AN ILL-POSED BOUNDARY VALUE PROBLEM FOR A MIXED TYPE SECOND-ORDER DIFFERENTIAL EQUATION WITH TWO DEGENERATE LINES Текст научной статьи по специальности «Математика»

Математические заметки СВФУ Январь—март, 2023. Том 30, № 1

UDC 517.946

AN ILL-POSED BOUNDARY VALUE PROBLEM FOR A MIXED TYPE SECOND-ORDER DIFFERENTIAL EQUATION

WITH TWO DEGENERATE LINES K. S. Fayazov and Y. K. Khudayberganov

Abstract: This work is devoted to the study of ill-posed boundary value problem for a second-order mixed type differential equation with two degenerate lines. Boundary value problems for mixed type equations are applicable in various fields of the natural sciences: in problems of laser physics, in plasma modelling, and in mathematical biology. In this paper, based on the idea of A. N. Tikhonov, the conditional correctness of the problem, namely, uniqueness and conditional stability theorems are proved, as well as approximate solutions that are stable on the set of correctness are constructed. In obtaining an a priori estimate for the solution to the equation, we used the logarithmic convexity method and results for the spectral problem considered by S. G. Pyatkov. The regularization parameter is determined by the minimum value estimate for the norm of the difference between exact and approximate solutions.

DOI: 10.25587/SVFU.2023.31.16.004 Keywords: mixed type equation with two degenerate lines, ill-posed problem, a priori estimate, conditional stability, uniqueness of solution, set of correctness, approximate solution.

1. Introduction

The theory of boundary value problems for mixed type equations is one of the most important field of the theory of partial differential equations of mathematical physics.

As a rule, boundary value problems for mixed type equations are well-posed problems and their study began with the works of Tricomi [1] and Gellerstedt. They proposed to study boundary value problems for a mixed type equation with one degenerate line.

F. I. Frankl [2] investigated important applications of this problem. In recent years, O. S. Ryzhov, L. D. Pilia and V. P. Fedorov, E. G. Shifrin, G. G. Cherny, A. G. Kuzmin [3] studied different problems for the mixed type equations.

We should note that the works of F. I. Frankl, A. V. Bitsadze, and K. I. Babenko laid a foundation for the modern theory of mixed type equations, with the research of new boundary problems for the mixed type equations. Afterwards, these types of boundary value problems have been studied by many authors, in particular by V. F. Volkodavov, V. N. Vragov [4], V. I. Zhegalov, T. D. Juraev, T. Sh. Kalmenov,

© 2023 K. S. Fayazov and Y. K. Khudayberganov

A. I. Kozhanov, Y. M. Krikunov, O. A. Ladyzhenskaya, M. E. Lerper, V. P. Mikhai-lov, E. I. Moiseev, A. M. Nakhushev [5], S. M. Ponomarev, S. P. Pulkin, K. B. Sabi-tov, M. S. Salakhitdinov, M. M. Smirnov, A. P. Soldatov, L. I. Chibrikova, R. S. Khai-rullin, B. N. Burmistrov, S. Agmon, L. Nirenberg, M. N. Protter, C. S. Morawetz, P. Germain, R. Bader, P. O. Lax, R. P. Phillips, M. Schneider, G. D. Karatoprakliev, N. I. Polivanov, G. D. Dachev and others.

Boundary value problems for the mixed type equations with two degenerate lines can be found in the works of M. M. Zainulabidov, V. F. Volkodavov, V. V. Azovsky, O. I. Marichev, A. M. Ezhov, N. I. Popivanov, T. B. Lomonosov, Khe Kan Cher, S. I. Makarov, S. S. Ismukhamedov, J. Oramov, M. S. Salakhitdinov, K. B. Sabitov [6], B. Islomov and other authors.

Well-posed and ill-posed boundary value problems for parabolic type equations were studied by many authors, including E. M. Landis, S. P. Shishatsky, and problems of elliptic type equations were investigated by M. M. Lavrent'ev [7], Landis, John, Hermander and others. We should cite the works of S. G. Krein [8], H. A. Le-vine [9], and others, where boundary value problems for abstract differential-operator equations were studied. A characteristic feature of these works is determined with the fact that the proof of the uniqueness of a solution and obtaining estimates characterizing the conditional stability of these problems were provided in these works. Correct boundary value problems for various non-classical equations were investigated in the works of A. V. Bitsadze, S. A. Tersenov, V. N. Vragov, A. M. Nakhushev [5] and other authors. The problems for these types of equations were the subject of researches conducted by N. Kislov, S. G. Pyatkov [10,11], A. I. Kozhanov [12], K. B. Sabitov [6], A. A. Gimaltdinova [13] and others. Ill-posed boundary value problems were studied by a number of accomplished authors, including A. L. Bukh-geim [14], V. Isakov, M. Klibanov, K. S. Fayazov. The works of K. S. Fayazov [15], K. S. Fayazov and I. O. Khajiev [16,17], I. O. Khajiev [18], K. S. Fayazov and Y. K. Khudayberganov [19], were dedicated to the construction of approximate solutions for non-classical equations.

This work is dedicated to the study of an ill-posed boundary-value problem for a second-order mixed hyperbolic-elliptic type partial differential equation with two degenerate lines.

Consider the equation

in the domain Q = Q0 x Q, where Q0 = {(x,y) : (-1; 1)2, x = 0, y = 0}, Q = {0 < t < T, T < ro}.

Statement of the problem. Find a solution of equation (1) in the domain Q so that the initial

utt(x, y, t) + sign(x)uxx(x, y, t) + sign(y)uyy (x, y, t) = 0

(1)

boundary

and gluing

u(x,y,t)\x=±1 = 0, (y, t) G [-1; 1] x Q, u(x, y, t)\y=±i = 0, (x, t) G [-1; 1] x Q,

(3)

dlu(x, y, t)

dxl dlu(x, y, t)

dlu(x, y, t)

dyl

x=-0

y=-o

dxl dlu(x, y, t)

dyl

x=+0

y=+0

t) G [—1; 1] x Q,

(x,t) G [—1; 1] x Q,

(4)

conditions are satisfied, where i = 0,1 and tpi(x,y) are given sufficiently smooth functions and satisfied wherein tpi(x,y)|go0 = 0.

In this paper, problem (1)-(4) is investigated for conditional correctness, namely, uniqueness and conditional stability theorems are proved. Moreover, approximate solutions, which are stable on the correctness set, are constructed.

2. Spectral problem

Find such values of A for which the following problem has a nontrivial solution:

sign(x)^xx (x, y) + sign(y)tfyy (x,y)+ A$(x, y) = 0, (x,y) G Mo, (5) tf(x,y)|x=-i,x=+i = 0, y G [-1; 1], $(x,y)|y=-i,y=+i = 0, x G [-1; 1],

dl,d(x, y)

dx dl,d(x, y)

dl,d(x, y)

x=-0

dx1

y g [-1;1],

x=+0

(6)

dyl

dl,d(x, y)

y=-0

dyl

x G [-1; 1], ¿ = 0,1.

y=+0

Using the methods of S. G. Pyatkov [11], we can prove that problem (5), (6)

has a non-decreasing sequence {A$}^^ {-A^}k°,=1, {X^}k°i=1> ki=i

(2h

k(3)\ ~

(4h ~

of eigenvalues and the corresponding eigenfunctions j

eigenvalues

\(1) 2 , 2 x(2) 2 2

Ak j = Mk + Ak i = Mk -

1,4. The

A(3) = „2 + o2 A (4) Ak, i = -„k + °i , Ak ,i

22 -„k - O ,

thus correspond to the eigenfunctions

41) (x, y) = (x) x (y), êkï (x, y) = XkK' (x) x Yf> (y)

(1)

(1)

(2)

(1)

(2)

^k3) (x, y) = X^> (x) x Y^>(y), Ci(x,y)= X^'(x) x Yr (y), k,l G N

(2)

(1)

(4)

(2)

(2)

where

Xk1) (x)

f sin „k(x - 1)/cos „k, 0 < x < 1, \ sh „k (x + 1)/ch „k, -1 < x < 0, sin oi (y - 1)/ cos o sh oi (y + 1)/ch oi,

Y(i)( ) = isinoi(y -1)/cosoi, 0 < y < 11,

1 (y) I sh oi (y + 1)/ch oi, -1 < y < 0,

k G N,

l G N,

(( 2)(x) f sh m k(x - 1)/ch ^k, 0 < x < 1, k \ sin Mk(x + 1)/ cos flk, —1 < x < 0,

(2) , , ( sh ri(y — 1)/ch ri, 0 < y < 1,

^(2)(y)= • 1 ( 1 1 < "< 0 l G N,

^ sinri(y + 1)/ cosri, —1 < y < 0,

¡j>k, r2 are eigenvalues corresponding to eigenfunctions X^, y/i) (i = 1,2), respectively.

In both cases, , t, k, l G N, are the positive roots of the transcendental equation tan a = — tanh a. Let ||u||2 = (u, u) where the inner product is

i i

(u,u) = //uudxdy.

Moreover,

(sign(x) (x, y),'dfj{x, y)) = 0, p ^ q, p,q= 1,4, Vk,l,i,j,

m = 1, 4,

(sign(x) sign(y)tfk™) (x,y),tij) (x,y)) = ^ k = ! A l = j (sign(x) sign(y)-$k™) (x,y),tf(™)(x,y)) = |

«)(*.»),<?W)) = {"1, k = I Al = j, m = «

where k, l, i, j G N.

Then, according to S. G. Pyatkov [10],

l|u(x,y,t)||0 = {Ksign(x) sign(y)u(x, y, t), ti^)1 (x, y)) |

k,i=1

+ Ksign(x) sign(y)u(x, y, t), ^k2)) (x, y))|2 + |(sign(x) sign(y)u(x, y, t), ti^ (x, y)) |2

|2,

+ |(sign(x)sign(y)u(x,y,t),tik))(x,y))| }. (7)

According to the results of S. G. Pyatkov [11], the eigenfunctions of problem (5), (6) form a Riesz basis in H0 and the norm in the space L2(—1,1)2, which is defined by equality (5), (6), is equivalent to the original one.

-1 -1

2

3. A priori estimate

A generalized solution to problem (l)-(4) is such a function u(x,y,t) G C(Q; L2(—1, 1)2) that for any arbitrary function V(x,y,t) G W22(Q), V(x,y,T) = 0, Vt(x,y,T) = 0, V(—1,y,t) = 0, V(+1,y,t) = 0, V(x, —1,t) = 0, V(x, +1,t) = 0,

satisfies the following integral identity:

T 1 1

u(x, y, t)(sign(x) sign(y)Vtt + sign(y)Vxx + sign(x)Vw) dxdydt

0 -1 -1

1 1

= J J sign(x) sign(y)V(x,y, 0)^1(x,y) dxdy

-1 -1

1 1

— J J sign(x) sign(y)Vt(x, y, 0)<^(x,y) dxdy. (8)

-1 -1

There the following lemma occurs:

Lemma 1 (see [7, p. 825, 826]). Let v(t) be a solution to the equation

v ''(t) — Av(t) = 0

and satisfy the conditions v(0) = p1 and v'(0) = p2. Then the following inequality is valid:

v2(t) < e2t(T_t)(¡/2(0) + (v2(T) + |a|)T - |a|, i G Q,

where A is a certain constant and a = ^(Ai/2(0) — v2(0)). By M we denote

M = {u(x, y, t) : ||u(x, y,T)||0 < m, m< ro}.

Theorem 1. Let u(x, y,t) be a solution of the equation

utt(x, y, t) + sign(x)ura(x, y, t) + sign(y)uyy (x, y, t) = 0,

and satisfy conditions (2)-(4). Then, for a solution to this equation for t G Q, we obtain the inequality

\\u(x, y, t)II2 < 4e2^T-r> {\\u(x, y, 0)||2 + a) ^ {\\u(x, y, T)||2 + a) * - a, (9)

wherea=±(\\<po\\i + yi\\2o)-

Proof. If a solution exists and belongs to M, then it has the form

u(x, y, t) = £ £ (ug (t)tik1)) (x, y) + uk2)) (t)tik2) (x, y)) k=1i=1

OO OO

+ E E (uk3) (t)tik3) (x, y)+(t)tik4) (x, y», k=1i=1

where {^'¡(a;, , j = 1,4, the eigenfunctions of the spectral problem (5), (6)

and

ukji(t) = (sign(x)sign(y)u(x,y,t),tikj)) (x,y)), j = 14,

uk,)(t) = -(sign(x)sign(y)u(x,y,i),^kf'](x,y)), j = 2, 3, k,l G N,

^Iki = (sign(x) sign(y)^i(x,y),^k ) (x,y)), j = 1,4,

VikA =-(smMx)sign(y)ipl(x,y),^k3j(x,y)), j = 2,3, ¿ = 0,1, k,l€N.

In (8), let

V(x,y,t)=uk,l(t)0$(x,y), J = 1,4 where Uk,i(T) = u'k l(T) = 0, (t) G W|(Q). Then

T 1 1

u(x y +)tsgn(x)sgn(y),.''k i ( + )., k i

III u(x,y,t)(sign(x)sign(y)w"k ,i (t)^kf ) (x,y) 0 -1 -1

- sign(x) sign(y)^k , I (t)Akf)$kji (x,y)) dxdydt 1 1

= ^,i (0) j J sign(x) sign(y)^kf ) (x,y)v1(x,y) dxdy

-1 -1

1 1

- w'k, i (0) | J sign(x) sign(y)^kf) (x,y)vo (x,y) dxdy. (10)

-1 -1

From (10), we have

T

J«g(t)Kk,i(t) - Akj)^k ,i(t)) dt = ^k,i(0V1i),i - ^k,i(0)^0fo),i, k,l G N. o

Thus, for uk\{t) G W|(<3), j = 1,4, we have the following sequence of solutions:

(ug(t))tt = Akf') ukf') (t), (11)

45(0) = <i, ("j(0))t = i, k,l G N. (12)

It is easy to see that

( 7 ) / ( 7 )

)(t) =

+ Ag=0, j =2,3

^cos A^'] <0, k,l £ N, j = 274.

According to (7), we have

Hx,y,t)ii2 = EE ((41)(t))2 + (uk2)(t))2) + ££ ((uk3)(t))2 + (uk4)(t))2)

k=1 i=1 k=1 i=1

where uk\(t), j = 1,4, are solutions (11), (12), respectively.

We introduce the norm

OO

m? = E E (Ak1) 2 + |Ak2)) K 2 + |Ak3)) K A)2 + |aK k ^i) 2).

k=1i=1

According to Lemma 1, for solutions (11), (12), for each fixed k,l, k, l G N,the following inequality is valid

(Wg(i))2<e2t^((-3( 0))2+ag)1^((-g(T))2 + ag)--ag, (13)

t G Q, j =I~4,

where

After an elementary transformation from (14), we can write

= ^(Ag(«g(0))2 - (^¿(0))2), J=TA- (14)

«i^ < + (^.(O))2), , = M.

We sum up inequalities (13) with respect to k, l, k, l G N, and, taking into account the Holder inequality, we obtain

EEiKlat))2 + (ugw)2 + (uk»2 + (uk4)(t^2) k=1i=1

(OO OO \

EE ((uk?)(0))2 + (uk2)(0))2 + (uk3)(0))2 + (uk4)(0))2

k=1i=1 ) OO

+ee(I«SI+k2]I+K3]I+K:]|))1^

k=1i=1

OO

x ( EE ((uk1)(T))2 + (uk2)(T))2 + (uk3)(T))2 + (uk4)(T))2)

t_

CO CO \ T

+ EE (|ak1)| + |ak2,))| + |ak3,))| + |ak^,) |) )

k=1 l=1

OO

— EEGaSMaSMagl + |ak4)|),

Vk=1 i = 1

and, summing up the above inequality, we finally get

IK*, y, t) II2 < Ae2t{T-^ (IK*, y, 0)||2 + a) ^ (H*, y, T) ||2 + af - a, where a = ^(||</?o||i + ll</?i|lo)- The theorem is proved.

4. Uniqueness and conditional stability

Theorem 2. If the solution of the problem (1)-(4) exists then it is unique.

Proof. Let ui(x, y, t) and u2(x, y, t) be solutions to problem (1)-(4). Then the difference u(x, y, t) = u1(x, y, t) — u2(x, y, t) becomes a solution to the homogeneous problem (1)-(4). Applying the estimates of Theorem 1, we find that ||u(x,y, t)||0 = 0, and it follows that u(x, y, t) = 0 for any (x, y, t) G O, or for V (x, y, t) € O, we have u1(x, y, t) = u2(x, y, t). The theorem is proved.

Theorem 3. Let a solution u(x, y,t) of the problem (1)-(4) exist and u(x, y,t) G M. Let ||<0 — ^0e|1 < e and ||< — <1e||0 < e. Then for solutions of the problem (1)-(4) the estimate

\\u(x,y,t)\\20<4e2t(T-t\2e2)1-^(2m2+£2 f -e2, is valid for t G Q .

Proof. Let u1(x, y,t) be a solution to (1)-(4) with exact data and u2(x, y,t) be a solution to that problem with approximate data. Then u(x, y, t) = u1(x, y, t) — u2(x, y, t) is a solution to equation (1) satisfying the boundary and the gluing conditions (3) and (4) with the initial data u(x,y, 0) = <0(x,y) — <0e(x,y), ut(x, y, 0) = ^1(x, y) — <1e(x, y), and ||<0 — ^0e|1 < e, ||< — <1e||0 < e. Then using the estimates of Theorem 1 and elementary transformations for the norm of the function u(x, y, t) we have t t

IK*, y, t)\\2 < Ae2ti-T-€1 (2e2)^* (2m2 + e2)* - e2. The theorem is proved.

5. Approximate solution

Let <0(x, y) = 0 and ^1(x, y) = 0 in (1)-(4). Then we can represent the solution u(x, y, t) in the form

oo oo ___

u(x, y,t) = Y/H (*>$« ch (*> y) + Vot cos (*> y))

k=1 1=1

OO fc ___CO CO __

+ + E E A cos ^t4}(x,y)

k=1 1=1 k=1 1=k+1,

CO CO __CO fc _

+ E E Vo3i ch v/a^O y) + E E Vot cos yf^tdfl (*> vl

k=11=k+1 k=1 1 = 1

where

<0m = (sign(x) sign(y)^0 (x,y),^kj] (x,y^, j = 1,4,

<0m = —(sign(x)sign(y)<0(x,y),tf$ (x,y^, j = 2, 3, M G N.

Then an approximate solution of the problem with exact data has the form

JV JV __AT AT _

y-i) = EE ch y) + EE cos ^Xi1 I/)

k=1 1 = 1 k=1 1=1

N k ,__N-1 N

+E E Ach (*> y) + E E cos ^M2] (x, „)

k=1 2=1 k=1 l=k+1, jv k ___at-1 jv

.,2 cos y ^k,lvuk,V k=1 2 = 1 k=1 2=k+1

where N is (N integer number) regularization parameter.

The approximate solution with approximate data has the form N N ___

uf (x, y, t) = £ £ (^li ch V Ak>k1)2 (x, y) + <44Jm cos ^ -Ak^'i^k^] (x, y)) k=12=1

+ £ E A««»

k=1 2=1 k=1 2=k+1,

jv k __at-1 at _

where

;,2 ^ y Z^ ' 0ek,2 y k,2 k,2

k=12=1 k=12=k+1

^O'e'k, 2 = (sign(x) sign(y)^0e (x,y),-dj (x,y)) , j = 1, 4,

<4ek,2 = — ( Sign(X)Sign(y)^oe(X,y),^kj2) Ox y^ , j = 3, ^ I G N. Let ||^0(x, y) — ^0e(x,y)||0 < e and u(x,y,t) G M. Then, for the norm of the difference between the exact and approximate solutions, the inequality is as follows:

||u(x,y,t) — uf (x,y,t)||0

< ||u(x,y,t) — uN(x,y,t)|0 + ||uN(x,y,t) — uf(x,y,t)||0. (15)

Let us estimate the second term on the right-hand side of (15), while we made some elementary transformations, and the conditions for estimating the norm of the difference between exact and approximate data are as follows:

||uN (x,y,t) — uf (x,y,t)|0

at at ___

= EE((4 - ^i,)2 ch2 ^ + «« - ^4i,2)2 cos2 yfJfit) k=12=1 at k

+ EE((4 " a,)2 ch2 ^ + «2 - ^3i,2)2 cos2 ^gt) k=12=1

at-1 at ___

+ E E (M - A«)2 ch2 ^+«2 - ^¿fcosz^lt)

k=12=k+1

_ at at

< ch2 TOEE (Mil, - + UZ - A«)2)

at at

k=1 2 = 1 _ at k

+ Ch2 ^EE((il " A,)2 + «2 - ^3i,2)2) k=12=1

N— 1 N

+ chE (M-+

k=1 ,=k+1 N N

< ch2 " ^l,«)2 + (A "

k=1,= 1

+ - <41,)2 + - <41,) ^ < Co£2e

22^

or

„ jLN(x,y,t)-u?(x,y,t)\\l < Coe2^V,

here C0— is a positive constant.

Next, we estimate the first term on the right-hand side of inequality (15) provided that, u(x, y,t) and (x, y,t) G M

N to

i(x, y, t) — uN(x, y,

TO to

k=1 ,=N +1

k=N+1 , = 1 to k— 1

k=N+1 ,= 1

N oo ___

+ E E (('Pohch\fiSt)2 + (vo&.icos /-Âk2^)2)

k=1 ,=N +1

TO TO

k=N +1 ,=k

We estimate the expression

N

TO TO

E E c^Ai»2 + £ ^(^okl, c^Ak1

(1)t^2 k,, '

k=1 ,=N +1

k=N +1 ,= 1

oo k —1 .__N oo ._

+ E E (foich TÂgi)2 + E E «^h^gt)2

k=N +1 , = 1 k=1 ,=N +1

oo oo

k=N+1 ,=k

+ E E (fohch vi

(16)

according to the condition

EE^Oki, c^Ak1) t)2 + £ E c^Ak2

k—1

(2) 2

k,l

k=1,=1

k=N+1 l=1

N

TO TO

+ E E («ffi, c^Ak»2 + E E«, c^Ak3)T)2 < m2. (17)

k=1 ,=N +1

k=N+1 ,=k

O

We estimate (16) under the condition (17) by the method of Lagrange multipliers. As a result, we get

N oo .__oo oo _

E E x

k=1 1=N+1 k=N +1 1=1

oo k — 1 __N oo _

E E UZ ch ^Ägi)2 + E E (^¿ch^gi)2

k=N +1 2 = 1 k=1 2=N+1

+ £ £ M ch X^Stf ^ C1m2(2e2\^(t-T^ + 2e2\/^(t-T) + l)

k=N +1 2=k

where C1 is a positive constant.

(1) 2 (2) 2 (3) 2 (4) 2

Assume the series E E ((^Ofc,0 + (Vok,i) + + ) to be con-

k=1 2=1

verging. Let

N o oo o oo k—1

E E GÄ)2 + E £«2)2+ E E(^kV2

k=1 2=N+1 k=N+1 2=1 k=N +1 2=1

N o

+E E «2)2+ £ Ztâlf - ),

N œ œ œ

^)2+ E E(<^,2)2-■

k=1 2=N+1 k=N+1 2=k

where 7(N) ^ 0 at N ^ ro. Thus,

Summing up the estimates, we have

0.5||u(x, y,t) — u^(x, y, t) ||2

< Cim2(2e2\f^{t-T) + 2e2\f^{t-T) + l) + C„e2V^V + 7(JV).

Minimizing evaluation at the right side of e > 0, we obtain a formula for the regu-larization parameter N. Here m is chosen arbitrarily and usually it is determined depending on the specific model.

Conclusion. An ill-posed boundary-value problem for a second-order mixed hyperbolic-elliptic type partial differential equation was investigated for conditional correctness. The ill-posed boundary value is expressed in the absence of continuous dependence of the solution on the data. The main result of the paper is the proof of the conditional stability theorem for the solution of the problem. Using this, a sequence of approximate solutions which is stable on the defined set of correctness with respect to the accuracy of the initial data was constructed.

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Submitted January 20, 2023 Revised February 7, 2023 Accepted February 28, 2023

Kudratillo S. Fayazov

Turin Polytechnic University in Tashkent,

17 Kichik Khalka Yuli Street, 100195 Tashkent, Uzbekistan

[email protected]

Yashin K. Khudayberganov National University of Uzbekistan, 4 Universitet Street, 100174 Tashkent, Uzbekistan [email protected]