Investigation of the uniqueness solution of the Showalter-Sidorov problem for the mathematical Hoff model. Phase space morphology

Extensive class of models of mathematical physics is based on semilinear non-classical equations or systems of partial derivative equations unsolved with respect to the time derivative

Lu = Mu + N (u),                            (1)

which are commonly called Sobolev type equations. Equations of this class and initial problems for them cannot be investigated by classical methods due to the possible degeneracy of the operator at the higher derivative, so their investigation requires the development of new and modifications of already known methods of investigation [1–5]. Let us consider a mathematical model based on the Sobolev type equation. Let Q C R ⁿ be a bounded domain with boundary of class C ^∞ . Let us consider the Hoff model [6]

(^ + A)u _t = au + ви³, x G Q, t G (0,T),                      (2)

with the Dirichlet condition

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

The Hoff model describes the deformation dynamics of an I-beam. The unknown function u = u(x,t), x G Q, t G (0,T), has the physical meaning of the deflection of the beam from the equilibrium position. The parameter ^ G R characterises the longitudinal load on the beam and the parameters а, в G R characterise the material properties of the beam. Studies for equation (2) on graphs are presented in [7, 8], on manifolds in [9].

One of the first to study the initial boundary value problem for equation (2) was N.A. Sidorov [10]. In this work, the principal insolvability of the Cauchy problem (u(x, 0) — u ₀ (x) = 0) at an arbitrary initial value in the case of degeneracy of equation (2) was noted. Consideration of the Showalter–Sidorov condition [11]

L (u(x, 0) — u o (x)) = 0                                 (4)

allows one to avoid difficulties in solving the Cauchy problem, but non-uniqueness of the solution of problems (2) – (4) is possible [12].

The questions of non-uniqueness of solutions of equations and systems of equations reduced to semilinear equations of the form (1) with the Showalter–Sidorov condition (4) and the connection of non-uniqueness of the solution with the existence of Whitney assemblies and folds in the phase space of equation (1) were devoted to the following works: T.A. Bokareva and G.A. Sviridyuk for the model of nerve impulse propagation in the membrane and for the model of autocatalytic reaction with diffusion showed the existence of 2-Whitney assemblies and 1-Whitney folds, respectively [13], A.F. Gilmutdinova for the Plotnikov mathematical model revealed the conditions for the existence of non-uniqueness of the solution [14].

To take advantage of earlier results obtained in [12–14] for semilinear abstract Sobolev type equations, let us reduce the problem (2), (3) to equation (1). For this purpose, we О assume U =W21 (Q), F = W2-1(Q). The operators L, M are defined by the formulas

(Lu, v) = (ци + Au, v) Vu, v G U, (Mu, v) = (au, v) Vu, v G U, where (•, •) is the scalar product in L2(Q). We note that the operator L G L(U, F), M G L(U, F). The spectrum a(L) of the operator L is real, discrete, finite-edge, and condensed to —ro. Now let us construct the operator N

( N(u),v ) = (j 3u3,v^ V u,v G U.

By virtue of Gelder’s inequality

|(N (u),v)|<|e ||u||i4(n) |М|Мя), operator N : L4 (Q) G (L4(Q))* = L4/3 (Q). When n < 4 the embedding W21 (Q) ч L4 (Q) is dense and continuous, and hence the embedding L4/3 (Q) ч W2-1(Q). Thus, the action of the operator N : U ч F. By virtue of the given operators, the Showalter-Sidorov problem (4) for equation (2) will take the form

(ц + A) (u(x, 0) — u ₀ (x)) = 0, x G Q.                          (5)

G.A. Sviridyuk and his successors developed a method [1, 15], based on the study of the morphology of the set of admissible initial values B

B = { u G U : ( ( I — Q) (Mu + N (u)) ) = 0 } ,                     (6)

understood as the phase space of equation (1). In [16] the projector

m

Q = I -'^(• ,^ i ) ^ i k=1

was constructed and it is shown that the phase space (6) for model (2), (3) take the form

B = < u G U : J (a + eu^u^ l dx = 0 / ,                    (7)

Ω where ϕl are the eigenfunctions of the homogeneous Dirichlet problem of the operator L.

In the case when the phase space of the model (2), (3) has singularities, the nonuniqueness of the solution of the Showalter–Sidorov problem arises, and the simplicity of the phase space of the model (2), (3) implies the singularity of the solution. In [16] it was shown that the phase space of equation (2) is a simple Banach C ^∞ -manifold in case ав >  0 , in case ав <  0 the phase space of equation (2) may contain a 2-Whitney assembly [17]. In [18], conditions on the parameters α, β , were found under which the phase space of equation (2) has singularities in case dimker(^ + A) = 1 .

As the paper, in addition to theoretical studies, also contains the results of numerical experiments, it is necessary to mention the Galerkin method, which is the basis for the computational experiments. Obtaining an analytical solution of Sobolev type equations (1) is not always possible, so the construction of algorithms for numerical methods is in demand. For degenerate semilinear equations, the Galerkin method is the most appropriate one, since it allows to incorporate the degeneracy of the equation for some parameters. Using the Galerkin method, approximate solutions of the problem are constructed, whose coefficients satisfy the system of algebro-differential equations with appropriate initial conditions [19–21].

The purpose of this study is to investigate the model (2), (3) and to identify the conditions imposed on the parameters α, β , under which the phase space has singularities and there are several solutions to the Showalter–Sidorov problem (2), (3), (5) at dimker(^ + A) = 1 and dimker(^ + A) = 2 .

• 1.    Features of Phase Space

Let us study the morphology of the phase space of the model (2), (3) in the case Q C R ² . Let us find the conditions imposed on the parameters of the equation а, в , in which the phase space has singularities and it follows that the solution to the Showalter– Sidorov problem is non-unique (2), (3), (5).

Let us consider the homogeneous Dirichlet problem for the Laplace operator ( — A) . Let us denote by { A _k 1 , _k 2 } the family of eigenvalues of the problem under consideration. In case

• a)    if ^ = A _{k 1} , _k 2 and k 1 = k ₂ , then dimker(^ + A) = 1 and the considered eigenvalue A _{k 1} , _k 2 corresponds to one eigenfunction ^ _{k 1} , _k 2 (x,y ) , then u can be represented as u = S i ^ k i , k 2 + u ^± , u ^± G U ^± = { u G L 4 (Q) : { u, ^ k i , _{k 2} ) = 0 } ;

• b)    if ^ = A _k 1 , _k 2 and k 1 = k ₂ , then dimker(^ + A) = 2 and the considered eigenvalue A _k 1 , _k 2 corresponds to two eigenfunction ^ _k 1 , _k 2 (x,y ) and ^ _k 2 , _k 1 (x,y ) , then u can be

represented as u = S i P k i , _k 2 + s ₂ ^ _k 2 , k i + u ¹ , u ¹ £ U ¹ = { u £ L 4 (^) : ( u,k k i , k 2 ) = 0, { u,P k 2 , k i ) = 0 } .

Example 1. As an example, consider Q = (0,1 1 ) x (0,1 2 ) , then the Dirichlet condition (3) takes the following form:

u(0, y, t) = u(1 1 ,y, t) = 0, u(x, 0, t) = u(x, 1 2 , t) = 0.

At the same time

(  ' П~ . nkix . nk2y ,      NkA2  (AA2

^ k i , k 2 ^(x,y ) = lljji sin —j— sin —j— ,Xk i ,k 2 =1 — ] +1 -^A ^,k = 1, 2,...

In case 1 1 = 1 2 = n,

• a)    if ^ = Л ₁ , ₁ = 2 (k 1 = k ₂ = 1 ), then the considered eigenvalue Л ₁ , ₁ corresponds to one eigenfunction p 1 1 (x,y ) = — sin x sin y and dimker(^ + A) = 1 ;

,            n

• b)    if ^ = Л 1, ₂ = 5 (k 1 = k ₂ ), then the considered eigenvalue Л 1, ₂ corresponds to two eigenfunctions p 1, ₂ (x, y) = — sin x sin2y , p ₂ , 1 (x, y) = — sin2x sin y and dimker(^ + A) =

  π

  
  

  π

  
  

In case a) u = s 1 p _{k i} , _k 2 + u ¹ , then set B C “ -diffeomorphic to the set

B i = {(s^u ¹ ) £ R x L 4 (Q) : s

^: lll k k i . k 2 K, ₍₄ ⁺ 3s 2 J f^ k i , k ₃ ^{u ±} dxdy ⁺

⁺ s i (³ ff V k i , k 2 (u ^{1 ) 2 dxd}y ^{+ ав} Ω

Ω

1) + ff P k i , k 2 (u ¹ ) ³ dxdy = 0^ .

In case b) u = s 1 p _{k i} , _k 2 + s ₂ p _{k 2} , _{k i} + u ¹ , then set B C “ -diffeomorphic to the set

B 2 = {(s i ,S 2 ,u ^± ) £ R ² x L 4 (Q) :

в П P k 2 , k i (u ¹ ) ³ dxdy + в П P k i , k 2 (u ¹ ) ³ dxdy + as i + as 2 +

ΩΩ

^{+ 3}^ 1//^k _i , k ₂ ⁽u ^{1 ) 2 dxd}y ^{+ 3e}s 2 J f^, k 2 P k 2 , ki ⁽u ^{1 ) 2 dx}dy ⁺

ΩΩ

^{+ 3}^ 1/ I kA, k 2 P k 2 , ki ⁽u ^{1 ) 2 dxd}y ^^J f plk k _i ⁽u ^{1 ) 2 d}xdy ⁺

ΩΩ

^{+ 3e}s1 / f P ki, k 2 u ¹ dxdy ^{+ 3e}s2 / f P k i , k 2 P k 2 , k i u ^{1 dxd}y +

ΩΩ

⁺³№/№2, k i u ^{1 d}xdy ⁺ Wslff Pk^ k 2 Pk2 , k i u ^{1 dxd}y +

ΩΩ

^{+ 3e}s i s 2 f f k >³_kl, k 2 P k 2 , k i ^dxdy ^{+ 3 e} s 1 s 2 / fP^, k 2 P k 2 , k i ^dxdy ⁺

Ω

^{+ 3e}s 1 s 2 / f P k i , k 2 Pk 2 , k i ^dxdy ^{+ 3e}s 1 s

Ω ^es 1 ^{| p} k i , k 2 II L 4 (Q)

^{+ в}^2іf Pki, k 2 Pk 2 , k i ^dxdy + Ω

Ω

'2/ fPb, k2 P^, k i ^dxdy ⁺ Ω

^{+ e}s 2 ^| P k 2 , k i II L 4 (Q) ^{+ e}s 1 ff P k i, k 2 P k 2 , k i ^dxdy = ⁰ .

Ω

The sets B 1 and B₂ describe the phase space of the model (2), (3) in the case of k 1 = k ₂ and k 1 = k ₂ , respectively. Let us give a definition of the phase space containing the k -Whitney assembly.

Definition 1. [14] Let V be a Banach space, function G G C “ ( R x V; R ) . The equation G(s, v) = 0 defines a k-Whitney assemblys over the open set V С V if there exist functions g ₀ , g 1 ,..., g _k G C “ (V ; R ) such that this equation is equivalent to the equation

0 = g ₀ (v) + g 1 (v)s + ... + g k (v')s^k + s k +1 V v g V ’ .

Let us formulate a theorem about the structure of the phase space of the model (2), (3).

• Theorem 1.

• (i)    Let ав > 0, then the phase space of the model (2), (3) is a simple Banach C “ - manifold modeled by a subspace complementary to ker(^ + A) .

• (ii)    Let ав <  0 and k 1 = k ₂ , then the set B 1 forms a Whitney 2-assembly.

• (iii)    Let ав <  0 and k 1 = k ₂ , then the set B 2 forms a k-Whitney assembly.

Proof. The proof of item 1 is given in the work [16], the proof of item 2 is given in the work [17]. The validity of item 3 follows from the construction of the set B 2 (9) and Definition 1. The set B 2 contains the k -Whitney assembly, k = 2... 8 . The degree of assembly depends on the model parameters and the type of domain Q .

□

The equation defining the set B1 is a cubic equation of general form asl + bs1 + csi + d — 0.

Let’s define

Q i (u)

R i (u)

/ 3ac — b ² \ ³    1 / 2b ³      bc

= \ 9a ² J ⁺ 4 ^270 3 —

= s lhk l , k 2 ||L 4 (Q) + ²s i У^ v k i , k 2 u ^ ^dxdy +

3a ^{2 +}

d

a

,

Ω

У^ ^ k i , k 2 ^{(u ± ) 2} dxdy,

Ω

where a = ll^ki, k2 |Ц4 (Q) , b = 3ff^, k2u' dxdy,

Ω

^c = ³JI k^, k ₂ ^{(u ± ) 2} dxdy + ^ав-1,d = /f Рь, k 2 ^{(u ± ) 3} dxdy,

ΩΩ and consider the following sets

(U 1 ) o = { u G U ^± : R i (u) = 0 } ,

(U 1 ) + = { u G U ^± : Q 1 (u) >  0 } , (U 1 ) - = { u G U ^± : Q 1 (u) <  0 } .

• Theorem 2.    Let ав <  0 . Then

• (i)    for any u G (U₁) ^± П (U 1 ) + there is one solution to the equation (10);

• (ii)    for any u G (U₁) ^± П (U 1 ) + П (U 1 ) q there are two solutions equations (10);

• (iii)    for any u G (U₁) ^± П (U 1 ) - there are three solutions to the equation (10).

Proof. The theorem is valid due to the Cardano formulas (11), (12) for the equation (10).

□

Theorem 3. Let ав <  0 and ц = X _{k i} , _k 2 ,k 1 = k 2 . Then

• (i)    for any u ₀ E (U_i) ¹ П (U 1 ) - there are three solutions to the problem (2), (3), (5);

• (ii)    for any u ₀ E (U_i) ¹ П (U_i) + П (U_x) o there are one or two solutions to the problem (2), (3), (5);

• (iii)    for any u ₀ E (U 1 ) ¹ П (U 1 ) + there is only one solution to the problem (2), (3), (5).

Proof. (i) Let’s take point u ₀ E (U 1 ) ¹ П (U 1 ) - . According to Theorem 2, there are three solutions s ¹ ₁ , s ² ₁ , s ³ ₁ to the equation (10), which means that the point u ₀ serves as the image of three points u 0 = s^ k 1 , k ₂ +U ¹ E B i , u 2 = s2^ k _i , k ₂ +U ¹ E B i , u 3 = s 3 ^ k _{i 5} k ₂ +U ¹ E B i . According to the Theorem on the existence of a solution to the Showalter–Sidorov problem [22] the problem (2), (3), (5) should have three different solutions.

• (ii)    Let’s take point u ₀ E (U_i) ¹ П (U_i) ¹ ₊ П (U_i) ₀ ¹ . According to Theorem 2, there are two s ¹ ₁ , s ² ₁ or one s ₁ solutions to the equation (10), which means that the point u ₀ serves as the image of three points u ₀ = s 1 ^ _{k i} , _k 2 + u ¹ E B 1 or for two points u 0 = s{^ _k 1 , k ₂ + u ^{1 E} B i , u 2 = s 2 ^ k i , k 2 + u ¹ E B i .

• (iii)    Let’s take point u ₀ E U ¹ П U + . According to Theorem 2, there is only one solution s ₁ to the equation (10), which means that the point u ₀ serves as the image of only one point u ₀ = s 1 ^ _{k i} , _k 2 + u ¹ E B_i. According to the Theorem on the existence of a solution to the Showalter–Sidorov problem the problem (2), (3), (5) should have only one solution.

□

Remark 1. In the case of Q C R , similar conditions were obtained in the work [18].

Let’s move on to studying case b), in which ц = A _{k i} , _k 2 ( k 1 = k ₂ ) и dimker(^ + △ ) = 2.

To do this, we construct the set (9) in an equivalent form:

B 2 = {(S 1 ,S 2 ,U ¹ ) E R ² x L 4 (Q) :

^as 1 + ^es 1 \ ^ k 1 , k 2 ^ L 4 (Q) +

+ ^es2 /f V k 1 , k 2 ^2, k 1 dxdy + ^в П V k 1 , k 2 ⁽u ^{1 ) 3} dxdy + ΩΩ

+ 3^es 1 s 2 f f ^ kl, k 2 V k 2 , k i dxdy + 3^es 1 s 2 f f ^ k 1 , k 2 <2 , k 1 dxdy +

ΩΩ

+ ³^ 1 / f ^ ki, k 2 u ^{1 d}xdy + ^3es 1 / f ^ k i , k 2 ⁽u ^{1 ) 2 d}xdy +

ΩΩ

+ ^s lf /^ k i , k 2 ^k₂, k! u ^{1 d}xdy ^{+ 3e}s 2 / f k^, k 2 V k 2 , k i ⁽u ^{1 ) 2 d}xdy = ⁰,

Q              ^,                       Q                                          (13)

^as 2 + ^es 3 \ ^ k 2 , k i \ l ₄ (Q) +

+ ^es ³ ff ^ k _i , k ₂ ^ k 2 , k i dxdy + ^в ff P k 2 , k i ⁽u ^{1 ) 3} dxdy +

ΩΩ

+ ^3es 1 s2 / f ^ k i , k 2 ^ k 2 , k i dxdy + ^3es 1 s 2 / f P k i , k 2 ^ k 2 , k i dxdy + ΩΩ

+ ^3es 1 / f ^ k _i , k₂ ^ k 2 , k i u ^{1 d}xdy + ^3es 1 ff ^ k i , k 2 V k 2 , k i ⁽u ^{1 ) 2 d}xdy + ΩΩ

+ ^{3 e} s 2 / f $2, k i u¹ Ω

dxdy + 3es 2 ff ^ k₂, Q ^,

_{k i} (u ¹ ) ² dxdy = 01

Approach 1. In the general case, identifying the number of solutions to the problem (2), (3), (5) causes difficulties. It is impossible to apply one general method. Let us consider one of the cases in which it is possible to identify the number of solutions to the problem being studied. Suppose there are pairs а1,в1 G R and а2,в2 G R such that upon substitution v1 = a1s1 + e1s2, v2 = a2s1 + e2s2, in the equations of the system (13), defining the set B2, we can obtain a system of equations of the following form:

J a i v 3 + b i v 2 + c i v i + d i = 0, [ a2 V 3 + b 2 v 2 + C 2 V 2 + d 2 = 0.

Let us define for the first equation of the system (14) the Cardano formulas i^ f3ai^

Q ^{2 (U)}   \   9a ²    J ⁺ 4 \27a ³

R 2 ^(u) = ^v 2 \\ ¥ k i , k 2 Ht (Q) +²v i У^<

b i c i , d i 3a i

a 1

,

^h, k 2 ^{U ± d}xdy +

Ω

У^ V k i , k 2 (^ ^{) 2} dxdy,

Ω

for the second equation of the system (14) of the Cardano formulas

(^{3 a} 2 c 2 — b ² V 1 f ^2b 3

^Q 2 ^IU'   <9.,²    ) +4^21

R l (^u) = ^v 2 \ V k- ₂ . k i |Ц(П) + ^2v 2 JI ■

^b 2 c 2 . d 2 3a 2

^ k 2 , k 1 ^{u 1}

2 _, a 2 dxdy ₊ Ц

v l ,, k i (^{u ± ) 2} dxdy

ΩΩ

and introduce the following sets

(U 2 ) + = { u G U ¹ : Q 2 (u) > 0, Q 2 (u) > 0 } ,

⁽U 2 ⁾ o i = ^{{ u G} U ^{1 :} R ^{i (u)} = ^{0 }} ,

⁽U 2 ) o 2 = ^{{ u G} U ^{1 : R} 2 (^u) = ^{0 }} ,

(U₂) - = { u G U ¹ : Q 2 (u) < 0, Q 2 (u) < 0 } .

Theorem 4. Let ав <  0 and ^ = A _{k 1} , _k 2 , k _i = k ₂ and 3 a _i , e _i , a ₂ , в 2 , such that (14 ). Then (i) for any u _o G (U 2 ) ¹ П (U₂) - there are nine solutions to the problem (2), (3), (5);

• (ii)    for any u _o G (U 2 ) ¹ П (U₂) 0i U (U₂) oi exists from two up to eight solutions to the problem (2), (3), (5);

• (iii)    for any u _o G (U₂) ¹ П (U₂) + there is a unique solution to the problem (2), (3), (5).

Proof. (i) Let’s take a point uo G (U2)1 П (U2)-. According to Theorem 2, there are three solutions v11 , v12, v13 of the first equation from the system (14) and three solutions v21 , v22, v23 of the second equation from system (14), which means that point u0 serves as the image of three points U1 = V1Vki к; + V1Vk; + + u1 G Bl, U2 = V2Vk, k; + V2Vk; + + u1 G Bl, o i i, 2        2     2, i                        , o i i, 2        2     2, i                        , u3 = v3vk1, k2 + v3Vk2, k1 + u1. But when reversely replaced by s1 and s2, the point uo should serve as the image of nine points. According to the Theorem on the existence of a solution to the Showalter–Sidorov problem [22], the problem (2), (3), (5) should have nine different solutions.

• (ii)    Let’s take the point u _o G (U 2 ) ¹ П (U₂) 01 ^{U ( U}2 )o i - According to Theorem 2, there are from two to eight (s 1 , s ₂ ) solutions to the system of equations (14), which means that the point u 0 serves as the image of two to eight points belonging to B 2 .

• (iii)    Let’s take the point u ₀ E U ¹ П U + . According to Theorem 2, there is one solution v ₁ and v ₂ to the equation from the system (14), which means that the point u ₀ serves as the image of only one point u ₀ = v 1 ^ _k 1 , _k 2 + v ₂ ^ _k 2 , _k 1 + u ¹ E B 2 or after reverse substitution u ₀ = s ₁ ^ _k 1 , k ₂ + s ₂ ^ _k 2 , _k 1 + u ¹ E B 2 . According to the Theorem on the existence of a solution to the Showalter–Sidorov problem, the problem (2), (3), (5) should have only one solution.

• 2. Numerical Experiment