proosition A. Let < q and E be UMD sace. Suose Ψ h C n (R n \ {};B (E and there is a ositive constant K such that ({ } su R ξ β +n( q D β Ψ h (ξ: ξ R n \ {}, β k {, } h Q K. Then Ψ h is a uniformly bounded collection of Fourier multilier from L (R n ; E to L (R n ; E. 2. Estimates for linearized equation Here, we make the necessary estimates for solutions of Cauchy roblem for the linearized abstract Boussinesq equation (2. u tt u tt + Au = g (x, t, x R n, t (, T, (2.2 u (x, = ϕ (x, u t (x, = ψ (x. Let X = L (R n ; E, Y s, = H s, (R n ; E, Y s, = H s, (R n ; E L (R n ; E, Y s, = Hs, (R n ; E L (R n ; E. Condition 2.. Assume E is an UMD sace and the oerator A is ositive in E. Moreover suose ϕ, ψ Y s, s,, g (., t Y for t (, T and s > n First we need the following lemmas for < <. Lemma 2.2. Suose the Condition 2. hold. Then roblem (2. (2.2 has a unique generalized solution. Proof. By using of the Fourier transform we get from (2. (2.2 (2.3 û tt (ξ, t + A ξ û (ξ, t = ξ 2 ( + ξ 2 ĝ (ξ, t, û (ξ, = ˆϕ (ξ, û t (ξ, = ˆψ (ξ, ξ R n, t (, T, where û (ξ, t is a Fourier transform of u (x, t with resect to x and A ξ = ( + ξ 2 A, ξ R n.

5 ABSTRACT BOUSSINESQ EQUATIONS 3 By virtue of [,] we obtain that A ξ is a generator of a strongly continuous cosine oerator function and roblem (2.3 has a unique solution for all ξ R n, moreover, the solution can be written as (2.4 û (ξ, t = C (t ˆϕ (ξ + S (t ˆψ (ξ + S (t τ, ξ, A ξ 2 ĝ (ξ, τdτ, t (, T, where C (t = C (t, ξ, A is a cosine and S (t = S (t, ξ, A is a sine oerator-functions (see e.g. [] generated by arameter deendent oerator A ξ. From (2.4 we get that, the solution of the roblem (2. (2.2 can be exressed as u (x, t = S (t, A ϕ (x + S 2 (t, A ψ (x (2.5 + (2π n e ixξ S (t τ, ξ, A ξ 2 ĝ (ξ, τdτdξ, R n where S (t, A and S 2 (t, A are linear oerators in E defined by S (t, A ϕ = (2π n e ixξ C (t ˆϕ (ξdξ, R n (2.6 S 2 (t, A ψ = (2π n e ixξ S (t ˆψ (ξdξ. R n t (, T, Lemma 2.3. Suose the Condition 2. hold. Then the solution (2. (2.2 satisfies the following uniform estimate (2.7 ( ( u X + u t X C ϕ Y s, + ϕ X Proof. Let N N and + ψ Y s, + ψ X + ( g (., τ Y s, + g (., τ X dτ. Π N = {ξ : ξ R n, ξ N}, Π N = {ξ : ξ R n, ξ N}. It is clear to see that (2.8 u (., t L (R n ;E F C (t ˆϕ (ξ F L (Π ;E + S (t ˆψ (ξ N L (Π ;E. N Using Hölder inequality we have (2.9 F C (t ˆϕ(ξ F + L S (t ˆψ (ξ (Π N C [ ] ϕ ;E L X + ψ X. (Π N ;E By using the resolvent roerties of oerator A, reresentation of (2. ξ α + n D α C (t, ξ B(E C, ξ α + n D α S (t, ξ B(E C 2, for s > n and all α = (α, α 2,...,α n, α k {, }, ξ R n, ξ, t [, T], here C (t, ξ = ( + ξ 2 s 2 C (t, S (t, ξ = ( + ξ 2 s 2 S (t.

6 4 V. B. SHAKHMUROV By Proosition A from (2.9 (2. we get that the oerator-valued functions C (t, ξ and S (t, ξ are L (R n ; E L (R n ; E Fourier multiliers uniformly in t [, T]. Then by Minkowski s inequality for integrals, the semigrous estimates (see e.g. [, ] and (2.9 we obtain (2. F C (t ˆϕ(ξ F L (Π ;E + S (t ˆψ (ξ N L (Π ;E C [ ϕ Y + ψ s, Y s,]. N By reasoning as the above we get (2.2 F S (t τ, ξ, Aĝ (ξ, τ dτ C X ( g (., τ Y s + g (., τ X dτ. By differentiating, in view of (2.6 we obtain from (2.5 the estimate of tye (2.9, (2., (2.2 for u t. Then by using (2.9, (2., (2.2 we get the estimate (2.7. Lemma 2.4. Assume the Condition 2. hold. Then the solution of (2. (2.2 satisfies the following uniform estimate ( (2.3 ( u Y s, + u t Y s, C ϕ Y s, + ψ Y s, + g (., τ Y s, dτ. Proof. From (2.4 we have the following estimate ( F ( + ξ 2 s 2 û + F ( + ξ 2 s 2 X û t X { F C C (t, ξ ˆϕ F + X S (t, ξ ˆψ (2.4 X + F ξ 2 ( + ξ 2 s } 2 S (t τ, ξ, Aĝ (., τ dτ. X By construction and in view of Proosition A we get that C (t and S (t are L (R n ; E Fourier multiliers uniformly in t [, T]. So, the estimate (2.4 by using the Minkowski s inequality for integrals imlies (2.3. From Lemmas we obtain Theorem 2.5. Let the Condition 2. hold. Then roblem (2. (2.2 has a unique solution u C (2 ([, T];Y s, and the following uniform estimates hold (2.5 u X + u t X C ( ϕ Y s, + ϕ X + ψ Y s, + ψ X + (2.6 u Y s, + u t Y s, C ( ϕ Y s, + ψ Y s, + ( g (., τ Y s, + g (., τ X dτ, g (., τ Y s, dτ.

7 ABSTRACT BOUSSINESQ EQUATIONS 5 Proof. From Lemma 2.2 we obtain that, roblem (2. (2.2 has a unique generalized solution. From the reresentation of solution (2.5 and Lemmas 2.3, 2.4 we get that there is a solution u C (2 ([, T];Y s and estimates (2.5, (2.6 hold. 3. Initial value roblem for nonlinear equation In this section, we will show the local existence and uniqueness of solution for the Cauchy roblem (. (.2. For the study of the nonlinear roblem (. (.2 we need the following lemmas Lemma 3. (Abstract Nirenberg s inequality. Let E be an UMD sace. Assume that u L (R n ; E, D m u L q (R n ; E,, q (,. Then for i with i m, m > n we have q (3. where D i u r C u µ n Dk m u µ q, k= r = i ( m + µ q m + ( µ n, i m µ. Proof. By virtue of interolation of Banach saces [33,.3.2], in order to rove (3. for any given i, one has only to rove it for the extreme values µ = i and µ =. m For the case of µ =, i.e., = i + m the above estimate is obtained from r m q n Theorem.. The case µ = i is derived by reasoning as in [36, 2] and in relacing m absolute value of comlex valued function u by the E-norm of E-valued function. Note that, for E = C the lemma considered by L. Nirenberg [35]. Using the chain rule of the comosite function, from Lemma 3. we can rove the following result Lemma 3.2. Let E be an UMD sace. Assume that u W m, (R n ; E L (R n ; E, and f (u ossesses continuous derivatives u to order m. Then f (u f ( W m, (Ω; E and there is a constant C > such that f (u f ( f ( (u u, D k f (u C k f (j (u u j D k u, k m. j= For E = C the lemma coincide with the corresonding inequality in [35]. Let X = L (R n ; E, Y = W 2, (R n ; E (A, E, E = (X, Y 2,, where (X, Y θ,, < θ <, denotes the real interolation [33].

8 6 V. B. SHAKHMUROV Remark 3.3. By using J. Lions-I. Petree result [8,.8.] we obtain that the ma u u (t, t [, T] is continuous from W 2, (, T; X, Y onto E and there is a constant C such that u (t E C u W 2, (,T;X,Y,. Let Y (T = C ([, T];Y 2, be the sace equied with the norm defined by u Y (T = max u Y 2, + max u X, u Y (T. t [,T] t [,T] It is easy to see that Y (T is a Banach sace. For ϕ, ψ Y 2,, let M = ϕ Y 2, + ϕ X + ψ Y 2, + ψ X. Definition 3.4. For any T > if υ, ψ Y 2, 2, and u C ([, T];Y satisfies the equation (. (.2 then u (x, t is called the continuous solution or the strong solution of the roblem (. (.2. If T <, then u (x, t is called the local strong solution of the roblem (. (.2. If T =, then u (x, t is called the global strong solution of the roblem (. (.2. Condition 3.5. Assume the oerator A generates continuous cosine oerator function in UMD sace E, ϕ, ψ Y 2, and < < for n < 2. Moreover, suose the function u f (u: R n [, T] E E is a measurable in (x, t R n [, T] for u E, f (x, t,.,. is continuous in u E for x R n, t [, T] and f (u C (3 (E ; E. Main aim of this section is to rove the following result: Theorem 3.6. Let the Condition 3.5 hold. Then roblem (. (2.2 has a unique local strange solution u C (2 ([, T ;Y 2,, where T is a maximal time interval that is aroriately small relative to M. Moreover, if then T =. ( su u Y 2, + u X + u t Y 2, + u t X < t [, T Proof. First, we are going to rove the existence and the uniqueness of the local continuous solution of the roblem (. (.2 by contraction maing rincile. Suose that u C (2 ([, T];Y 2, is a strong solution of the roblem (. (.2. Consider a ma G on Y (T such that G(u is the solution of the Cauchy roblem (3.2 G tt (u G tt (u + AG (u = f (G (u, x R n, t (, T, G (u(x, = ϕ (x, G t (u(x, = ψ (x.

9 ABSTRACT BOUSSINESQ EQUATIONS 7 From Lemma 3.2 we know that f(u L (, T; Y 2, for any T >. Thus, by Theorem 2.5, roblem (3.2 has a unique solution which can be written as (3.3 G (u(t, x = S (t, Aϕ(x + S 2 (t, A ψ (x + S (t, τ, ξ = (2π n F ξ 2 S (t τ, ξ, A. R n S (t, τ, ξ ˆf (u(ξ, τdξdτ, For the sake of convenience, we assume that f ( =. Otherwise, we can relace f (u with f (u f (. Hence, from Lemma 3.2 we have f (u Y 2, iff f C 2 (R; E. Consider the oerator in Y (T defined as (3.4 Gu = S (t, A ϕ (x + S 2 (t, A ψ (x + R n S (t, τ, ξ ˆf (u(ξ, τdξdτ. From Lemma 3.2 we get that the oerator G is well defined for f C (2 (R; E. Moreover, from Lemma 3.2 it is easy to see that the ma G is well defined for f C (2 (X ; E. We ut Q (M; T = { } u u Y (T, u Y (T M +. By reasoning as in [9] let us rove that the ma G has a unique fixed oint in Q (M; T. For this aim, it is sufficient to show that the oerator G mas Q (M; T into Q (M; T and G : Q (M; T Q (M; T is strictly contractive if T is aroriately small relative to M. Consider the function f(ξ : [, [, defined by { f (ξ = max f ( (x x ξ E, f (2 (x } E, ξ. It is clear to see that the function f (ξ is continuous and nondecreasing on [,. From Lemma 3.2 we have f (u Y 2, f ( (u X u X + f ( (u X Du X (3.5 2C f (M + (M + u Y 2,. By using the Theorem 2.5 we obtain from (3.4 (3.6 G (u X ϕ X + ψ X + (3.7 G (u Y 2, ϕ Y 2, + ψ Y 2, + Thus, from (3.5 (3.7 and Lemma 3.2 we get f (u (τ X, f (u (τ Y 2, dτ. G (u Y (T M + T (M + [ + 2C (M + f (M + ]. If T satisfies (3.8 T { (M + [ + 2C (M + f (M + ]}

10 8 V. B. SHAKHMUROV then Gu Y (T M +. Therefore, if (3.8 holds, then G mas Q (M; T into Q (M; T. Now, we are going to rove that the ma G is strictly contractive. Assume T > and u, u 2 Q (M; T given. We get [ G (u G (u 2 = S (t, τ, ξ ˆf (u (ξ, τ ˆf (u 2 (ξ, τ] dξdτ. R n By using the mean value theorem and using Hölder s and Nirenberg s, Minkowski s inequalities for integrals, Fourier multilier theorems for oerator-valued functions in X saces and Young s inequality, we obtain Gu Gu 2 Y (T 2 u u 2 Y (T. That is, G is a constructive ma. By contraction maing rincile, we know that G(u has a fixed oint u(x, t Q (M; T that is a solution of the roblem (. (.2. From (2.5 we get that u is a solution of the following integral equation u (t, x = S (t, A ϕ (x + S 2 (t, Aψ (x + S (t, τ, ξ ˆf (u(ξ, τdξdτ. R n Let us show that this solution is a unique in Y (T. Let u, u 2 Y (T be two solution of the roblem (. (.2. Then, by Lemmas 2.4, Minkowski s inequality for integrals and Theorem 2.5 we obtain from (3.8 (3.9 u u 2 Y 2, C 2 (T u u 2 Y 2, dτ. From (3.9 and Gronwall s inequality, we have u u 2 Y 2, =, i.e. roblem (. (.2 has a unique solution which belongs to Y (T. That is, we obtain the first art of the assertion. Now, let [, T be the maximal time interval of existence for u Y (T. It remains only to show that if (3.4 is satisfied, then T =. Assume contrary that, (3.4 holds and T <. For T [, T, we consider the following integral equation (3. υ (x, t = S (t, Au(x, T + S 2 (t, A u t (x, T + By virtue of (3.4, for T > T we have ( u Y 2, + u X + u t Y 2, + u t X <. su t [,T R n S (t, τ, ξ ˆf (υ(ξ, τdξdτ. By reasoning as a first art of theorem and by contraction maing rincile, there is a T (, T such that for each T [, T, the equation (3. has a unique solution υ Y (T. The estimates (3.8 and (3.9 imly that T can be selected indeendently of T [, T. Set T = T T and define 2 u (x, t, t [, T] ũ (x, t = υ (x, t T, t [ ]. T, T + T 2

11 ABSTRACT BOUSSINESQ EQUATIONS 9 By construction ũ (x, t is a solution of the roblem (. (.2 on [ ] T, T + T and in view of local uniqueness, ũ (x, t extends u. This is against to the maximality of [, T, i.e we obtain T =. 4. The Cauchy roblem for the system of Boussinesq equation of infinite order Consider the Cauchy roblem for the following nonlinear system N (4. (u m tt (u m tt + a mj u j (x, t = f m (u, x R n, t (, T, j= (4.2 u m (x, = ϕ m (x, (u m t (x, = ψ m (x, m =, 2,..., N, N N, where u = (u, u 2,...,u N, a mj are comlex numbers, ϕ m (x and ψ m (x are data functions. Let l q = l q (N (see [33,.8] and A be the oerator in l q (N defined by ( N D (A = u = {u j}, u l s q (N = 2 sj u q j j= q A = [a mj ], a mj = g m 2 sj, m, j =, 2,..., N. <, Let X q = L (R n ; l q,y s,,q = H s, (R n ; l q and ( E q = B 2( 2,q R n ; l s( 2 q, l q, Y s,,q = Hs, (R n ; l q L (R n ; l q. From Theorem 2.5 we obtain the following result Theorem 4.. Assume ϕ m, ψ m Y 2,,q and < < for n < 2. Suose the function u f (u: R n [, T] E q l q is a measurable function in (x, t R n [, T] for u E q ; f (x, t.,. and this function is continuous in u E q for x, t R n [, T]; moreover f (u C (3 (E q ; l q. Then roblem (4. (4.2 has a unique local strange solution u C (2 ([, T ;Y 2,,q, where T is a maximal time interval that is aroriately small relative to M. Moreover, if (4.3 su t [,T then T =. ( u Y 2,,q + u X,q + u t Y 2,,q + u t X,q < Proof. It is known that l q (N is a UMD sace. It is easy to see that the oerator A is R-ositive in l q (N. Moreover, by interolation theory of Banach saces [33,.3], we have E q = ( H 2, ( R n ; l s q, l q, L (R n ; l q 2 = 2 B2(,q,q ( R n ; l s( 2 q, l q. By using the roerties of saces Y s,,q, Y s,,q, E q we get that all conditions of Theorem 2.5 are hold, i,e., we obtain the conclusion. 2

12 2 V. B. SHAKHMUROV 5. The Wentzell-Robin tye mixed roblem for Boussinesq equations Consider the roblem (.6 (.8. Let ( E = L, E = L, H 2, = B 2,,, Y s, = H s, L, Y s, = Hs, L. Suose ν = (ν, ν 2,..., ν n are nonnegative real numbers. In this section, we resent the following result: Condition 5.. Assume ϕ, ψ Y 2, and < < for n < 2. Moreover, the function u f (u : Ω [, T] B 2, L (, is a measurable for u B 2,, f (x, y, t,.,. is continuous with resect to u B 2, for x R n, y ( (,, t [, T] and f (u C (3 B 2, ; L (, for a.e. x R n, y (,. The main aim of this section is to rove the following result: Theorem 5.2. Suose the Condition 5. hold and a W, (,, a (x δ >, b, c L (,. Then roblem (.5 (.7 has a unique local strange solution u C (2 ([, T ; Y 2,, where T is a maximal time interval that is aroriately small relative to M. Moreover, if then T =. ( su u Y 2, + u X + u t Y 2, + u t X < t [,T Proof. Let E = L 2 (,. It is known [29] that L 2 (, is an UMD sace. Consider the oerator A defined by (.5. Then, the roblem (.6 (.8 can be rewritten in the form of (., where u (x = u (x,., f (x = f (x,. are functions with values in E = L 2 (,. By virtue of [3,32] the oerator A generates analytic semigrou in L 2 (,. Then in view of Hill- Yosida theorem (see e.g. [33,.3] this oerator is ositive in L 2 (,. Since all uniform bounded set in Hilbert sace is an R-bounded, we get that the oerator A is R-ositive in L 2 (,. Then from Theorem 2.5 we obtain the assertion. aer. Acknowledgement I would like to thank Assoc. Prof. Burak Kelleci for his hel on the style of this

13 ABSTRACT BOUSSINESQ EQUATIONS 2 REFERENCES [] V. G. Makhankov, Dynamics of classical solutions (in non-integrable systems, Phys. Lett. C 35, ( [2] G. B. Whitham, Linear and Nonlinear Waves, Wiley Interscience, New York, 975. [3] N. J. Zabusky, Nonlinear Partial Differential Equations, Academic Press, New York, 967. [4] C. G. Gal and A. Miranville, Uniform global attractors for non-isothermal viscous and nonviscous Cahn Hilliard equations with dynamic boundary conditions, Nonlinear Analysis: Real World Alications ( [5] J. T. Kirbay, Boussinesq models and alications nearshore wave roagation, surf zone rocessess and wave-induced currents, Advances in coastal modeling, Elsevier science B. V., 23. [6] A. J. Majda, Introduction to PDEs and waves for the atmoshere and ocean., Courant Lecture Notes in Mathematics, v. 9. AMS/CIMS, 23. [7] J. Pedlosky, Geohysical fluid dynamics, Sringer, New York, 987. [8] S. Wang, G. Chen, Small amlitude solutions of the generalized IMBq equation, J. Math. Anal. Al. 274 ( [9] S. Wang, G. Chen, The Cauchy Problem for the Generalized IMBq equation in W s, (R n, J. Math. Anal. Al. 266, ( [] H. O. Fattorini, Second order linear differential equations in Banach saces, in North Holland Mathematics Studies, V. 8, North-Holland, Amsterdam, 985. [] J. A. Goldstein, Semigrou of linear oerators and alications, Oxford, 985. [2] S. Piskarev and S.-Y. Shaw, Multilicative erturbations of semigrous and alications to ste resonses and cumulative oututs, J. Funct. Anal. 28 (995, [3] H. Amann, Linear and quasi-linear equations,, Birkhauser, Basel 995. [4] R. Agarwal R, D. O Regan, V. B. Shakhmurov, Searable anisotroic differential oerators in weighted abstract saces and alications, J. Math. Anal. Al. 338( [5] A. Ashyralyev, C. Cuevas and S. Piskarev, On well-osedness of difference schemes for abstract ellitic roblems in saces, Numer. Func. Anal.Ot., (29-2(28, [6] A. Favini, V. Shakhmurov, Y. Yakubov, Regular boundary value roblems for comlete second order ellitic differential-oerator equations in UMD Banach saces, Semigrou form, 79 (, 29, [7] A. Lunardi, Analytic semigrous and otimal regularity in arabolic roblems, Birkhauser, 23. [8] J. L Lions and J. Peetre, Sur une classe d esaces d interolation, Inst. Hautes Etudes Sci. Publ. Math., 9(964, [9] V. B. Shakhmurov, Embedding and maximal regular differential oerators in Banach-valued weighted saces, Acta. Math. Sin., (Engl. Ser., (22, 28 (9, [2] V. B. Shakhmurov, Embedding and searable differential oerators in Sobolev-Lions tye saces, Mathematical Notes, 28, v. 84, no 6, [2] V. B. Shakhmurov, Stokes equations with small arameters in half lane, Comut. Math., 24(67, 9 5. [22] V. B. Shakhmurov, Embedding theorems and maximal regular differential oerator equations in Banach-valued function saces, J. Inequal. Al., 2( 4, (25, [23] A Sahmurova, V. B. Shakhmurov, Singular degenerate roblems occurring in biosortion rocess, Bound. Value Probl., 23 (, 7.

14 22 V. B. SHAKHMUROV [24] V. B. Shakhmurov, Comact embedding in Besov saces and B-searable ellitic oerators, Sci. China Math., v. 53(4, 2, [25] V. B. Shakhmurov, Anisotroic differential oerators with arameters and alications, Ukrainian Math. J., 24(7, [26] P. Guidotti, Otimal regularity for a class of singular abstract arabolic equations, J. Differ. Equations, v. 232, 27, [27] R. Shahmurov, On strong solutions of a Robin roblem modeling heat conduction in materials with corroded boundary, Nonlinear Analysis, Real-wold alications, v.3, (, 2, [28] R. Shahmurov, Solution of the Dirichlet and Neumann roblems for a modified Helmholtz equation in Besov saces on an annuals, J. of Differential equations, v. 249(3, 2, [29] L. Weis, Oerator-valued Fourier multilier theorems and maximal L regularity, Math. Ann. 39, (2, [3] S. Yakubov and Ya. Yakubov, Differential-oerator Equations. Ordinary and Partial Differential Equations, Chaman and Hall /CRC, Boca Raton, 2. [3] A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, Degenerate Second Order Differential Oerators Generating Analytic Semigrous in L and W,, Math. Nachr. 238 (22, [32] V. Keyantuo, M. Warma, The wave equation with Wentzell-Robin boundary conditions on L-saces, J. Differential Equations 229 ( [33] H. Triebel, Interolation theory, Function saces, Differential oerators, North-Holland, Amsterdam, 978. [34] H. Triebel, Fractals and sectra, Birkhauser Verlag, Related to Fourier analysis and function saces, Basel, 997. [35] L. Nirenberg, On ellitic artial differential equations, Ann. Scuola Norm. Su. Pisa 3 (959, [36] S. Klainerman, Global existence for nonlinear wave equations, Comm. Pure Al. Math. 33 (98, 43.