The Quality Properties of Integral Type Problems for Wave Equations and Applications ()
1. Introduction, Definitions and Background
The aim here, is to study the existence, uniqueness, regularity properties of solutions of the integral problem (IP) for abstract wave equation (WE)
(1.1)
(1.2)
where A is a linear and
is a nonlinear operator in a Hilbert space H,
,
are measurable functions on
, a is a complex number,
. Here,
denotes the Laplace operator with respect to
,
and
are the given H-valued initial functions.
Wave type equations occur in a wide variety of physical systems, such as in the propagation of longitudinal deformation waves in an elastic rod, hydro-dynamical process in plasma, in materials science which describes spinodal decomposition and in the absence of mechanical stresses (see [1] [2] [3] [4] ). The nonlocal theory of elasticity was introduced (see [5] [6] [7] [8] [9] and the references cited therein). The global existence of the Cauchy problem for Boussinesq type equations has been studied by many authors (see [10] [11] [12] ). Note that, the existence and uniqueness of solutions and regularity properties of a wide class of wave equations were considered e.g. in [13] - [22]. The abstract evolution equations were studied e.g. in [23] - [32]. Unlike in these studies, in this paper the abstract wave equation (1.1) is considered. The
well-posedness of the Cauchy problem (1.1)-(1.2) depends crucially on the presence of the linear operator A and nonlinear operator
. Then the question that naturally arises is which of the possible forms of the operator functions and kernel functions are relevant for the global well-posedness of the Cauchy problem (1.1)-(1.2). We find the class of operator A such that provides the existence, uniqueness, regularity properties and blow up of solutions (1.1)-(1.2) in terms of fractional powers of operator A. By choosing the space H, operator A in (1.1)-(1.2), we obtain a wide class of wave equations which occur in application. Let we put
and consider the operator
defined by
(1.3)
where
are VMO functions (see definitions below),
,
are complex numbers.
Consider the following mixed problem for WE with discontinuous coefficients
(1.4)
where a is a complex number. From our results we obtain the existence, uniqueness, regularity properties and blow up of solutions of (1.4) in
with terms of fractional powers of the operator
, where
and
denotes the space of all
-summable complex-valued measurable functions f defined on
with the mixed norm
Let E be a Banach space.
denotes the space of strongly measurable E-valued functions that are defined on the measurable subset
with the norm
Let
and
be two Banach spaces.
for
,
denotes the real interpolation spaces defined by K-method ( [33], Section 1.3.2). Let
and
be two Banach spaces.
will denote the space of all bounded linear operators from
to
. For
it will be denoted by
.
Here,
A closed linear operator A is said to be sectorial in a Banach space E with bound
if
and
are dense on E,
and
for any
,
, where I is the identity operator in E,
and
denote domain and range of the operator A, respectively. It is known that (see e.g. [33], Section 1.15.1) there exist the fractional powers
of a sectorial operator A. Let
denote the space
with the graphical norm
A sectorial operator
is said to be uniformly sectorial in E for
, if
is independent of
and the following uniform estimate
holds for any
.
A function
is called a Fourier multiplier from
to
if the map
is well defined for
and extends to a bounded linear operator.
Definition 1.1. Let U be an open set in a Banach space X, let Y be a Banach space. A function
is called (Frechet) differentiable at
if there is a bounded linear operator
, called the derivative of f at a, such that
If f is differentiable at each
, then f is called differentiable. This function may also have a derivative, the second order derivative of f, which, by the definition of derivative, will be a map
Let E be a Banach space.
denotes E-valued Schwartz class, i.e. the space of all E-valued rapidly decreasing smooth functions on
equipped with its usual topology generated by seminorms.
denoted by S. Let
denote the space of all continuous linear functions from S into E, equipped with the bounded convergence topology. Recall
is norm dense in
when
. Let m be a positive integer.
denotes an E-valued Sobolev space of all functions
that have the generalized derivatives
with the norm
Let
denotes the fractional Sobolev space of order
, that is defined as:
It is clear that
. Let
and E be two Banach spaces and
is continuously and densely embedded into E. Here,
denote the Sobolev-Lions type space i.e.,
In a similar way, we define the following Sobolev-Lions type space:
Let
denote the space of all E-valued function space such that
Let
. Fourier-analytic representation of E-valued Besov space on
is defined as:
It should be noted that, the norm of Besov space does not depend on
(see e.g. [33], Section 2.3 for
).
Let A be a sectorial operator in H. Here,
where
denotes the real interpolation space between
and
for
,
(see e.g. [33], Section 1.3).
Remark 1.1. By Fubini’s theorem we get
Then by definition of spaces
,
and
we have
By J. lions-J. Peetre result (see e.g. [33], Section 1.8.2) for
the trace operator
is bounded from
into
Moreover, if
, then under some assumptions that will be stated in Section 3,
for all x,
and the map
is bounded from
into E. Hence, the nonlinear Equation (1.1) is satisfied in the Banach space H. Here,
denotes a domain of A equipped with graphical norm.
Sometimes we use one and the same symbol C without distinction in order to denote positive constants which may differ from each other even in a single context. When we want to specify the dependence of such a constant on a parameter, say
, we write
. Moreover, for
the relations
,
means that there exist positive constants
independent on u and
such that, respectively
The paper is organized as follows: In Section 1, some definitions and background are given. In Section 2, we obtain the existence of unique solution and a priory estimates for solution of the linearized problem (1.1)-(1.2). In Section 3, we show the existence and uniqueness of local strong solution of the problem (1.1)-(1.2). In Section 4, the existence and uniqueness of global strong solution of the problem (1.1)-(1.2) is derived. Section 5 is devoted to blow up property of the solution of (1.1)-(1.2). In Section 6, we show some applications of the problem (1.1)-(1.2).
Sometimes we use one and the same symbol C without distinction in order to denote positive constants which may differ from each other even in a single context. When we want to specify the dependence of such a constant on a parameter, say h, we write
.
2. Estimates for Linearized Equation
In this section, we make the necessary estimates for solutions of the integral problem for linear WE
(2.1)
(2.2)
where A is a linear operator in a Banach space E, a is a complex number and
,
are measurable functions on
.
Remark 2.1. By properties of real interpolation of Banach spaces and interpolation of the intersection of the spaces (see e.g. [33], Section 1.3) we obtain
In a similar way, we have
Remark 2.2. Let A be a sectorial operator in a Banach space E. In view of interpolation of sectorial operators (see e.g. [33], Section 1.8.2) we have the following relation
for
and
.
Note that from J. lions-J. Peetre result (see e.g. [33], Section 1.8.2) we obtain the following result.
Lemma A1. The trace operator
is bounded from
into
We assume that A is a sectorial operator in a Hilbert space H. Let A be a generator of a strongly continuous cosine operator function in a Banach space E defined by formula
(see e.g. [25], Section 11 or [23], Section 3). Then, from the definition of sine operator-function
we have
Remark 2.3. Let A be a densely defined operator in H. By virtue of ( [23], Theorem 3.15.3) if A be the generator of a cosine function
, i.e.
Let
(2.3)
Condition 2.1. Assume: 1)
(2.0)
2) A is a
-sectorial operator in the Hilbert space H and A is a generator of a cosine function; 3)
for
,
; 4)
and
.
Definition 1.1. Let
,
and
. The function
satisfies of the problem (1.1)-(1.2) is called the continuous solution or the strong solution of (1.1)-(1.2). If
, then
is called the local strong solution of (1.1)-(1.2). If
, then
is called the global strong solution of (1.1)-(1.2).
First we need the following lemmas:
Lemma 2.1. Let the Condition 2.1 holds. Then, problem (2.1)-(2.2) has a solution.
Proof. By using of the Fourier transform, we get from (2.1)-(2.2):
(2.4)
(2.5)
where
is a Fourier transform of
in x and
,
are Fourier transform of
and
, respectively and
Consider first, the Cauchy problem
(2.6)
where
for
. By virtue of ( [25], Section 11.2, 11.4) we obtain that
is a generator of a strongly continuous cosine operator function and the Cauchy problem (2.6) has a unique solution for all
. Moreover, the solution of (2.6) can be expressed as
(2.7)
where
is a cosine and
is a sine operator-functions generated by
, i.e.
Using the formula (2.7) and the first integral condition (2.5) we get
i.e. we obtain the first equation with respect to
,
:
(2.8)
where
Differentiating both sides of formula (2.7) and using the seconf integral condition (2.5), we have
i.e. we get the second equation with respect to
,
:
(2.9)
where
Now, we consider the system of Equations (2.8)-(2.9) in
and
. By assumption (2.0) and due to uniformly boundedness of
, the main determinant of this system
for all
. By solving the system (2.8)-(2.9) we get
(2.10)
By substituting the values
and
in (2.7), we obtain
(2.11)
i.e. problem (2.1)-(2.2) has a unique solution
(2.12)
where
,
, Q are linear operator functions defined by
Theorem 2.1. Assume the Condition 2.1 holds and
(2.13)
for
and for a
. Let
. Then for
,
,
for
and
for
problem (2.1)-(2.2) has a unique solution
. Moreover, the following estimate holds
(2.14)
uniformly in
, where the constant
depends only on A, the space H and initial data.
Proof. By Lemma 2.1, the problem (2.1)-(2.2) has a solution
for
,
and
. Let
and
From (2.12) we deduced that
(2.15)
By virtue of Remakes 2.1, 2.2 and the properties of sectorial operators we get the following uniform estimate
Hence, due to uniform boundedness of operator functions
,
, by (2.3), in view of (2.8)-(2.10) and by Minkowski’s inequality for integrals we get the uniform estimate
(2.16)
Let
Moreover, in a similar way, we deduced that
(2.17)
here, the space
is denoted by
. Let
(2.18)
By using the resolvent properties of sectorial operators, we have
(2.19)
Then by calculating
,
, we obtain
Let we show that
for some
and for all
, where
By embedding properties of Sobolev and Besov spaces it is sufficient to derive that
for some
. Indeed by contraction, by Condition 2.2 and by (2.18) we get
. Let
. For deriving the embedding relations
, it sufficient to show
Indeed, in view of (2.18),
are uniformly bounded for
. By virtue of (2.3), (2.19), by Condition 2.2 for
and
we have
Hence, by Fourier multiplier theorems (see e.g. [32], Theorem 4.3) we get that the functions
are Fourier multipliers from
to
. In a similar way we obtain that
are
Fourier multipliers. Then by Minkowski’s inequality for integrals, from (2.3), (2.16)-(2.18) and by Remake 2.3 we have
(2.20)
Moreover, by virtue of Remakes 2.1 - 2.3 and by reasoning as the above, we have the following estimate
(2.21)
uniformly in
. Thus, from (2.12), (2.20) and (2.21) we obtain
(2.22)
By differentiating (2.12) in a similar way, we get
(2.23)
Then from (2.22) and (2.23) in view of Remarks 2.1, 2.2 we obtain the estimate (2.14).
Let now show that problem (2.1) has a unique solution
. Let’s admit it is the opposite. So let’s assume that the problem (2.1) has two solutions
. Then by linearity of (2.1), we get that
is also a solution of the corresponding homogenous equation
Moreover, by (2.7) we have the following estimate
Since
, the above estimate implies that
, i.e.
.
Theorem 2.2. Assume the Condition 2.1 and (2.13) is satisfied. Let
. Then for
,
,
for
and
for
problem (2.1)-(2.2) has a unique solution
and the following estimate holds
(2.24)
for all
.
Proof. From (2.11) and (2.17) we get the following uniform estimate
(2.25)
By using the Fourier multiplier theorem ( [32], Theorem 4.3) and by reasoning as in Theorem 2.1 we get that
,
and
are Fourier multipliers in
uniformly with respect to
. So, the estimate (2.25) by using the Minkowski’s inequality for integrals implies (2.24).
The uniquness of (2.1)-(2.2) is obtained by reasoning as in Theorem 2.1.
3. Local Well Posedness of IVP for Nonlinear WE
In this section, we will show the local existence and uniqueness of solution of the nonlinear problem (1.1)-(1.2).
For this aim we need the following lemmas. By reasoning as in [7] [18] [35], we show the following lemmas concerning the behaviour of the nonlinear term in E-valued space
. Here, let E be a Banach algebra.
Lemma 3.1. Let
,
with
. Then for any
, we have
. Moreover, there is some constant
depending on M such that for all
with
,
(3.1)
Proof. For
in view of
, we get
It follows that
If s is a positive integer, we have
(3.2)
By calculation of derivative and applying Holder inequality, we get
(3.3)
where
Applying Gagliardo-Nirenberg’s inequality in E-valued
spaces, we have
(3.4)
Hence, from (3.3) and (3.4) we get
(3.5)
Then combining (3.2), (3.3) and (3.5) we obtain (3.1).
Let s is not integer number and
. From the above proof, we have
Then using interpolation between
and
yields (3.1) for all
.
By using Lemma 3.1 and properties of convolution operators we obtain.
Corollary 3.1. Let
,
with
. Moreover, assume
. Then for any
we have,
. Moreover, there is some constant
depending on M such that for all
with
,
Lemma 3.2. Let
,
. Then for any M there is some constant
depending on M such that for all u,
with
,
,
,
,
Corollary 3.2. Let
,
. Then for any positive M there is a constant
depending on M such that for all
with
,
,
Lemma 3.3. If
, then
is an algebra. Moreover, for
,
By using, the Corollary 3.1 and Lemma 3.3 we obtain.
Lemma 3.4. Let
,
and
for
,
be a positive integer. If
and
, then
Corollary 3.3. Let
,
and
for
,
be a positive integer. Moreover, assume
. If
and
, then
Lemma 3.5. Let
,
and
for
. Moreover, let
be a positive integer. If
,
,
and
,
, then
Let
denotes the real interpolation space between
and
with
, i.e.
Remark 3.1. By using J. Lions-J. Peetre result (see e.g. [33], Section 1.8) we obtain that the map
,
is continuous and surjective from
onto
and there is a constant
such that
(3.6)
Let
Condition 3.1. Assume:
1) the Condition 2.1 holds for
,
, for a
and
;
2) the function
: continuous from
into H,
with k an integer,
and
for
,
be a positive integer.
Let
Main aim of this section is to prove the following results:
Theorem 3.1. Let the Condition 3.1 holds. Then there exists a constant
such that for any
and
satisfying
(3.7)
problem (1.1)-(1.2) has a unique local strange solution
. Moreover,
(3.8)
where the constant C depends only on A, E, g, f and initial values.
Proof. By (2.5), (2.6) the problem of finding a solution u of (1.1)-(1.2) is equivalent to finding a fixed point of the mapping
(3.9)
where
,
are defined by (2.6) and
is a map defined by
We define the metric space
equipped with the norm defined by
where
satisfies (3.7) and
is a constant in Theorem 2.1 and 2.2. It is easy to prove that
is a complete metric space. From imbedding in Sobolev-Lions space
(see e.g. [27], Theorem 1) and trace result (3.6) we got that
if we take that
is enough small. For
and
, let
So, we will find T and M so that G is a contraction in
. By Theorems 2.1, 2.2 and Corollary 3.3
. So, problem (1.1)-(1.2) has a solution that satisfies the following
(3.10)
where
,
are defined by (2.5) and (2.6). By assumptions, it is easy to see that the map G is well defined for
. First, let us prove that the map G has a unique fixed point in
. For this aim, it is sufficient to show that the operator G maps
into
and G is strictly contractive if
is suitable small. In fact, by (2.7) in Theorem 2.1, Corollary 3.3 and in view of (3.7), we have
(3.11)
On the other hand, by (2.17), Corollary 3.3 and (3.7), we get
(3.12)
Hence, combining (3.11) with (3.12) we obtain
(3.13)
So, taking that
is enough small such that
, by Theorems 2.1, 2.2 and (3.13), G maps
into
.
Now, we are going to prove that the map G is strictly contractive. Let
given. From (3.10) we get
By (2.7) in Theorem 2.1 and Corollary 3.3, we have
(3.14)
On the other hand, by (2.17) in Theorem 2.2, Corollary 3.3 and (3.7), we get
(3.15)
Combining (3.14) with (3.15) yields
(3.16)
Taking
is enough small, from (3.16) we obtain that G is strictly contractive in
. Using the contraction mapping principle, we get that
has a unique fixed point
and
is the solution of (1.1)-(1.2).
Let us show that this solution is a unique in
. Let
are two solutions of (1.1)-(1.2). Then for
, we have
(3.17)
Hence, by Minkowski’s inequality for integrals and by Theorem 2.2 from (3.17) we obtain
(3.18)
From (3.18) and Gronwall’s inequality, we have
, i.e. problem (1.1)-(1.2) has a unique solution in
.
Consider the problem (1.1)-(1.2), when
and
. Let
4. Application
Consider the problem (1.4). Let
Let
,
be roots of equation
. Let
Here,
From Theorem 3.1 we obtain the following result.
Theorem 4.1. Suppose the following conditions are satisfied:
1)
for
,
,
and
for all
;
2)
,
and
for a.e.
,
;
,
,
.
3)
,
and
for
for
,
and
.
4) The function
is continuous in
for
; moreover
.
Then problem (1.9)-(1.10) has a unique local strange solution
where
is a maximal time interval that is appropriately small relative to M. Moreover, if
then
.
Proof. By virtue of [30],
is a Fourier type space. By virtue of [30], the operator
defined by (1.3) is sectorial in
. Moreover, by interpolation of Banach spaces ( [33], Section 1.3), we have
Then, by using the properties of spaces
,
,
we get that all conditions of Theorem 3.1 are hold, i.e., we obtain the conclusion.
5. Conclusion
Here, assuming enough smoothness on the initial data in terms of interpolation spaces
, H and the sectorial operators, the existence, uniqueness, regularity properties of solutions are established. By choosing the space H and A, the regularity properties of solutions of a wide class of wave equations in the field of physics are obtained.
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