# The cauchy problem for generalized abstract boussinesq equations.

1. Introduction, Definitions and BackgroundThe subject of this paper is to study the local existence and uniqueness of solution of the Cauchy problem for the following Boussinesq-operator equation

(1.1) [u.sub.tt] - [DELTA][u.sub.tt] + Au = [DELTA]f (u), x [member of] [R.sup.n], t [member of] (0,T),

(1.2) u (x, 0) = [phi] (x), [u.sub.t] (x, 0) = [psi](x),

where A is a linear operator in a Banach space E, u(x,t) denotes the E-valued unknown function, f(u) is the given nonlinear function, [phi](x) and [psi](x) are the given initial value functions, subscript t indicates the partial derivative with respect to t, n is the dimension of space variable x and [DELTA] denotes the Laplace operator in [R.sup.n].

This is a first paper that devoted to initial value problem for abstract Boussinesq equations. Since the equation contain generally, unbounded operator in a abstract Banach space E and the problem (1.1) is considered in UMD-valed function space, the methods applied for Boussinesq equations in scalar case does not pass here. By this reason, in this paper, the methods of proofs naturally differs to those used in scalar case. Since the Banach space E is arbitrary and A is a possible linear operator, by choosing E and A we can obtain numerous classis of Boussinesq type equations which have a different applications (see [1-7] and the references therein).

Here, by inspiring [8] and [9] in this paper, we obtain the local existence and uniqueness of small-amplitude solution of the problem (1.1)-(1.2). Note that, differential operator equations were studied e.g. in [10-31]. The Cauchy problem for abstract hyperbolic equations were treated e.g. in [10-12]. The strategy is to express the abstract Boussinesq equation as an integral equation with operator coefficient, to treat in the nonlinearity as a small perturbation of the linear part of the equation, then use the contraction mapping theorem and utilize an estimate for solutions of the linearized version to obtain a priori estimates on E-valued [L.sup.p] norms of solutions. The key step is the derivation of the uniform estimate for the solutions of the linearized Boussinesg-operator equation. If we choose the UMD space E as a abstract Hilbert space H, then we obtain the existence and uniqueness results of the Cauchy problem for abstract Boussinesg equation defined in Hilbert space valued classes. For example, if we choose E a concrete Hilbert space, for example E = [L.sup.2] ([R.sup.n]) and A = -[DELTA], we obtain the scalar Cauchy problem for generalized Boussinesq type equation

(1.3) [u.sub.tt] - [DELTA][u.sub.tt] - [DELTA]u = [DELTA]f (u), x [member of] [R.sup.n], t [member of] (0,T),

(1.4) u (x, 0) = [phi] (x), [u.sub.t] (x, 0) = [psi] (x).

The equation (1.3) arise in different situations (see [1,2]). For example, equation (1.3) for n =1 describes a limit of a one-dimensional nonlinear lattice [3], shallow-water waves [4,5] and the propagation of longitudinal deformation waves in an elastic rod [6]. In [8] and [9] the existence of the global classical solutions and the blow-up of the solution for the initial boundary value problem and the Cauchy problem (1.3)-(1.4) are obtained. Moreover, let we choose E = [L.sub.p1] (0,1) and A to be differential operator with generalized Wentzell-Robin boundary condition defined by

(1.5) D (A) = {u e [W.sup.2.sub.p1] (0,1), [B.sub.j]u = Au (j) + [1.summation over (i=0)] [[alpha].sub.ji] [u.sub.(i)] (j) , j = 0, 1},

Au = [au.sup.(2)] + [bu.sup.(1)] + cu,

in (1.1)-(1.2), where [[alpha].sub.ji] are complex numbers, a, b, c are complex- valued functions. Then, we get the following Wentzell-Robin type mixed problem for Boussinesq equation

(1.6) [u.sub.tt] - [[DELTA].sub.x] [u.sub.tt] + a [[partial derivative].sup.2]u/[partial derivative][y.sup.2] + b- [partial derivative]u/[partial derivative]y + cu = [[DELTA].sub.x] f(u), x [member of] [R.sup.n], y [member of] (0,1), t [member of] (0,T),

(1.7) [B.sub.j]u = Au (x,t,j) + [1.summation over (i=0)] [[alpha].sub.ji] [u.sup.(i)] (x,t,j) = 0, j = 0,1,

(1.8) u (x, y, 0) = [phi] (x, y), [u.sub.t](x, y, 0) = [psi](x, y),

Note that, the regularity properties of Wentzell-Robin type BVP for elliptic equations were studied e.g. in [31,32] and the references therein. Here [??] = [R.sup.n] x (0,1), p = ([p.sub.1], p) and [L.sup.p] ([??]) denotes the space of all p-summable complex-valued functions with mixed norm i.e., the space of all measurable functions f defined on [??], for which

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By applying the general Theorem 2.5 obtained here, we established the local existence and uniqueness of small-amplitude solution of the problem (1.6)-(1.8) in mixed [L.sup.p] ([??]) space.

In order to state our results precisely, we introduce some notations and some function spaces.

Let E be a Banach space. [L.sup.p] ([OMEGA]; E) denotes the space of strongly measurable E-valued functions that are defined on the measurable subset [OMEGA] [subset] [R.sup.n] with the norm

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let [E.sub.0] and E be two Banach spaces and [E.sub.0] is continuously and densely embeds into E. Let [H.sup.s,p] ([R.sup.n]; E), -[infinity] < s < [infinity] denotes the E-valued Sobolev space of order s which is defined as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let [B.sup.s.sub.p,q] ([R.sup.n]; E) denote E-valued Besov space (see e.g. [34, [section] 15]). Let [B.sup.s.sub.p,q] ([R.sup.n]; [E.sub.0], E) denote the space [L.sup.p] ([R.sup.n]; [E.sub.0]) [intersection] ([R.sup.n]; E) with the norm

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For estimating lower order derivatives we use following embedding theorem that is obtained from [22, Theorem 1]:

Theorem 1.1. Suppose the following conditions are satisfied:

(1) E is a UMD space and A is an R-positive operator in E (see e.g. [29] for definitions );

(2) [alpha] = ([[alpha].sub.1], [[alpha].sub.2], ..., [[alpha].sub.n]) is a n- tuples of nonnegative integer number and s is a positive number such that

[??] = 1/s [absolute value of ([alpha]) + n(1/p - 1/q) [less than or equal to] 1, 0 [less than or equal to] [mu] [less than or equal to] 1 - [??], l < p [less than or equal to] q < [infinity], 0 < h [less than or equal to] [h.sub.0],

where [h.sub.0] is a fixed positive number.

Then the embedding [D.sup.[alpha]][H.sup.s'p] ([R.sup.n]; E (A), E) [subset] [L.sup.q] ([R.sup.n]; E ([A.sup.1-N- [mu])]) is continuous and for u [member of] [H.sup.s'p] ([R.sup.n]; E (A), E) the following uniform estimate holds

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In a similar way as [19, Theorem [A.sub.0]] we obtain:

Proposition [A.sub.1]. Let 1 < p [less than or equal to] q [less than or equal to] [infinity] and E be UMD space. Suppose [[PSI].sub.p] [C.sup.n] ([R.sup.n]\ {0} ; B (E)) and there is a positive constant K such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then [[PSI].sub.h] is a uniformly bounded collection of Fourier multiplier from [L.sup.p] ([R.sup.n]; E) to [L.sup.[infinity]] ([R.sup.n]; E).

2. Estimates for linearized equation

Here, we make the necessary estimates for solutions of Cauchy problem for the linearized abstract Boussinesq equation

(2.1) [u.sub.tt] - [DELTA][u.sub.tt] + Au = [DELTA]g (x, t), x [member of] [R.sup.n], t [member of] (0,T),

(2.2) u (x, 0) = [phi] (x), [u.sub.t] (x, 0) = [psi] (x). Let

[X.sub.p] = [L.sup.p] ([R.sup.n]; E) , [Y.sup.s,p] = [H.sup.s,p] ([R.sup.n]; E) , [Y.sup.s,p.sub.1] = [H.sup.s,p] ([R.sup.n]; E) [intersection] [L.sup.1] ([R.sup.n]; E),

[Y.sup.s,p.sub.[infinity]] = [H.sup.s,p] ([R.sup.n]; E) [intersection] ([R.sup.n]; E).

Condition 2.1. Assume E is an UMD space and the operator A is positive in E. Moreover suppose [phi], [psi] [member of] [Y.sup.s,p.sub.[infinity]], g (., t) [member of] [Y.sup.s,p.sub.[infinity]] for t [member of] (0, T) and s > [n/p] for 1 < p < .[infinity]

First we need the following lemmas

Lemma 2.2. Suppose the Condition 2.1 hold. Then problem (2.1)--(2.2) has a unique generalized solution.

Proof. By using of the Fourier transform we get from (2.1)-(2.2)

(2.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [??] ([xi], t) is a Fourier transform of u (x, t) with respect to x and

[A.sub.[xi]] = [(1 + [[absolute value of ([xi])].sup.2]).sup.-1] A, [xi] [member of] [R.sup.n].

By virtue of [10,11] we obtain that [A.sub.[xi]] is a generator of a strongly continuous cosine operator function and problem (2.3) has a unique solution for all [xi] [member of] [R.sup.n], moreover, the solution can be written as

(2.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where C (t) = C (t, [xi], A) is a cosine and S (t) = S (t, [xi], A) is a sine operator-functions (see e.g. [11]) generated by parameter dependent operator [A.sub.[xi]]. From (2.4) we get that, the solution of the problem (2.1)-(2.2) can be expressed as

(2.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [S.sub.1] (t, A) and [S.sub.2] (t, A) are linear operators in E defined by

(2.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Lemma 2.3. Suppose the Condition 2.1 hold. Then the solution (2.1)--(2.2) satisfies the following uniform estimate

(2.7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. Let N [member of] N and

[[PI].sub.N] = {[xi] : [xi] [member of] [R.sup.n], [absolute value of ([xi])] < N}, [[PI]'.sub.N] = {[xi] : [xi] [member of] [R.sup.n], [absolute value of ([xi])] > N}.

It is clear to see that

(2.8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Using Holder inequality we have

(2.9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By using the resolvent properties of operator A, representation of

(2.10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

for s > [n/p] and all [alpha] = {[[alpha].sub.1], [[alpha].sub.2], ..., [[alpha].sub.n]), [[alpha].sub.k] [member of] {0,1}, [xi] [member of] [R.sup.n], [xi] [not equal to] 0, t [member of] [0,T], here

[C.sub.1] {t, [xi]) = [(1 + [[absolute value of ([xi])].sup.2]).sup.-s/2] C (t), [S.sub.1] (t, [xi]) = [(1 + [[absolute value of ([xi])].sup.2]).sup.-s/2] S (t).

By Proposition [A.sub.1] from (2.9)-(2.10) we get that the operator-valued functions [C.sub.1] (t, C) and [S.sub.1] (t, C) are [L.sup.p] ([R.sup.n]; E) [right arrow] ([R.sup.n]; E) Fourier multipliers uniformly in t [member of] [0, T]. Then by Minkowski's inequality for integrals, the semigroups estimates (see e.g. [10, 11]) and (2.9) we obtain (2.11)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By reasoning as the above we get

(2.12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By differentiating, in view of (2.6) we obtain from (2.5) the estimate of type (2.9), (2.11), (2.12) for [u.sub.t]. Then by using (2.9) , (2.11), (2.12) we get the estimate (2.7).

Lemma 2.4. Assume the Condition 2.1 hold. Then the solution of (2.1)-(2.2) satisfies the following uniform estimate

(2.13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. From (2.4) we have the following estimate

(2.14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By construction and in view of Proposition [A.sub.1] we get that C (t) and S (t) are [L.sup.p] ([R.sup.n]; E) Fourier multipliers uniformly in t [member of] [0,T]. So, the estimate (2.14) by using the Minkowski's inequality for integrals implies (2.13).

From Lemmas 2.2-2.4 we obtain

Theorem 2.5. Let the Condition 2.1 hold. Then problem (2.1)-(2.2) has a unique solution u [member of] C(2) ([0,T]; [Y.sup.s,p.sub.1]) and the following uniform estimates hold

(2.15) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(2.16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. From Lemma 2.2 we obtain that, problem (2.1)-(2.2) has a unique generalized solution. From the representation of solution (2.5) and Lemmas 2.3, 2.4 we get that there is a solution u [member of] [C.sup.(2)] ([0, T]; [Y.sub.1.sup.s]) and estimates (2.15), (2.16) hold.

3. Initial value problem for nonlinear equation

In this section, we will show the local existence and uniqueness of solution for the Cauchy problem (1.1)-(1.2). For the study of the nonlinear problem (1.1)- (1.2) we need the following lemmas

Lemma 3.1 (Abstract Nirenberg's inequality). Let E be an UMD space. Assume that u [member of] [L.sup.p] ([R.sup.n]; E), [D.sup.m]u [member of] [L.sup.q] ([R.sup.n]; E), p, q [member of] (1, [infinity]). Then for i with 0 [less than or equal to] i [less than or equal to] m,

(3.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where

[1/r] = [i/m] + [mu] ([1/q] - [m/n]) + (1 - [mu])[1/p], [i/m] [less than or equal to] [mu] [less than or equal to] 1.

Proof. By virtue of interpolation of Banach spaces [33, [section] 1.3.2], in order to prove (3.1) for any given i, one has only to prove it for the extreme values [mu] = [i/m] and [mu] = 1. For the case of [mu] = 1, i.e., [1/r] = [i/m] + [1/q] - [m/n] the above estimate is obtained from Theorem 1.1. The case [mu] = [i/m] is derived by reasoning as in [36, [section] 2] and in replacing absolute value of complex 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 composite function, from Lemma 3.1 we can prove the following result

Lemma 3.2. Let E be an UMD space. Assume that u [member of] [W.sup.m,p] ([R.sup.n]; E) [intersection] ([R.sup.n]; E), and f (u) possesses continuous derivatives up to order m [greater than or equal to] 1. Then f (u) - f (0) [member of] [W.sup.m,p] ([OMEGA]; E) and there is a constant [C.sub.0] > 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

For E = C the lemma coincide with the corresponding inequality in [35]. Let

X = [L.sup.p] ([R.sup.n], E), Y = [W.sup.2,p] ([R.sup.n], E(A),E), [E.sub.0] = [(X,Y).sub.[1/2p],p],

where [(X, Y).sub.[theta],p], 0 < [theta] < 1, 1 [less than or equal to] p [less than or equal to] [infinity] denotes the real interpolation [33].

Remark 3.3. By using J. Lions-I. Petree result [18, [section] 1.8.] we obtain that the map u [right arrow] u ([t.sub.0]), [t.sub.0] [member of] [0,T] is continuous from [W.sup.2,p] (0,T; X, Y) onto E0 and there is a constant [C.sub.1] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let Y (T) = C ([0, T]; [Y.sup.2,p.sub.[infinity]]) be the space equipped with the norm defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It is easy to see that Y (T) is a Banach space. For [phi], [psi] [member of] [Y.sup.2,p], let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Definition 3.4. For any T > 0 if u, [psi] [member of] [Y.sup.2,p.sub.[infinity]] and u [member of] C ([0,T]; [Y.sup.2,p.sub.[infinity]]) satisfies the equation (1.1)-(1.2) then u (x,t) is called the continuous solution or the strong solution of the problem (1.1)-(1.2). If T < [infinity], then u (x,t) is called the local strong solution of the problem (1.1)-(1.2). If T = [infinity], then u (x, t) is called the global strong solution of the problem (1.1)-(1.2).

Condition 3.5. Assume the operator A generates continuous cosine operator function in UMD space E, [phi], [psi] [member of] [Y.sup.2,p.sub.[infinity]] and 1 < p < [infinity] for [n/p] < 2. Moreover, suppose the function u [infinity] f (u): [R.sup.n] x [0, T] x [E.sub.0] [right arrow] E is a measurable in (x, t) [member of] [R.sup.n] x [0, T] for u [member of] [E.sub.0], f (x, t,.,.) is continuous in u [member of] [E.sub.0] for x [member of] [R.sub.n], t [member of] [0, T] and f (u) [member of] [C.sup.(3)] ([E.sub.0]; E).

Main aim of this section is to prove the following result:

Theorem 3.6. Let the Condition 3.5 hold. Then problem (1.1)-(2.2) has a unique local strange solution u [member of] [C.sup.(2)] ([0, [T.sub.0]); [Y.sup.2,p.sub.[infinity]]), where [T.sub.0] is a maximal time interval that is appropriately small relative to M. Moreover, if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

then [T.sub.0] = [infinity].

Proof. First, we are going to prove the existence and the uniqueness of the local continuous solution of the problem (1.1)-(1.2) by contraction mapping principle. Suppose that u [member of] [C.sup.(2)] ([0, T]; [Y.sup.2,p.sub.[infinity]]) is a strong solution of the problem (1.1)-(1.2). Consider a map G on Y (T) such that G(u) is the solution of the Cauchy problem

(3.2) [G.sub.tt] (u) - [DELTA][G.sub.tt] (u) + [DELTA]G (u) = [DELTA]f (G(u)), x [member of] [R.sup.n], t [member of] (0,T), G(u) (x, 0) = [phi] (x), [G.sub.t] (u) (x, 0) = [psi] (x).

From Lemma 3.2 we know that f(u) [member of] [L.sup.p] (0,T; [Y.sup.2,p.sub.[infinity]]) for any T > 0. Thus, by Theorem 2.5, problem (3.2) has a unique solution which can be written as

(3.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For the sake of convenience, we assume that f (0) = 0. Otherwise, we can replace f (u) with f (u) - f (0). Hence, from Lemma 3.2 we have f (u) [member of] [Y.sup.2,p] iff f [member of] [C.sup.2] (R; E). Consider the operator in Y (T) defined as

(3.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From Lemma 3.2 we get that the operator G is well defined for f [member of] [C.sup.(2)] (R; E). Moreover, from Lemma 3.2 it is easy to see that the map G is well defined for f [member of] [C.sup.(2)] ([X.sub.0]; E). We put

Q (M; T) = |u | u [member of] Y (T), [parallel]u[[parallel].sub.Y(T)] [less than or equal to] M + 1}.

By reasoning as in [9] let us prove that the map G has a unique fixed point in Q (M; T). For this aim, it is sufficient to show that the operator G maps Q (M; T) into Q (M; T) and G : Q (M; T) [right arrow] Q (M; T) is strictly contractive if T is appropriately small relative to M. Consider the function [bar.f]([xi]) : [0, [infinity]) [right arrow] [0, to) defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It is clear to see that the function [bar.f] ([xi]) is continuous and nondecreasing on [0, [infinity]). From Lemma 3.2 we have

(3.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By using the Theorem 2.5 we obtain from (3.4)

(3.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(3.7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus, from (3.5)-(3.7) and Lemma 3.2 we get

[parallel]G (u)[[parallel].sub.Y(T)] [less than or equal to] M + T (M + 1) [1 + 2 [C.sub.0] (M + 1) [bar.f](M + 1)] *

If T satisfies

(3.8) T [less than or equal to] [{(M +1) [1 + 2[C.sub.0] (M + 1) [bar.f](M + 1)]}.sup.-1]

then [parallel]Gu[[parallel].sub.Y(T)] [less than or equal to] M + 1. Therefore, if (3.8) holds, then G maps Q (M; T) into Q (M; T). Now, we are going to prove that the map G is strictly contractive. Assume T > 0 and [u.sub.1], [u.sub.2] [member of] Q (M; T) given. We get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By using the mean value theorem and using Holder's and Nirenberg's, Minkowski's inequalities for integrals, Fourier multiplier theorems for operator-valued functions in [X.sub.p] spaces and Young's inequality, we obtain

[parallel][Gu.sub.1] - [Gu.sub.2][[parallel].sub.Y(T)] [less than or equal to] [1/2] [parallel][u.sub.1] - [u.sub.2][[parallel].sub.Y(T)].

That is, G is a constructive map. By contraction mapping principle, we know that G(u) has a fixed point u(x, t) [member of] Q (M; T) that is a solution of the problem (1.1)-(1.2). From (2.5) we get that u is a solution of the following integral equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let us show that this solution is a unique in Y (T). Let [u.sub.1], [u.sub.2] [member of] Y (T) be two solution of the problem (1.1)-(1.2). Then, by Lemmas 2.4, Minkowski's inequality for integrals and Theorem 2.5 we obtain from (3.8)

(3.9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

From (3.9) and Gronwall's inequality, we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], i.e. problem (1.1)(1.2) has a unique solution which belongs to Y (T). That is, we obtain the first part of the assertion. Now, let [0, [T.sub.0]) be the maximal time interval of existence for u [member of] Y ([T.sub.0]). It remains only to show that if (3.4) is satisfied, then [T.sub.0] = [infinity]. Assume contrary that, (3.4) holds and [T.sub.0] < [infinity]. For T [member of] [0, [T.sub.0]), we consider the following integral equation

(3.10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By virtue of (3.4), for T' > T we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By reasoning as a first part of theorem and by contraction mapping principle, there is a [T.sup.*] [member of] (0, [T.sub.0]) such that for each T [member of] [0, [T.sub.0]), the equation (3.10) has a unique solution u [member of] Y ([T.sup.*]). The estimates (3.8) and (3.9) imply that [T.sup.*] can be selected independently of T [member of] [0, [T.sub.0]). Set T = [T.sub.0] - [T.sup.*]/2 and define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By construction [??](x,t) is a solution of the problem (1.1)-(1.2) on [T, [T.sub.0] + and in view of local uniqueness, u(x, t) extends u. This is against to the maximality of [0, [T.sub.0]), i.e we obtain [T.sub.0] = [infinity].

4. The Cauchy problem for the system of Boussinesq equation of infinite order

Consider the Cauchy problem for the following nonlinear system

(4.1) [([u.sub.m]).sub.tt] - [DELTA] [([u.sub.m]).sub.tt] + [N.summation over (j=1)] [a.sub.mj] [u.sub.j] (x, t) = [DELTA][f.sub.m] (u), x [member of] [R.sup.n], t [member of] (0, T),

(4.2) [u.sub.m] (x, 0) = [[phi].sub.m] (x), [([u.sub.m]).sub.t] (x, 0) = [[psi].sub.m] (x), m =1, 2, ..., N, N [member of] N,

where u = ([u.sub.1], [u.sub.2], ..., [u.sub.N]), [a.sub.mj] are complex numbers, [p.sub.m] (x) and [[psi].sub.m] (x) are data functions. Let [l.sub.q] = [l.sub.q] (N) (see [33, [section] 1.18]) and A be the operator in [l.sub.q] (N) defined

by 1

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

From Theorem 2.5 we obtain the following result

Theorem 4.1. Assume [[phi].sub.m], [[psi].sub.m] [member of] [Y.sup.2,p,g.sub.[infinity]] and 1 < p < [infinity] for [n/p] < 2. Suppose the function u [right arrow] f (u): [R.sup.n] x [0, T] x [E.sub.0q] [right arrow] [l.sub.q] is a measurable function in (x, t) [member of] [R.sup.n] x [0, T] for u [member of] [E.sub.0q]; f (x, t., .) and this function is continuous in u [member of] [E.sub.0q] for x, t [member of] [R.sup.n] x [0, T]; moreover [DELTA]f (u) [member of] [C.sup.(3)] ([E.sub.0q]; [l.sub.q]). Then problem (4.1)-(4.2) has a unique local strange solution u [member of] [C.sup.(2)] ([0, [T.sub.0]); [Y.sup.2,p,q.sub. [infinity]]), where [T.sub.0] is a maximal time interval that is appropriately small relative to M. Moreover, if

(4.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then [T.sub.0] = [infinity].

Proof. It is known that [l.sub.q] (N) is a UMD space. It is easy to see that the operator A is R-positive in [l.sub.q] (N). Moreover, by interpolation theory of Banach spaces [33, [section] 1.3], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By using the properties of spaces [Y.sup.s,p,q], [Y.sup.s,p,q.sub.[infinity]], [E.sub.0q] we get that all conditions of Theorem 2.5 are hold, i,e., we obtain the conclusion. ?

5. The Wentzell-Robin type mixed problem for Boussinesq equations

Consider the problem (1.6)-(1.8). Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Suppose v = ([v.sub.1], [v.sub.2], ..., [v.sub.n]) are nonnegative real numbers. In this section, we present the following result:

Condition 5.1. Assume [phi], [psi] [xi] [member of] [Y.sup.2,p.sub.[infinity]] and 1 < p < [infinity] for [n/p] < 2. Moreover, the function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a measurable for u [member of] [B.sup.1/p.sub.2,p] ([??]), f (x, y, t, ., .) is continuous with respect to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. s

The main aim of this section is to prove the following result:

Theorem 5.2. Suppose the Condition 5.1 hold and a [member of] [W.sup.1,[infinity]] (0,1), a (x) [greater than or equal to] [delta] > 0, b, c [member of] [L.sup.[infinity]] (0,1). Then problem (1.5)-(1.7) has a unique local strange solution u e [C.sup.(2)] ([0, [T.sub.0]); [Y.sup.2,p.sub.[infinity]]), where [T.sub.0] is a maximal time interval that is appropriately small relative to M. Moreover, if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then [T.sub.0] = [infinity].

Proof. Let E = [L.sup.2] (0,1). It is known [29] that [L.sup.2] (0,1) is an UMD space. Consider the operator A defined by (1.5). Then, the problem (1.6)-(1.8) can be rewritten in the form of (1.1), where u (x) = u (x,.), f (x) = f (x,.) are functions with values in E = [L.sup.2] (0,1). By virtue of [31,32] the operator A generates analytic semigroup in [L.sup.2] (0,1). Then in view of Hill- Yosida theorem (see e.g. [33, [section] 1.13]) this operator is positive in [L.sup.2] (0,1). Since all uniform bounded set in Hilbert space is an R-bounded, we get that the operator A is R-positive in L2 (0,1). Then from Theorem 2.5 we obtain the assertion.

Received March 30, 2016

Acknowledgement

I would like to thank Assoc. Prof. Burak Kelleci for his help on the style of this paper.

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VELI B. SHAKHMUROV

Department of Mechanical Engineering, Okan University, Akfirat, Tuzla 34959 Istanbul, Turkey veli.sahmurov@okan.edu.tr Khazar University, Baku, Azerbaijan

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Author: | Shakhmurov, Veli B. |
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Publication: | Dynamic Systems and Applications |

Article Type: | Report |

Date: | Mar 1, 2016 |

Words: | 5616 |

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