Semilinear Sobolev type mathematical models

Mathematical models based on semilinear Sobolev type equations are called semilinear Sobolev type mathematical models.

The necessity to study semilinear Sobolev type mathematical models caused by the need to study applied problems related to the dynamics of viscoelastic fluids [16], vibrations in the DNA molecule [6], the theory of metal creep [5], wave propagation in shallow water [25], the propagation of ion-acoustic waves in plasma [1], the theory of electric circuits [14], the theory of heat conduction with two temperatures [4], filtration in a fractured-porous medium [2, 28] and others [19, 39].

The study of semilinear Sobolev type equations was initiated in [29,37] and the concept of a quasistationary trajectory was introduced. Later, on the basis of abstract results, the Oskolkov [33, 35] and Hoff [32] mathematical models were investigated, and the structure of the phase space of the Hoff and Oskolkov equations was studied. In parallel with the solution of the problem of existence and uniqueness, the theory of optimal control of solutions to the semilinear Sobolev type equations arose and was developed [23, 36, 44], the theory of stability of solutions to the semilinear Sobolev type equations [22, 34] was studied and the phenomenon of non-uniqueness of solutions was explained [18, 31].

Probably, the first work devoted to the study of equations unsolvable with respect to the highest time derivative belongs to A. Poincare (1885) [20]. However, the regular study of initial-boundary value problems for such equations began with the works of

• S .L. Sobolev [27]. At present, the theory of Sobolev type equations is actively developing both in breadth and in depth, as evidenced by many scientific directions around which the scientific schools have developed [1, 7, 10, 14, 15, 21, 24, 26]. In this paper, we trace the stages in the development of the theory of semilinear Sobolev type equations and consider its applications to the study of three semilinear Sobolev type mathematical models.

The first one is the Oskolkov mathematical model (a mathematical model of nonlinear filtration in a fractured-porous medium). Let Q C Rn be a bounded domain with boundary dQ of class Cro. In a cylinder Q x R, consider the equation ut — $Aut = vAu — K (u)(1)

with the Cauchy–Dirichlet conditions

u(x, 0) = u0(x), x G Q,(2)

u(x,t) = 0, (x, t) G dQ x R.(3)

Equation (1) describes many processes and phenomena occurring with the participation of a viscoelastic fluid, including filtration. It was obtained by A.P. Oskolkov [19]. The nonlinear term in (1) is such that K (0) = 0, ( K(u),u ) >  0 ( (• , •) is an inner product in L ² (Q)). In particular, it can take the form K (u) = u ^2m+1 or K (u) = sh u. Generally speaking, the nonlinearity can be represented by the series

∞

K(u) = ^ a m u ^2m+ 1 ,   a m G R ₊ .

m =0

The parameters ж, v G R ₊ characterize the elastic and viscous properties of the fluid, respectively.

The second one is a mathematical model of ion-acoustic waves in plasma. Let Q = (0, a) x (0, b) x (0, c) C R ³ . In a cylinder Q x R consider the equation

∂ ² u

^{(A — A)}u tttt + ^{(A — A )u} tt + ^атг"2 = ^A(u )

∂x ² ₃

with the Cauchy–Dirichlet conditions

u(x, 0) = u0(x), ut(x, 0) = u1(x), utt(x, 0) = u2(x),   uttt(x, 0) = u3(x),   x G Q,                     (5)

u(x,t) = 0,   (x, t) G dQ x R .

Equation (4) describes the ion-acoustic waves in a plasma in an external magnetic field. Here function u is a generalized potential of the electric field, the constants λ, λ ^′ , α characterize ion gyrofrequency, Langmuir frequency and Debye radius.

The third one is a mathematical model of wave propagation in shallow water. It is based on a modified Boussinesq equation. Let Q C R n be a domain with boundary dQ of class C “ , T G R ₊ . In a cylinder C = Q x (0, T), consider the modified Boussinesq equation

(A — A)u _tt — a ² Au + u ³ = 0,   (x, t) G Q x (0, T)                 (6)

with the Cauchy–Dirichlet conditions

u(x,t) = 0, (x,t) G dQ x (0,T)                           (7)

u(x, 0) = u ₀ (x), u t (x, 0) = u 1 (x), x G Q,                       (8)

where λ, α ∈ R . Other modifications of the Boussinesq equation with nonlinearities of the form A | u | p u have also become widespread [4,5,17]. The equation has many applications in various fields of natural science. For example, it models the propagation of waves in shallow water, taking into account capillary effects. In this case, the function u = u(x, t) determines the height of the wave.

The article, in addition to the Introduction, Conclusion and References, includes seven paragraphs. The first section contains the main results of the theory of p-bounded operators obtained by G.A. Sviridyuk, necessary for further presentation. The second section presents results on the solvability of the Cauchy problem for semilinear first-order equations obtained by V.O. Kazak. The third paragraph contains some results of the study of Oskolkov generalized mathematical model. The fourth section presents the main statements of the theory of relatively polynomially bounded operator pencils, obtained by A.A. Zamyshlyaeva. The fifth section contains results on the solvability of the Cauchy problem for high-order semilinear Sobolev-type equations obtained in the authors’ papers. The sixth paragraph is devoted to the study of the semilinear model of ion-acoustic waves. The seventh paragraph contains the results of the study of the mathematical model of shallow water wave propagation.

1.    Relatively p-Bounded Operators

A detailed exposition of the theory of relatively p-bounded operators can be found in [30]. Let U , F be Banach spaces and operators L, M G L ( U ; F).

Definition 1. The set pL(M) = {p g C : (pL - M)-1 G L(F; U)} is called a resolvent set of the operator M with respect to operator L (in short, L-resolvent set of the operator M). The set C\pL(M) = aL(M) is called a spectrum of the operator M with respect to the operator L (in short, the L-spectrum of the operator M).

Definition 2. Operator-functions

(pL - M ) ^-1 ,   R L = (pL - M ) ^-1 L,   L L = L(pL - M ) ^-1

with domain p^L (M ) are called respectively resolvent, right resolvent, left resolvent of the operator M with respect to the operator L (in short, the L-resolvent, right L-resolvent, left L-resolvent of the operator M ).

Theorem 1. [30] Let the operator L G L (U, F) , and the operator M : dom M C U ^ F be linear and closed. Then the L-resolvent, the right and the left L-resolvent are analytic in the set p ^L (M ) .

Definition 3. The operator M is said to be spectrally bounded with respect to the operator L (in short, (L, a)-bounded), if

3 a > 0 V p G C : ( | p | > a) ^ (p G p ^L (M )).

Lemma 1. [30] Let the operator M be (L, a)-bounded. Then the operators

P = — [ R L (M)dX and Q = — [ L L (M)dX 2ni J                      2ni J

Γ

Γ

are projectors. Here Г = { X G C : | X | = r > a } .

Set U ^o = ker P , F ^o = ker Q, U ¹ = im P , F ¹ = im Q. Denote by L k (M k ) the restriction of the operator L (M ) to the subspace U k , k = 0,1.

Theorem 2. [30] Let the operator M be (L, a)-bounded. Then

• (i)    L k ,M k : U k ^ F ^k ,k = 0,1;

• (ii)    Mo¹ G L (F 0 ,U ⁰ );

• (iii)    operator L ^— G L (F 1 , U ¹ ) exists;

• (iv)    operator M 1 G L ( U 1 , F ¹ ) exists.

Let ^ ₀ G ker L \{ 0 } be an eigenvector of the operator L.

Definition 4. An ordered set { ^ 1 , ^ 2 ,... } C im L is called a chain of M-adjoined vectors of an eigenvector ϕ ₀ if

L^ q+1 = M^ q , q = 0,1, 2 ...,    ^ q G ker L for q = 1, 2,...

The chain is said to be finite if there exists an M -adjoined vector ϕ p such that either ^ _p G domM or M^ p G imL. The power of the final chain is called its length. The linear span of all eigenvectors and M -adjoined vectors of the operator L is called the M -root lineal of the operator L . Under the conditions of the theorem 2, we construct the operators H = M ^-1 L _o G L (U ^o ) and S = L -- 1 M 1 G L (U 1 ). Since the operator-function (mL _o — M _o ) ^-1 is an entire function, it can therefore be expanded into a Taylor series

⁽M^L o ^{— M} o ) 1

= (_MH — I ) -¹ M o ^-1 = ( — E ^ ^k H ^k) M o ^-1 - k =0

absolutely and uniformly convergent on any compact set in C . Let’s do the same with the operator-function (^L 1 — M 1 ) ^-1 .

(^L 1 — M 1 ) ^-1 = (д ! — S ) ^-1 L ^-1 = ^ ^-1 ( I — ^ ^-1 S) ^-1 L ^-1 =

= M

(52 M^-k s k) k =0

L 1

where д G p(S) or what is the same as д G p^L (M ). Hence, for the (L, a)-bounded operator, by virtue of the last two expansions, we have

( ^                  ^

— E M k H ^k M o ^-1 ( I — Q) + E M ^-k S k^-1^ '

k =0                           k =1

Let the operator M be (L, a)-bounded.

Definition 5. The infinity point of the L -resolvent of the operator M is called

• •    a removable singular point if H = O ;

• •    a pole of order p if H ^p = O , H ^p+1 = O , p E N ;

• •    an essentially singular point if H q = O , V q E N .

Further the removable singular point will be called a pole of order zero.

Remark 1. In what follows the (L, a)-bounded operator M will be called (L,p)-bounded, if the point to is a pole of order p E { 0 } U N of its L-resolvent.

Theorem 3. [30] Let L be a Fredholm operator (that is, ind L = 0 ). Then the following statements are equivalent:

• 1.    The operator M is (L, 0)-bounded;

• 2.    Any eigenvector of the operator L does not have M -adjoined vectors.

2.    Semilinear Sobolev Type Equations of the First Order

Let the operators L, M E L (U; F), N E C k (U; F), k E N U {to} , and the operator M be ( L, p )-bounded, p E { 0 } U N . Consider the Cauchy problem

u(0) = u o

for a semilinear Sobolev type equation

Lu = Mu + N (u).

The vector-function u E C ^k (( - T,T );U) is called a solution to equation (11) if for some T E R ₊ it satisfies this equation. The solution u = u(t) of equation (11) is called a solution to problem (10), (11) if it satisfies the initial condition (10).

Example 1. Let U = F = R^ _n ) , the operators L, M , and N be defined by formulas

L =

( 0 0 )’ M = I , N ^: u ^(Д 2 )’ u =«,n).

Then the Cauchy problem with u ₀ = (0, 0), for equation (11) will have two solutions (0, 0) and (t/2, t ² /4). If instead of the operator N in this case we take the operator N : u ^ ^₂ ^, then the same problem will not have a solution at all. This simple example shows the need to narrow down the definition of a solution to equations (11).

By virtue of theorem 2 equation (11) can be reduced to an equivalent system

Hut ⁰ = u ⁰ + M ^-1 ( I - Q)N(u),                        (12)

u ¹ = Su ¹ + L—^QN (u),                            (13)

where u ¹ = Pu, u ⁰ = u — u ¹ .

Definition 6. Solution u = u(t) of problem (10), (11) is called a quasi-stationary trajectory for equation (11) passing through the point u ₀ , if Hu(t) = 0 for all t E ( — T, T).

Obviously, any stationary solution of problem (10), (11) is a quasi-stationary trajectory, however the opposite is not true. In example 1 a stationary solution is the only quasi-stationary trajectory passing through the point (0, 0), and in this sense such solution is unique. Further only quasi-stationary trajectories are considered.

To this end, introduce a set which is called the phase space of the equation (11):

M = { u E U : ( I — Q)(Mu + N (u)) = 0 } .

By virtue of theorem 2 and (12) any quasi-stationary trajectory u = u(t) lies in M, that is u(t) E M for all t E ( — T,T).

Let the point u ₀ E M, put u 0 = Pu ₀ E U ¹ . The set M at the point u ₀ is a Banach C k -manifold if there are neighborhoods O M C M and O 0 C U ¹ of the points u ₀ and u 0 , respectively, and a C k -diffeomorphism 5 : O 0 ^ O M such that 5 ^-1 is equal to the contraction of P onto O M . The set M is called a Banach C k -manifold modelled by the space U ¹ , if it is a Banach C k -manifold at each point of U ¹ .

Theorem 4. [29] Let the set M be a Banach C ^k -manifold at the point u ₀ . Then there is a unique quasi-stationary trajectory of equation (11) passing through the point u ₀ .

3.    Mathematical Model of Oskolkov

Reduce problem (1) – (3) to the Cauchy problem (10) for the semilinear Sobolev type equation (11). To do this, put U = W m⁺² П W 2 , F = W m , m E N . All functional spaces are defined on the domain Q. The operators L and M are defined by the formulas L = 1 — жЛ, M = vЛ. Obviously, L,M E L ( U ; F), and L is a Fredholm operator for all ж E R \ { 0 } .

Lemma 2. [33] For all ж, v E R \ { 0 } the operator M is (L, 0)-bounded.

Denote by {Ak} the set of eigenvalues of the homogeneous Dirichlet problem for the Laplace operator Л in the domain Q, numbered in non-increasing order, taking into account multiplicity, and by {yk} denote the set of orthonormal (in the sense of L2) of the corresponding eigenvectors. Then ker L = J {0}, if ж-1 E {Ak}, [ span{yl : ж 1 = Al}.

In case ker L = { 0 } there is an operator L ^-1 E L (F;U), and so the operator M E L ( U ; F) (L, 0)-bounded. If ker L = { 0 } take the vector у E ker L \ { 0 } , i.e.

• У = У^ a l У 1 , a l E R ,    У2 к I > 0.

$ ^{- 1} =X l                       $ ^{- 1} = A l

Since

My = v ^ A i a i y i / imL,

$-1 =Xi then the vector ϕ has no M -adjoint vectors, and the assertion follows from theorem 3. In what follows, we need regularity theorem [12]:

Lemma 3. Let f E C “ ( R ) and m > n/2 . Then F E C“(W 2m ), where the operator F : u ^ f (u).

Lemma 4. Let the function K E C “ ( R ) and let m + 2 >  n/2. Then the operator N : u ^ K (u) belongs to the class C “ (U; F) .

By virtue of lemma 3 the operator N E C“ (U), and in view of the continuity of embedding U ^ F , N E C “ (U; F).

So, reduction of problem (1) – (3) to (10), (11) is finished. By lemma 1 construct a projector

Г I , if ж E { A l } ,

P = Г I - 52 ^(,* >

I       $-1=Ai where (•, •) is an inner product in L2. The projector Q has the same form, but is defined on the space F. Fix m > n/2 — 2 and construct the set

M = Г ^U, ^if ^ ^{1 E { A} i } ,

[ {u E U : ( Mu — N(u), pl) = 0, ж ^-1 = A l } and the space

U i = f ^U, ^{if ж} ' ^{E { A} i ^} ,

[ {u E U : ( Mu — N(u), y l ) = 0, ж ¹ = A l } .

In the case of ж ¹ / { A l } the set M is obviously a smooth Banach C “ -manifold. In the case of ж ^-1 E { A l } this is yet to be proven.

Theorem 5. [33] (i) For any ж ^-1 / { A k } , v E R \ { 0 } , m > n/2 — 2 , u ₀ E U and some T E R ₊ there exists a unique solution u E C“(( — T,T ); U) of problem (1) - (3).

• (ii)    Let for ж E {Ak}, v E R \ {0}, m > n/2 — 2 set M at the point u₀be a Banach C“-manifold. Then for some T E R₊there exists a unique solution u E C“((—T,T); M) of problem (1) – (3).

4.    Relatively Polynomially Bounded Operator Pencils

Theorem 5 follows directly from theorem 4. We only note that in our case the operator H = O , and therefore, any solution to problem (1) - (3) necessarily turns out to be quasi-stationary trajectory.

The statements presented in this paragraph were obtained in the works of A.A. Zamyshlyaeva [38,45]. Let U, F be Banach spaces and operators A, B q , B 1 ,..., B _n-1 E L (U; F). By H denote the pencil formed by operators B _n-1 , ..., B 1 , B q . The sets p ^A ( B ) = { ^ E C : (^ ⁿ A — ц^{п -1} В _п-1 — ... — цВ 1 — B o ) ^-1 E L (F; U) } and o ^A ( B ) = C \ p ^A (B3 ) are called an A - resolvent set and an A - spectrum of the pencil B respectively. The operatorfunction of a complex variable R A ( B ) = (^ ⁿ A — ^ ^n-1 B _n-1 — ... — ^B 1 — B _o ) ^-1 with the domain p ^A (( ^ ) is called an A-resolvent of the pencil (^.

Definition 7. The operator pencil B^{^} is called polynomially bounded with respect to an operator A ( or polynomially A -bounded ) if

3 a G R +   V ^ G C (Ы > a) ^ (R A (B ) G L (F;U)).

Remark 2. If there exists an operator A ¹ G L (F; U) then the pencil B is A-bounded.

Lemma 5. [38] Let the operator pencil B^ be polynomially A-bounded and condition j^RA. . O, k = 0,1.....П- 2,                 (14)

γ where the circuit y = {^ G C : |^| = r > a}, be fulfilled. Then the operators

P = Л [ RA(B^"-'Ad?, Q = 2- f ^ARAIBW 2П i J '                          2ni J

γ

γ

are projectors in spaces U and F respectively.

Denote U 0 = ker P, F ⁰ = ker Q, U ¹ = im P, F ¹ = im Q. According to lemma 5 U = U 0 Ф U ¹ , F = F Ф F ¹ . By A k (B ^ ) denote restriction of operators A (B l ) onto U ^k , k = 0, 1; l = 0, 1,...,n - 1.

Theorem 6. [38] Let the operator pencil B^{^} be polynomially A-bounded and condition (14) be fulfilled. Then

• (i)    A ^k G L ( U ^k ; F ^k ), k = 0, 1;

• (ii)    B^k G L (U ^k ; F ^k ), k = 0,1, l = 0,1,...,n - 1 ;

• (iii)    operator ( A ¹ ) ^-1 G L (F ¹ ; U ¹ ) exists;

• (iv)    operator ( B ₀ ⁰ ) ^-1 G L (F ⁰ ; U ⁰ ) exists.

Using theorem 6 construct operators H ₀ = (B 0 ) ^-1 A 0 G L (U ⁰ ), H 1 = (B 0 ) ^-1 B 0 G L (U ⁰ ),..., H n-1 = (B ₀ ⁰ ) ^-1 B n_-1 G L (U ⁰ ) and S o = (A ¹ ) ^-1 B ₀ 1 G L (U 1 ), S 1 = (A ¹ ) ^-1 B ₁ ¹ G L (U 1 ), ..., S _n-1 = (A 1 ) ^-1 B n- 1 G L (U 1 ).

Definition 8. Define the family of operators { K _q ¹ , K _q ² , . . . , K _q ⁿ } as follows:

K 0 ^s = O , s = n, K 0n = I ,

K 11 = H o , K 2 = - H 1 ,...,K 1 ^s = - H s-1 ,...,K = H n-1 ,

1 n            2       1           n                s       s - 1        n

^K q = ^K q-1 ^H 0 ^{, K} q = ^K q-1 ^{- K} q-1 H 1 ^, . . . ^,K q = ^K q-1 ^{- K} q-1 H s-1 ^, . . . , K q = K^ - K n- 1 H n- 1 , q = 1, 2,....

The A -resolvent can be represented by a Laurent series

(^A - ^n-1Bn-1 - ... - MB1 - Bo)-1 = - 52 MqKqn(Bo0)-1(I - Q)+ q=0

+ 52 ^ ^-q (^ ^n-1 S n- 1 + ••• + ^S 1 + S 0 ) q A ¹ 1 Q.

q =1

Using this representation we classify the character of the infinity point of the A-resolvent of the operator pencil B.

Definition 9. The point to is called

• •    a removable singular point of an A-resolvent of the pencil B , if K ® = O , s = 1,2,...,n;

• •    a pole of order p E N of an A-resolvent of the pencil B , if 3 p such that K p = O , s = 1, 2,..., n, but K p     = O , s = 1, 2,..., n;

• •    an essential singular point of an A-resolvent of the pencil B , if K n = O for all q E N .

Further a removable singular point of an A-resolvent of the pencil B^ will be called a pole of order 0, for brevity. If the operator pencil B is polynomially A-bounded and the → point to is a pole of order p E {0} U N of an A-resolvent of the pencil B then the operator pencil B is called polynomially (A,p)-bounded.

Theorem 7.    [42] Let A, B _n-1 , ... , B ₁ , B ₀ E L ( U , F) and A be a Fredholm operator.

Then the following statements are equivalent:

• (i)    The lengths of all chains of the B^{^}-adjoined vectors of the operator A are bounded by number (p + n — 1) E {0} U N and the chain of length (p + n — 1) exists.

• (ii)    The operator pencil 13 is polynomially (A, p)-bounded.

5. Semilinear Sobolev Type Equations of Higher Order

Consider the Cauchy problem

u ^(k) (0) = u k ,   k = 0,1,...,n — 1,                          (15)

for a semilinear Sobolev type equation of higher order

Au ⁽n = B n-i u ^n^-11 + B n -^ ^-2 + ... + B o u + N (u),            (16)

where operators A, B _n-1 , B _n-2 , ..., B _o E L ( U ; F ),N E C “ (U; F), and U, F are Banach spaces.

Definition 10. If a vector-function u E C “ (( — т,т );U), т E R . satisfies equation (16) then it is called a solution of this equation . If the vector-function satisfies in addition condition (15) then it is called a solution of problem (15), (16).

Definition 11. The set P is called a phase space of (16), if

• (i)    for all (u ₀ ,u ₁ ,..., u _n-1 ) E T ^n-1 p there exists a unique solution of (15), (16);

• (ii)    a solution u = u(t) of (16) lies in P as a trajectory, i.e. u(t) E P for all t E ( — т,т ).

If ker A = { 0 } then equation (16) can be reduced to an equivalent equation

u(n) = F (u,u ,...,u(n-1)), where F(u,u,..., u(n-1)) = A-1(Bn-1u(n-1) + Bn-2u(n-2') +... + Bou + N(u)) is a mapping of class C∞ by construction. The existence of a unique solution u of (15), (16) for all (u0 , u1 , . . . , un-1 ) follows from the classical Cauchy theorem.

Let ker A — { 0 } and operator pencil B be (A, 0)-bounded, then by theorem 6 equation (16) can be reduced to an equivalent system of equations

0 — ( I — Q )( B q + N )(u ⁰ + u ¹ ),

Jn                      Jn^-1           Jn- 2                                         (17)

—u1 — A-1 Q(Bn_1-—- + Bn_2-—- + ... + Bo + N )(u0 + u1), dtn        1 v n 1 dtn-1     n 2dtn—           0 a v, where u1 — Pu, u0 — (I — P)u.

Now consider a set M — { u E U : (I — Q)(B ₀ u + N (u)) — 0 } . Let the set M be not empty, i.e. there is a point u ₀ E M. Denote u ₀ 1 — Pu E U ¹ . The set M is called a Banach C k -manifold at point u ₀ if there exist neighborhoods O C M and O ¹ C U ¹ of points u ₀ and u 0 respectively and a C k -diffeomorphism 5 : O ¹ ^ O such that 5 ¹ is a restriction of projector P on O . The set M is called a Banach C ^k -manifold modelled by the space U ¹ if it is a Banach C ^k -manifold at any point.

Let the following condition be fulfilled

( I — Q)(B ₀ + N 0 ) : U ⁰ ^ F ⁰ is a toplinear isomorfism.

According to the implicit function theorem there exist neighborhoods O ⁰ C U ⁰ and O ¹ C U ¹ of points u 0 — ( I — P )u ₀ , u 0 — Pu ₀ respectively and the operator B E C “ ( O ¹ ; O ⁰ ) such that u 0 — B(u0). Lets construct an operator 5 — I + B : O ¹ ^ M, 5(u0) — u ₀ . Then the operator 5 ^-1 together with the set O ¹ makes a map of M and is a restriction of P on 5[ O 1 ] — O C M. Thus, we proved

Lemma 6. [42] The set M — { u E U : ( I — Q)(B ₀ u + N (u)) — 0 } under condition (18) is a C “ -manifold at point u ₀ .

Lets act with the Frechet derivative 5 ⁽ⁿ⁾ i i of order n on the second equation of ( ^u 0 ,u 1 ,...,^u n-i )

system (17). Since 5(u 1 ) — u and

5 (1,1 „1 )« ¹⁽” — d W) ( ^u o ,^u i ,...,^u n-i )               dtn ^x '

we obtain equation u ⁽ⁿ⁾ — F (u, u ,..., u ^(n-1) ), where

F(u,u, ... ,u ^(n-1) ) — 5 ₍⁽n 0 ,_u 1 ,...,_u i-1 ) A ^-1 Q(B n- 1 u (n^-1^ + B n- 2 u ^(n-2) + ...

+B 0 u + N (u)) E C “ (U).

Therefore, we get

Theorem 8. [42] Let the operator pencil BB be (A, 0)-bounded, N E C “ (U; F) and condition (18) be fulfilled. Then for any (u ₀ ,u 1 ,..., u _n-1 ) E T ^n-1 M there exists a unique solution of problem (15), (16) lying in M as trajectory.

6.    Mathematical Model of Ion-Acoustic Waves in Plasma

As a model example, consider problem (4), (5). In order to reduce mathematical model (4), (5) to problem (15), (16) set

U — { u e W 2 ⁺² (Q) : u(x) — 0,x E dQ } , F — W 2 (Q).

Define operators A = A — A, B ₂ = (A ‘

A), B o =

— a-A?, B ₃ = B 1 = O . Operators

A, B3, B2, B1, Bo are G L(U; F) for all l G {0} U N. Denote the eigenvectors of the Dirichlet problem (5) for the Laplace operator by ^kmn = < sin П_Х1 sin П_X2 sin ППХ3 I where abc k, m, n G N and denote the eigenvalues by Akmn = — J("/)' + (П-)2 + ()2. The spectrum a(A) is negative, discrete, finite and tends only to —to. Since {^kmn} C C^(Q) we obtain

^ 4 A — ^ ³ B 3 — ^ 2 B 2 — ^B i — B o =

∞2

E у [(Akmn   A)^ + (Akmn A )P'    a ^ ^ ] < Vkmn, • > Vkmn, k m n=1

where <  • , • > is an inner product in L 2 (Q).

Remark 3. In the case when (i) A G a(A) the A-spectrum of pencil B a ^A (B) = { ^ Г_тп :

r,m,n G N , j = 1,

..

., 4 } , where ^ ^j _rmn are the roots of equation

^(A rmn ^{— A)}^ 4 + ^(A rmn ^{— A} ‘ )^ 2 ^{— a}       = ⁰

In the case when (ii) (A G a(A)) Л (A = A ’ ) the A-spectrum of pencil B a ^A (B ) = { ^ j _k : k G N } , where ^ j'k are the roots of equation (19) with A = A l . In the case when (iii) (A G a(A)) Л (A = A ’ ) the A-spectrum of pencil B a ^A (B) = { ^ ^j _k : k G N , k = l } .

Check condition (14). In case (i) there exists A ^-1 G L (F 1 ;U ¹ ) therefore condition (14) is fulfilled.

In case (ii)

E

²ⁿⁱ J I ™ л ^(Akmn _γ k m n =1

=— E

2ni j

^ < ^ kmn > • >  V kmn dB

^{— A)}^ 4 + ^(A kmn ^{— A}G ^ 2 ^{— a( n}n ^{) 2}

^ <  V kmn i • >  V kmn d^

^(A kmn

A> ² — a( ⁿn ) ²

= o,

γ when r = 1, therefore condition (14) is not fulfilled and this case is excluded from further considerations. In case (iii) (A G a(A)) Л (A = A’) condition (14) is fulfilled.

Lemma 7. [42] Let (i) A G a(A)) or (ii) (A G a(A)) Л (A = A ’ ) . Then the pencil B is polynomially (A, 0)-bounded.

In case (i) ker A = { 0 } that is, the operator A has no eigenvectors and, by remark 2 the pencil B is (A, 0)-bounded.

In case (ii) A G a(A) and A = A ’ construct the chain of .B-adjoined vectors of an eigenvector ^ 0 =   ^2 a kmn V kmn G ker A \ { 0 } . Since B ₃ = B 1 = O the first three

λ = λ _kmn

B^ -adjoined vectors can be taken equal to zero. On the fourth we obtain

Bo^o = Bo( E^ akmn ^kmn) a c ) E akmn^kmn G imA, λ=λkmn                          λ=λkmn since У |akmnI > °.

Therefore the eigenvector ϕ ₀ doesn’t have a B^{^} -adjoined vector of order four, the length of the chains of B^{^} -adjoined vectors of operator A is bounded by three, and the chain of length three exists.

Construct projectors. In case (i) P = I and Q = I . In case (ii)

P = I - £

λ = λ _kmn

< V kmn ^, ' >  ^ kmn ^,

and the projector Q has the same form but it is defined on space F. Construct the set

M =

{ u e U : £

λ = λ _kmn

<α

πcn

u + A(u ³ ),^ kmn

> ^ kmn ° } .

By theorem 8 we have

Theorem 9. [42] (i) Let A e ^(A) , (u o ,u ₁ ,... ,u _n-1 ) e U n . Then for some т = т (u ₀ , u ₁ ,..., u _n-1 ) > 0 there exists a unique solution u e C ⁿ (( — т, т ), U) of problem (4), (5).

(ii) Let (A e ^(A)) Л (A = A ’ ) , (u ₀ , u ₁ ,..., u _n-1 ) e T ^n-1 M and condition (18) be fulfilled. Then for some т = т (u ₀ , u ₁ ,..., u n- 1 ) > 0 there exists a unique solution u e C ^п ( ( — т,т ), M ) of problem (4), (5).

7.    Mathematical Model of Waves Propagation in Shallow Water

In some particular cases of a non-linear term in equation, one can not only answer the question of the existence and uniqueness of a solution, such as [41], but also find this solution. A detailed algorithm is described in [3], in this section we present only the main steps in finding a solution to problem (6) – (8).

For the solution, we need several function spaces. Let Q C R ⁿ be a domain with the boundary dQ of class C ” , denote Q = Q x (0,T). Define spaces L ⁴ (Q), HJ^Q) and denote B = L ⁴ (Q) П H q (Q) , D = H ¹ (Q) П coim L (where coim L = H ¹ (Q) 0 ker L).

The operator A : H ¹ (Q) ^ H ^—1 (Q) is given by formula

( ^Au,v ) = j

Ω

( V u V v)dx.

Introduce the notation of the operators

( Lu,v ) = y*(V u V v + Auv)dx, ( Mu,v ) = a ² ^V u V v)dx, ( N(u),v ) = У u ³ vdx.

Ω

Ω

Ω

In addition, define distribution spaces (functions with values in a Banach space) U ” (0,T; B) and U ” (0,T; L ² (Q)). Construct dual spaces using the Dunford-Pettis theorem: (L ” (0,T; B)) * ~ L ¹ (0,T; L4 )(Q) U H ^-1 (Q)) and (L ” (0,T; D)) * ~ L ¹ (0,T; D * ).

Let λ _k be the eigenvalues of the homogeneous Dirichlet problem (7)) for the operator A, numbered in nonincreasing order with multiplicity taken into account, and ^ _k be the corresponding eigenvectors. Moreover, the linear span span { ^ ₁ , ^ 2 ,..., ^ _m } is dense in B for m ^ to and is orthonormal (in the sense of an inner product in L ² (Q)).

Theorem 10. [3] Let A E [A 1 , +to) , (u ₀ ,u 1 ) E T P , where u ₀ E B = H(Q) П L 4 (Q) and u 1 E L 2 (Q) П coimL . Then there exists a solution u = u(x,t) of problem (6) - (8) such that u E L ” (0,T ; H 0 (Q) П L ⁴ (Q)) and u E L ” (0,T ; L 2 (Q) П coimL) .

Below we give a sketch of the proof. The solution of problem (6) – (8) will be sought in the form of the Galerkin approximation

m um(t) = 52 am (t^.(20)

k =1

Substituting the approximate solution (20) into equation (6) and multipliyng scalarly by the basis functions { ^ _k } m = ₁ , we get

(Lum, W) - a2(Aum, W) + ((um)3, W) = 0, 1 < k < m.(21)

Using the series expansions of the initial functions in terms of basis functions, we obtain the initial conditions for the system of algebraic differential equations (21)

am(0) = em, am(0) = Ym, 1 < k < m,(22)

where um = £ em^k ^ uo in B for m ^ to, and um = £ YmVk ^ Ui in L2(Q) as k=1

m ^ to . The existence of a unique local solution u ^m = u ^m (t, x), t E [0,t ^m ] was proved.

After that, a priori estimates were obtained as follows. Multiplying equation (21) by a m (t) (1 <  k <  m) and summing over k from 1 to m, we get

( Lu m , u m - a ² ( Au ^m , u m + ( (u ^m ) ³ , u m = 0.

Introduce a norm in the space D (L 2 (Q) = coim L ф ker L) | u | H 1 = ( Lu, u ) . By the Courant principle, this norm is equivalent to the norm induced by the space H ¹ (Q). Using the self-adjointness of L, A and integrating it on the segment [0,t],t <  t _m we obtain

| u m l 2 + a ² ^ u ^m ^ H i + 2 H u mHb <  C.                       (23)

The constant C does not depend on t _m and hence t _m = T .

Remark 4. Due to inequality (23) for m ^ to , the sequence of functions йл _m is bounded in the space U ” (0,T ; L ² (Q)) and u is bounded in U ” (0,T ; B ).

Since the sequence { (u ^m l ) ³ } is bounded in the space L ” (0,T; L ^4/3 (Q)), we have

(u ^m l ) ³ ^ z *-weakly in L ” (0, T ; L ^4/3 (Q)).                     (24)

Moreover, it can be shown that z = u ³ .

Now we can pass term by term to the limit in equality (21), setting m l = l. Let k be fixed and l > k , we get

• (i)    due to the density of the system of functions { ^ _k } m =1 in the space B for m ^ to , and the arbitrariness of the choice of ^ _k , we have equality for arbitrary v E B

d ²

— ( Lu,v\ + a ² (V u, V v ) + ( u ³ ,v ) = 0.                    (25)

dt ²

• (ii)    u(0) = u o ;

• (iii)    ( u(0),^ k ) = ( u i ,^ k ) , V k.

Remark 5. Under the conditions of the theorem 10 and the Rellich–Kondrashov theorem, the solution to problem (6) – (8) is unique.

Conclusion

Instead of the Cauchy condition in all mathematical models, one can consider the Showalter–Sidorov condition

P (u ^(k) (0) - u _k ) = 0, k = 0,1,...,n - 1,                      (26)

where P is a projector along the kernel of the operator at the highest derivative with respect to t. Condition (26) is a natural generalization of the Cauchy condition for Sobolev type equations.

Further directions of development are seen in the study of semilinear Sobolev type equations with additive “white noise” [8, 9, 13], nonlinear inverse problems [43], as well as the study of multipoint initial-final problems [11, 23, 40].

Acknowledgments. The authors express their sincere gratitude to Professor G.A. Sviridyuk for posing the problem, interest in the work, and provided opportunities.

This work was supported by a grant from Ministry of Science and Higher Education of the Russian Federation No. FENU-2020-0022 (2020072GZ).