In subsequent consideration we shall have frequently to use certain theorems of the theory of functions of complex variable, the statements of which we now quote. The proofs may be found in all text-book of the theory. Let
be smooth contour and
a function of position on it.

Definition 2.1.

The function

is said to satisfy on the curve

the Hölder condition, if for two arbitrary points of this curve

where
and
are positive numbers.

Theorem 2.2.

Let

be a complex function, which obeys the Hölder condition for all finite

and tends to a definite limit

as

, such that for large

the following inequality holds:

Then Cauchy type integral

constitutes a function analytic in the left and right semiplanes. Here and below these functions will be denoted by

and

, respectively. These functions have the limiting values

and

at all points of imaginary axis

, on approaching the contour from the left and from the right, respectively. These limiting values are expressed by Sokhotzki-Plemelj formulae:

Subtracting and adding the formula (2.4) we obtain the following two equivalent formulae:

which will be frequently employed hereafter.

We consider the following linear initial-boundary value problem on half-line:

Setting

,

, we define

where the function

is given by

for

. Here and below

, where

and

are a left and right limiting values of sectionally analytic function

given by

where for some fixed real point

,

All the integrals are understood in the sense of the principal values.

Proposition 2.3.

Let

. Then there exists a unique solution

for the initial-boundary value problem (2.6), which has an integral representation:

Proof.

In order to obtain an integral representation for solutions of the problem (2.6) we suppose that there exist a solution

, which is continued by zero outside of

:

Let

be a function of the complex variable

, which obeys the Hölder condition for all

, such that Re

. We define the operator

by

Using the Laplace transform we get

Since

is analytic for all Re

, we have

Therefore, applying the Laplace transform with respect to

to problem (2.6) and using (2.15) and (2.16), we obtain for

We rewrite (2.17) in the form

with some function

such that

Applying the Laplace transform with respect to time variable to (2.18), we find

where Re

and Re

. Here, the functions

and

are the Laplace transforms for

and

with respect to time, respectively. In order to obtain an integral formula for solutions to the problem (2.6) it is necessary to know the function

. We will find the function

using the analytic properties of the function

in the right-half complex planes Re

and Re

. Equation (

2.16) and the Sokhotzki-Plemelj formulae imply for Re

In view of Sokhotzki-Plemelj formulae via (2.21) the condition (2.22) can be written as

where the sectionally analytic functions

and

are given by Cauchy type integrals:

To perform the condition (2.23) in the form of a nonhomogeneous Riemann-Hilbert problem we introduce the sectionally analytic function:

Taking into account the assumed condition (2.19), we get

Also observe that from (2.24) and (2.26) by Sokhotzki-Plemelj formulae,

Substituting (2.23) and (2.28) into this equation we obtain for Re

Equation (2.30) is the boundary condition for a nonhomogeneous Riemann-Hilbert problem. It is required to find two functions for some fixed point
, Re
:
, analytic in the left-half complex plane Re
and
, analytic in the right-half complex plane Re
, which satisfy on the contour Re
the relation (2.30).

Note that bearing in mind formula (2.27) we can find the unknown function

, which involved in the formula (2.21), by the relation

The method for solving the Riemann problem
is based on the following results. The proofs may be found in [17].

Lemma 2.4.

An arbitrary function

given on the contour

, satisfying the Hölder condition, can be uniquely represented in the form

where

are the boundary values of the analytic functions

and the condition

holds. These functions are determined by

Lemma 2.5.

An arbitrary function

given on the contour

, satisfying the Hölder condition, and having zero index,

is uniquely representable as the ratio of the functions

and

, constituting the boundary values of functions,

and

, analytic in the left and right complex semiplane and having in these domains no zero. These functions are determined to within an arbitrary constant factor and given by

In the formulations of Lemmas 2.4 and 2.5 the coefficient

and the free term

of the Riemann problem are required to satisfy the H

lder condition on the contour Re

. This restriction is essential. On the other hand, it is easy to observe that both functions

and

do not have limiting value as

. So we cannot find the solution using

. The principal task now is to get an expression equivalent to the boundary value problem (2.30), such that the conditions of lemmas are satisfied. First, we introduce the function

where
,
, and
. We make a cut in the plane
from point
to point
through
. Owing to the manner of performing the cut the functions
and
are analytic for Re
and the function
is analytic for Re
.

We observe that the function

, given on the contour Re

, satisfies the Hölder condition and

does not vanish for any Re

. Also we have

Therefore in accordance with Lemma 2.5 the function

can be represented in the form of the ratio

From (2.37) and (2.39) we get

where

. We note that (2.41) is equivalent to

Now, we return to the nonhomogeneous Riemann-Hilbert problem defined by the boundary condition (2.30). We substitute the above equation in (2.30) and add

in both sides to get

On the other hand, by Sokhotzki-Plemelj formulae and (2.25),

. Now, we substitute

from this equation in formula (2.43); then by (2.41) we arrive to

In subsequent consideration we shall have to use the following property of the limiting values of a Cauchy type integral, the statement of which we now quote. The proofs may be found in [17].

Lemma 2.6.

If

is a smooth closed contour and

a function that satisfies the Hölder condition on

, then the limiting values of the Cauchy type integral

also satisfy this condition.

Since

satisfies on Re

the Hölder condition, on basis of Lemma 2.6 the function

also satisfies this condition. Therefore, in accordance with Lemma 2.4 it can be uniquely represented in the form of the difference of the functions

and

, constituting the boundary values of the analytic function

, given by

Therefore, (2.44) takes the form

The last relation indicates that the function

, analytic in Re

, and the function

, analytic in Re

, constitute the analytic continuation of each other through the contour Re

. Consequently, they are branches of a unique analytic function in the entire plane. According to Liouville theorem this function is some arbitrary constant

. Thus, we obtain the solution of the Riemann-Hilbert problem defined by the boundary condition (2.30):

Since

is defined by a Cauchy type integral, with density

, we have

, as

for

. Using this property in (2.48) we get

and the limiting values for

are given by

Now, we proceed to find the unknown function

involved in the formula (2.21) for the solution

of the problem (2.6). First, we represent

as the limiting value of analytic functions on the left-hand side complex semiplane. From (2.41) and Sokhotzki-Plemelj formulae we obtain

Now, making use of (2.49) and the above equation, we get

We observe that

is boundary value of a function analytic in the left-hand side complex semi-plane and therefore satisfies our basic assumption (2.19). Having determined the function

, bearing in mind formula (2.21) we determine the required function

:

Now we prove that, in accordance with last relation, the function

constitutes the limiting value of an analytic function in Re

. In fact, making use of Sokhotzki-Plemelj formulae and using (2.41), we obtain

Thus, the function
is the limiting value of an analytic function in Re
. We note the fundamental importance of the proven fact, the solution
constitutes an analytic function in Re
, and, as a consequence, its inverse Laplace transform vanishes for all
. We now return to solution
of the problem (2.6). Taking inverse Laplace transform with respect to time and space variables, we obtain
where the function
is defined by formula (2.8). Thus, Proposition 2.3 has been proved.

Now we collect some preliminary estimates of the Green operator
.

Lemma 2.7.

The following estimates are true, provided that the right-hand sides are finite:

where
,
,
,
, and
and
are given by (1.13) and (1.11), respectively.

Proof.

First, we estimate the function

We note that

, as

, and write

in the form

For first integral in (2.57), we obtain the estimate

where

, and for second integral we have

Therefore, substituting (2.58) and (2.59) in (2.57), we get for

Now, we estimate function

defined by

Using (2.60), we get for

the estimate

where Re

. Then, by (2.62) and Cauchy Theorem,

Equations (2.63) imply that we can write

in the form

Thus, for Re

,

where

satisfies

In fact, we use (2.62) and inequality

, where Re

Re

and

, to obtain

Making the change of variable

, (2.67) follows. Now, substituting (2.66) in (2.8), for Green function

, we obtain

The function

defined in (2.70) satisfies the estimate

In fact, using (2.67) we get

We have used inequality

, where

and

is some positive constant. Taking

and making the change of variables:

and

, we obtain (2.71). Now, let us split (2.69):

By Fubini's theorem and Cauchy's theorem, from the first and fourth summands we obtain

Now, we show that function

defined by (2.78) satisfies

In fact, using (2.62) and the inequality

, where Re

and

, we get

Then, taking

,

and making the change of variable

and

, we obtain (2.79). In the same way, we show that function

defined in (2.78) satisfies the inequality:

In fact, using the inequality

, where Re

and

, we get

Making the change of variable

, we arrive to

Thus, (2.81) follows. Finally, we show that

where

is given by (1.11). Making the change of variable

,

, and choosing

, we get

Now, making the change of variable

and

in equation for

we obtain

Therefore, (2.84) follows. Finally, using estimates (2.71), (2.79), (2.81), and (2.84), we get the asymptotic for the Green function

:

where

is given by (1.11) and

. By last inequality

where
and
are given by (1.13) and (1.11), respectively, and
. Thus, the first estimate in Lemma 2.7 has been proved.

Now, we are going to prove the second estimate in Lemma 2.7. First, for large

, using Sokhotzki-Plemelj formulae, we have for function

, defined in (2.61),

Substituting last equation in (2.8), we get

Making the change of variable

we get for

To estimate

, we consider an extension to the function

:

to obtain for

Let us write the function

, defined in (2.61), in the form

where Re

. Then, by Cauchy Theorem, for

the second summand in last equation is zero. Thus, using (2.62) we obtain for Re

From the last inequality and (2.97) we get

Taking

and making the change of variables

and

, in the last inequality, we obtain

From (2.92) and the estimates (2.94) and (2.101) we get the estimate

. Thus,

Now, for small

, we are going to prove the estimate

where

. First, we rewrite the Green function

in the form

The contours

and

are defined in (2.96) and

Moreover, we have extended the function

as in (2.95). Making the change of variable

and using the inequality

,

, we obtain the estimate

or

for

. Now, we estimate

. Using

, for Re

,

, and

, we get

Making the change o variables

and

, into the last inequality, we obtain

By (2.104) and the estimates (2.107) and (2.109) we get

Thus, we get (2.103) and the second estimate in Lemma 2.7 has been proved.

Let us introduce the operators

where

and

are defined in (2.104). Then, the operator

can be written in the form

Now, we are going to prove the third estimate in Lemma 2.7,

First, we estimate the operator

. Making the change of variable

, we get for the function

Now, we make the change of variable

:

Integrating by parts the last equation we obtain

for

, where

are defined as above. Thus, for

,

Therefore, from the inequalities (2.116) and (2.120) we have

We remember some well-known inequalities.

(i)

*Young's Inequality*. Let

and

, where

,

. Then, the convolution

belongs to

, where

and Young's inequality

holds.

(ii)

*Minkowski's Inequality*. Let

and

; then

(iii)

*Interpolation Inequality*. Let

with

; then

for any

, and the interpolation inequality holds:

where
and
.

(iv)

*Arithmetic-Geometric Mean Inequality*. If

and

are nonnegative, then

Then, by (2.121) and Young's inequality (2.122), we obtain

Finally, using the Interpolation Inequality (2.124) and the arithmetic-geometric mean inequality (2.125), we obtain

Now, we estimate the operator

. First, by Cauchy Theorem we get for Re

Then, using (2.131) and the inequalities

, where Re

and

, and

Then, using the inequalities (2.132) and

we obtain for

Thus, the last estimate and (2.129) imply the third estimate in Lemma 2.7.

Now, we are going to prove the fourth estimate in Lemma 2.7. We use (2.114). First, we estimate the operator

, defined in (2.112). Using the inequality

, where

, and Minkowski's inequality (2.123), we obtain

Then, Young's inequality (2.122) implies

where

,

,

, and

. Then, by the inequality (2.121) and the change of variables

, we get

Thus,

, provided

. Using

, it follows that

where

. We note that

, since

. Substituting (2.139) in (2.137), we get

where

,

, and

. Now, we estimate the operator

, defined in (2.113). First, we use that function

satisfies the following inequality:

Then, by the inequality

, we obtain

Substituting in the last inequality the estimate

where

, we obtain

Then, using

and

, we get

where
,
and
. Finally, from estimates (2.140) and (2.146) we obtain the fourth estimate in Lemma 2.7. Then, we have proved Lemma 2.7.

Theorem 2.8.

Let the initial data be

, with

. Then, for some

there exists a unique solution

to the initial boundary-value problem (1.1). Moreover, the existence time
can be chosen as follows:
, where
.