**5. Uniqueness of solutions of BVP**

Define the functions

$$W(\boldsymbol{u}) = \mathbf{0}, \mathsf{5C}\_{ij}^{nl} \boldsymbol{u}\_i, \boldsymbol{u}\_j \boldsymbol{u}\_l, \quad K(\boldsymbol{u}) = \mathbf{0}, \mathsf{5} \|\boldsymbol{u}\_i\|^2,$$

$$E(\boldsymbol{u}) = K(\boldsymbol{u}) + W(\boldsymbol{u}), \quad L(\boldsymbol{u}) = K(\boldsymbol{u}) - W(\boldsymbol{u}),$$

which are called the *densities of internal, kinetic, and total energy* of the system, respectively, and *L* is *the Lagrangian*.

**Theorem 5.1.** *If u is a classical solution of the Dirichlet (Neumann) boundary value problem, then*

$$\int\_{D\_t^-} L(u(\mathbf{x},t))dV(\mathbf{x},t) = \int\_{D\_t^-} G\_i(\mathbf{x},t)u\_i(\mathbf{x},t)dV(\mathbf{x},t) +$$

$$\int\_{D\_t} \left[ \mathbf{g}\_i(\mathbf{x},t)u\_i^S(\mathbf{x},t)d\mathbf{S}(\mathbf{x},t) - \int\_{\mathcal{S}^-} (u\_i(\mathbf{x},t)u\_{i\cdot\cdot t}(\mathbf{x},t) - u\_i^0(\mathbf{x})u\_i^1(\mathbf{x}))dV(\mathbf{x}) \right]$$

*Here and below, dV x*ð Þ¼ *dx*<sup>1</sup> … *dxN*, *dV x*ð Þ¼ , *t dV x*ð Þ*dt*; *dS x*ð Þ, *and*, *dS x*ð Þ , *t are the differentials of the area of S and D, respectively*.

**Proof**. Multiplying (1) by *ui* and summing the result over *i*, after simple algebra, we obtain the expression

$$L = \left(\mathbf{C}\_{ij}^{ml}\boldsymbol{u}\_j, \boldsymbol{u}\_m\boldsymbol{u}\_i\right), \boldsymbol{l} - (\boldsymbol{u}\_i\boldsymbol{u}\_i, \boldsymbol{l}), \boldsymbol{t} + \mathbf{G}\_i\boldsymbol{u}\_i.$$

This equality is integrated over *Dt* taking into account the front discontinuities and using the Gauss-Ostrogradsky theorem and initial conditions (33) and (34) to obtain

*Mathematical Theorems - Boundary Value Problems and Approximations*

$$\begin{aligned} \int\_{D\_{i}^{-}} L(u(\mathbf{x},t))dV(\mathbf{x},t) &= \int\_{D\_{i}^{-}} \left(C\_{ij}^{ml}u\_{j,m}u\_{i}\right)\_{,l} - (u\_{i}u\_{i:t})\_{,l}dV(\mathbf{x},t) + \\ &+ \int\_{D\_{i}^{-}} G\_{i}(\mathbf{x},t)u\_{i}(\mathbf{x},t)dV(\mathbf{x},t) = \int\_{D\_{i}} \sigma\_{i}^{l}u\_{l}(\mathbf{x})u\_{i}(\mathbf{x},t)dS(\mathbf{x},t) - \\ &- \int\_{S} u\_{i}u\_{i:t}(\mathbf{x},t) - u\_{i}u\_{i:t}(\mathbf{x},0))dS(\mathbf{x},t) + \int\_{D\_{i}^{-}} G\_{i}(\mathbf{x},t)u\_{i}(\mathbf{x},t)dV(\mathbf{x},t) + \\ &+ \sum\_{k} \int\_{F\_{k}\cap D\_{i}^{-}} u\_{i} \left[\nu\_{l}^{k}\sigma\_{l}^{l}(\mathbf{x},t) - \nu\_{l}^{k}u\_{i:t}(\mathbf{x},t)\right]\_{F\_{k}}dF\_{k}(\mathbf{x},t) \end{aligned}$$

Here, *ν<sup>k</sup> <sup>l</sup>* , *and ν<sup>k</sup> <sup>t</sup>* are the components of the unit normal vector to the front *Fk*ð Þ *<sup>x</sup>*, *<sup>t</sup>* in *RN*þ<sup>1</sup> , for which we have [17]

$$
\nu\_t^k = -c\_k / \left(\nu\_j^k \nu\_j^k\right)^{1/2},\tag{37}
$$

and *u*2, then their difference *u* ¼ *u*<sup>1</sup> � *u*<sup>2</sup> satisfies the system of equations with *G* ¼

*Singular Boundary Integral Equations of Boundary Value Problems for Hyperbolic Equations…*

*<sup>i</sup>* ð Þ¼ *x* 0 ð Þ *m* ¼ 0, 1 *:*

The vector *u* on the boundary *S* satisfies the homogeneous boundary conditions

Since the integrand is positive definite and by the conditions of the theorem, we

Let us assume that *S* is a smooth boundary with a continuous normal of a set *S*�.

The Heaviside function *H t*ð Þ is extended to zero by setting *H*ð Þ¼ 0 1*=*2. Define

*<sup>S</sup>* ð Þ *x* of a set *S*� is defined for *x*∈*S* as

. Similarly,

*<sup>i</sup>* ð Þ *x*, *t* denotes the Green's matrix, i.e. the fundamental solution of Eq. (1)

*<sup>i</sup>* ð Þ *<sup>x</sup>*, *<sup>t</sup>* <sup>∗</sup> *tH t*ðÞ ) *<sup>∂</sup>tV*^ *<sup>k</sup>*

Here and below, the star denotes the complete convolution with respect to ð Þ *x*, *t* , while the variable under the star denotes the incomplete convolution with respect to *x* or *t*, respectively. The convolution exists since the supports are semibounded with

*<sup>S</sup>* ð Þ¼ *x* 1*=*2 (38)

*<sup>S</sup>* ð Þ *x H t*ð Þ (39)

*<sup>D</sup>*ð Þ *x*, *t* , (40)

*<sup>D</sup>*ð Þ *x*, *t :* (41)

*<sup>i</sup> δ*ð Þ *x δ*ð Þ*t* and satisfies the conditions

*<sup>i</sup>* , *<sup>t</sup>*ð Þ¼ *x*, 0 0, *x* 6¼ 0 (42)

*<sup>i</sup>* <sup>¼</sup> *<sup>U</sup>*^ *<sup>k</sup>*

*<sup>i</sup> :* (43)

*<sup>i</sup> δ*ð Þ *x H t*ð Þ.

ð Þ¼ *x*, *t* 0*:*

ð Þ *K u*ð Þþ , *t W u*ð Þ , *t dS x*ð Þ¼ 0*:*

*ui*ð Þ¼ *x*, *t* 0 or *gi*

*S*

*H*�

*<sup>D</sup>*ð Þ¼ *x*, *t H*�

Accordingly, for *u* defined on *D*�, we introduce the generalized function

*u x* ^ð Þ¼ , *t uH*�

*<sup>G</sup>*^ *<sup>k</sup>*ð Þ¼ *<sup>x</sup>*, *<sup>t</sup> GkH*�

*H*�

*<sup>i</sup>* ð Þ¼ *<sup>x</sup>*, 0 0, *<sup>U</sup>*^ *<sup>k</sup>*

respect to *<sup>t</sup>*. Clearly, the convolution is the solution of Eq. (1) at *Fi* <sup>¼</sup> *<sup>δ</sup><sup>k</sup>*

For system (1), such a matrix was constructed in [10]. The primitive of Green's matrix with respect to *t* is defined as

*<sup>i</sup>* ð Þ¼ *<sup>x</sup>*, *<sup>t</sup> <sup>U</sup>*^ *<sup>k</sup>*

*um*

*E u*ð Þ , *<sup>t</sup> dS x*ð Þ¼ <sup>ð</sup>

**6. Analogues of the Kirchhoff and Green's formulas**

0 and the zero initial conditions, i.e.

*DOI: http://dx.doi.org/10.5772/intechopen.92449*

ð

*S*

have *u* � 0. The theorem is proved.

The characteristic function *H*�

the characteristic function of *D*� as

Let *<sup>U</sup>*^ *<sup>k</sup>*

**69**

which is defined on the entire space *R<sup>N</sup>*þ<sup>1</sup>

that corresponds to the function *Fi* <sup>¼</sup> *<sup>δ</sup><sup>k</sup>*

*V*^ *k*

*U*^ *k*

Theorem 5.2 yields

where *ck* is the velocity of the front. With the notation introduced, the relation (37) and the front condition (7) yield the assertion of the theorem.

It is easy to see that the following result holds true. **Corollary.** *If ui*ð Þ¼ *x*, 0 0, *ui*, *<sup>t</sup>*ð Þ¼ *x*, 0 0, *and*

$$\lim\_{l \to +\infty} u\_{i,l} \to 0, \quad \lim\_{l \to +\infty} u\_{i,l} \to 0, \quad x \in \mathbb{S}^-,$$

then

$$\int\_{D^{-}} L(u(\mathbf{x},t))dV(\mathbf{x},t) = \int\_{D^{-}} G\_i(\mathbf{x},t)u\_i(\mathbf{x},t)dV(\mathbf{x},t) + \int\_{D} \mathbf{g}\_i(\mathbf{x},t)u\_i^S(\mathbf{x},t)dS(\mathbf{x},t)$$

is proved in the following theorem [17]:

**Theorem 5.2.** *If u is a classical solution of the Dirichlet (Neumann) boundary value problem, then*

$$\int\_{\mathcal{S}^{-}} (E(\mathfrak{u}, t) - E(\mathfrak{u}, \mathbf{0}))dV(\mathfrak{x}) = $$

$$\int\_{D\_t^{-}} G\_i(\mathfrak{x}, t)u\_{i, t}(\mathfrak{x}, t)dV(\mathfrak{x}, t) + \int\_{D\_t} \mathfrak{g}\_i(\mathfrak{x}, t)u\_i^{\mathcal{S}},\_t(\mathfrak{x}, t)d\mathfrak{S}(\mathfrak{x}, t).$$

It is easy to see that this theorem implies the uniqueness of the solutions to the initial-boundary value problems in question.

**Theorem 5.3.** *If a classical solution of the Dirichlet (Neumann) boundary value problem exists and satisfies the conditions*

$$\lim\_{l \to +\infty} u\_{i,l} \to 0, \quad \lim\_{l \to +\infty} u\_{i,l} \to 0, \ \forall \mathfrak{x} \in \mathbb{S}^-,$$

*then this solution is unique*.

**Proof**. Since the problem is linear, it suffices to prove the uniqueness of the solution to the homogeneous boundary value problem. If there are two solutions *u*<sup>1</sup> *Singular Boundary Integral Equations of Boundary Value Problems for Hyperbolic Equations… DOI: http://dx.doi.org/10.5772/intechopen.92449*

and *u*2, then their difference *u* ¼ *u*<sup>1</sup> � *u*<sup>2</sup> satisfies the system of equations with *G* ¼ 0 and the zero initial conditions, i.e.

$$
\mu\_i^m(\mathbf{x}) = \mathbf{0} \quad (m = \mathbf{0}, \mathbf{1}).
$$

The vector *u* on the boundary *S* satisfies the homogeneous boundary conditions

$$u\_i(\mathbf{x}, t) = \mathbf{0} \quad \text{or} \quad \mathbf{g}\_i(\mathbf{x}, t) = \mathbf{0}.$$

Theorem 5.2 yields

ð

*Lux* ð Þ ð Þ , *t dV x*ð Þ¼ , *t*

Here, *ν<sup>k</sup>*

*Fk*ð Þ *<sup>x</sup>*, *<sup>t</sup>* in *RN*þ<sup>1</sup>

then

ð

*Lux* ð Þ ð Þ , *t dV x*ð Þ¼ , *t*

ð

*D*� *t*

*then this solution is unique*.

is proved in the following theorem [17]:

initial-boundary value problems in question.

*problem exists and satisfies the conditions*

*D*�

*problem, then*

**68**

*<sup>l</sup>* , *and ν<sup>k</sup>*

ð

*Cml ij u <sup>j</sup>*, *mui* � �

*Mathematical Theorems - Boundary Value Problems and Approximations*

*Gi*ð Þ *x*, *t ui*ð Þ *x*, *t dV x*ð Þ¼ , *t*

*uiui*, *<sup>t</sup>*ð Þ� *x*, *t uiui*, *<sup>t</sup>*ð ÞÞ *x*, 0 *dS x*ð Þþ , *t*

ð Þ� *<sup>x</sup>*, *<sup>t</sup> <sup>ν</sup><sup>k</sup> <sup>t</sup> ui*, *<sup>t</sup>*ð Þ *<sup>x</sup>*, *<sup>t</sup>* � �

> *j νk j* � �<sup>1</sup>*=*<sup>2</sup>

where *ck* is the velocity of the front. With the notation introduced, the relation

lim*<sup>t</sup>*!þ<sup>∞</sup> *ui*, *<sup>l</sup>* ! 0, lim*<sup>t</sup>*!þ<sup>∞</sup> *ui*, *<sup>t</sup>* ! 0, *<sup>x</sup>*∈*S*�,

*Gi*ð Þ *x*, *t ui*ð Þ *x*, *t dV x*ð Þþ , *t*

**Theorem 5.2.** *If u is a classical solution of the Dirichlet (Neumann) boundary value*

ð Þ *E u*ð Þ� , *t E u*ð Þ , 0 *dV x*ð Þ¼

ð

*Dt gi* ð Þ *<sup>x</sup>*, *<sup>t</sup> <sup>u</sup><sup>S</sup>*

It is easy to see that this theorem implies the uniqueness of the solutions to the

**Theorem 5.3.** *If a classical solution of the Dirichlet (Neumann) boundary value*

lim*<sup>t</sup>*!þ<sup>∞</sup> *ui*, *<sup>l</sup>* ! 0, lim*<sup>t</sup>*!þ<sup>∞</sup> *ui*, *<sup>t</sup>* ! 0, <sup>∀</sup>*x*∈*S*�,

**Proof**. Since the problem is linear, it suffices to prove the uniqueness of the solution to the homogeneous boundary value problem. If there are two solutions *u*<sup>1</sup>

*<sup>t</sup>* are the components of the unit normal vector to the front

, *<sup>l</sup>* � *uiui*, ð Þ*<sup>t</sup>* , *tdV x*ð Þþ , *t*

ð

*nl*ð Þ *x ui*ð Þ *x*, *t dS x*ð Þ� , *t*

*dFk*ð Þ *x*, *t*

, (37)

*<sup>i</sup>* ð Þ *x*, *t dS x*ð Þ , *t*

*Gi*ð Þ *x*, *t ui*ð Þ *x*, *t dV x*ð Þþ , *t*

ð

*D*� *t*

*Fk*

ð

*D gi* ð Þ *<sup>x</sup>*, *<sup>t</sup> uS*

*<sup>i</sup>* , *<sup>t</sup>*ð Þ *x*, *t dS x*ð Þ , *t :*

*Dt σl i*

*D*� *t*

þ ð

� ð

þ X *k*

*S*

ð

*ui ν<sup>k</sup> l σl i*

*νk*

(37) and the front condition (7) yield the assertion of the theorem.

*<sup>t</sup>* ¼ �*ck<sup>=</sup> <sup>ν</sup><sup>k</sup>*

*Fk* ∩ *D*� *t*

, for which we have [17]

It is easy to see that the following result holds true. **Corollary.** *If ui*ð Þ¼ *x*, 0 0, *ui*, *<sup>t</sup>*ð Þ¼ *x*, 0 0, *and*

ð

*D*�

ð

*S*�

*Gi*ð Þ *x*, *t ui*, *<sup>t</sup>*ð Þ *x*, *t dV x*ð Þþ , *t*

*D*� *t*

*D*� *t*

$$\int\_{S} E(u,t)dS(\mathbf{x}) = \int\_{S} (K(u,t) + W(u,t))dS(\mathbf{x}) = \mathbf{0}.$$

Since the integrand is positive definite and by the conditions of the theorem, we have *u* � 0. The theorem is proved.
