On the Solvability of Burgers-type Equation with Special Type of Non-linearity Текст научной статьи по специальности «Математика»

EDN: YWCEYV УДК 517.9

On the Solvability of Burgers-type Equation with Special Type of Non-linearity

Igor V. Frolenkov* Roman V. Sorokin^ Ivan E. Zubrov*

Siberian Federal University Krasnoyarsk, Russian Federation

Received 10.05.2023, received in revised form 18.06.2023, accepted 24.08.2023 Abstract. A one-dimensional parabolic Burgers equation of special form with Cauchy data is considered in this paper. To prove the theorem on the solvability of this problem the method of weak approximation developed by Yu. Ya. Belov is used. The results of this paper enhance the results obtained in [2].

Keywords: inverse problem, parabolic equation, Burgers type equation, Cauchy problem, method of weak approximation.

Citation: I.V. Frolenkov, R.V. Sorokin, I.E. Zubrov, On the Solvability of Burgers-type Equation with Special Type of Non-linearity, J. Sib. Fed. Univ. Math. Phys., 2023, 16(5), 690-699. EDN: YWCEYV.

1. Problem statement

Let us choose r different points a1,a2,... ,ar in the space E1. Consider the Cauchy problem

ut = a(t)uxx + b(t, x, u(t, x), w(t))ux + f (t, x, u(t, x), w(t)), (1)

u(0, x) = uo(x). (2)

In the strip G[0,T] = {(t, x)|0 < t < T,x G Ei}. Let us introduce the vector-function u(t) = ( dk \

[u(t, aj), dj-j u(t, aj )J, k = 0,... ,p1,j = 1,... ,r. Components of this function are the traces (depending only on the variable t) of function u(t, x) and all its derivatives with respect to x up to order p1 inclusive. Choose and fix the constant p > xaax{2,p1} > 2.

Definition 1. Let us denote the set of functions u(t,x) defined in G[o,t*j belonging to the class

( du dk u 1

CiXPGot,*]) = | u(t,x)l — , dxj G C (G[o,t*]), k = 0,...,pj,

* igor@frolenkov.ru trsorokin@sfu-kras.ru ti.zubrov@mail.ru © Siberian Federal University. All rights reserved

by Zp([0,t*}). Functions are bounded for (t,x) G G[0,t*] together with all derivatives satisfying inequalities

P d k u(t,x)

E

k=0

dxk

< C.

(3)

Definition 2. Classical solution of problem (1), (2) in G[oit*j is a function u(t,x) € Zp([0,t*]) that satisfies (1) and initial data (2) in G[0,t*]-

Here 0 <t* < T is a fixed constant. Let us assume that the following conditions are satisfied.

Condition 1. Functions b(t, x, u(t, x), v(t)), f (t, x, u(t, x), v(t)) are real-valued continuous functions that are defined for any values of their arguments. For all t* € (0,T] and for all u(t,x) € Zp+2([0,t*]) these functions, as functions of the variables (t,x) € G[0 t»], are continuous and have continuous derivatives involved in (5) and (6). Function a(t) > a0 > 0 is a continuous bounded function on the interval [0,T]. Function u0(x) has continuous derivatives satisfying inequalities

^^ dku0(x)

k=0

dxk

< C.

(4)

Condition 2. Let us introduce the following notations

dk

Uk(0)=sup — uo(x) , k = 0, l,...,p + 2,

xeEi dx

dk

Uk (t)

sup sup

x£Ei

dk

dxk u(t>x)

k = 0, l,...,p + 2,

p+2 p+2

U(t) = Y, Uk(t), U(0) = J2 Uk(0).

k=0 k=0

Let us assume that for all t* € (0,T], for all t € [0, t*] and for any u(t,x) € Zp+2([0,t*]) the following estimates hold

p+2 dk

dxk b(t,x,u(t,x),u(t))

E

k=0

p+2

E

k=0

dk

dxk f (t,x,u(t,x),^(t))

< Pyi (U(t)),

< Py2(U(t)).

(5)

(6)

Here 7i, 72 ^ 0 are some fixed integer constants and

Pe (y) = C (1 + \y\ + \y\2 + ... + \y\« ), where C > 1 is a constant independent of function u(t, x) and its derivatives.

Theorem 1 (Existence). Let us assume that Conditions (1) and (2) are satisfied and 0 ^ Yi < œ, 0 ^ y2 < œ. Then there exists the constant t* € (0,T] that depends on a0 from Condition (1) and C from inequalities (5), (6) such that classical solution u(t,x) of problem (1), (2) exists in the class Zp([0,t*]).

The proof of the theorem for 0 ^ y2 ^ 1 is given in [2]. The case 2 ^ y2 < œ is considered in this paper.

Proof. To prove the existence of a solution of Cauchy problem (1), (2) the weak approximation method [1] is used. Let us consider an auxiliary split problem with time shift (t — 3 j in unknown functions and non-linear terms

uT(t, x) = 3a(t)uTxx(t, x), nr <t ^ ^n + t; (7)

uTt (t,x) = 3b(t — 3 ,x,uT(t — 3 — 3) )uX(t,x), (n t <t < (n t ; (8)

uTt (t,x) = 3f(t - 3 ,x,uT (t - 3 (t - 3)) , (n +3) T<t < (n +1)t ; (9)

uT(t, x) 11^0 = uo(x). (10)

Let us prove a priori estimates that ensure the compactness of the family of solutions u(t, x) of problem (7)-(10) in the class Cj^C^,^]) for some constant 0 <t* < T.

At the first fractional step ^0 <t ^ 3 j for (n = 0) we apply the maximum principle to

problem (7), (10) and obtain the estimate for function uT(t,x)

luT(t,x)l < Uo(0), 0 <t < 3.

Differentiating problem (7), (10) k times with respect to x, we obtain similar estimates

d k

ex* uT (t,x)

< Uk(0), 0 <t < 3, k = 1,---,p + 2-

Summing up the obtained inequalities, we obtain the estimate

UT(t) < U(0), 0 <t < 3. (11)

At the second fractional step )3 <t ^ we solve equation (8). Since function

b (t — 3,x,uT (t — 3(t — 3^ is continuous and it is known from the previous fractional step solution of this equation exists ( [3], item 2.6). Let us consider the characteristic equation for equation (8)

§ =—^—3 'x'uT —3 'x)'wT —3)) •

Let us denote the characteristic function of the resulting characteristic equation by p(€,Z,n), that is, x = Z, n) is the integral curve passing through the point (Z, n). Then the solution at the second fractional step has the form

uT(t,x)= uT (33 <t < y. (12)

3 3 3 3

Therefore, the following estimate is true

t 2t

UT(t) < UT (3) < U(0), 3 <t < -T. (13)

Let us differentiate equation (8) with respect to x and introduce the following notations

ul(t, x) = zT(t, x),

b0 (t,x) =3b(t - 3 ,x,uT (t - 3 (t - 3)) ,

b1 (t'x)=3 - 3 ,x'uT ^- 3 >x)>uT k- 3)) •

Using new notations, the differentiated equation is written in the form

z1 = b0 (t,x)zX + b1(t,x)zT •

The solution of this equation can be written in the parametric form ([3], p. 4.3)

zT (t,x) = eF0 3 ,n)zT (3,V) , x = (t, 3 ,V) ,

where

F0T = FT(t,Z,n) = - j b\(Z,Z,ri)dt and x = pT (£, Z, n) is the characteristic function of the equation

ddx=-b0 (t,x)=-3b(t - 3 ^ ^- 3 ,x),„T k- 3)).

Therefore, the following estimate is true

\K(t,x)\ = \zT (t,x)\ < UT (3) e^ (t-3 ))t < UT (3) eP-n ^ • Now we take from the left and right parts of the resulting inequality sup for x € E1 and obtain

UT(t) < UT (3) e^(0))T, 3 <t < ^ (14)

Next, we differentiate equation (8) twice with respect to x and introduce the following notations uXx(t,x) = vT(t,x),

c0(t,x)=3b(t- 3,x,uT (t- 3x ,ut (t- 3)),

( 3 ( 3 ) ( 3

cl (t,x)=6 d-b (t - 3 ,x,uT (t - 3 ,x),^T (t - 3)), (t,x)=3 dx b(t - 3 ,x,uTit - 3 ,x),^T{t - 3)) •

33

Using new notations, the equation is written in the form

VT = c0 (t, x)vX + cl (t, x)vT + c2 (t, x)zT (t, x).

vt = c0 (t,x)vx + c1 (t,x)v + c2 (

The solution of this equation can be written in the parametric form ( [3], p. 4.3)

V = eG3n)(vT (3,n) + T c2 3zT 3eG°3

x = Z, n),

where

o = Co

CO = CO (t,Z,n) = - ci (t,(,n)dC

Note that estimate for function zT(t, x) is already available. Therefore, one can evaluate function

vT (t, x)

luUt,x)l = V (t,x)l < e2^ {u (0))(U; (3)+^ (U (0))e2TPYi{u {0)) i ^ (£)#)) <

"3

^ eCrPjl {um^yr + CrP11 (U(0)U (3) eTPYi{u{0)^ <

< eCTPYi{U{0)) U (3) + CTPYI (U(0))UJ (3) ) < < eCrPji{u{0)) (UT (3) + UT (3) ) (1 + CTP7i (U(0))) < eCrPYi{u{0)) (UJ (3) + UJ (3) ) .

Now we take from the left and right parts of the resulting inequality sup for x G E1 and obtain

UT (t) < eCTPYi{u {0)) (UT (3) + U[( 3)), 3 <t < f. (15)

Next, we differentiate equation (8) k = 3,.. .p + 2 times with respect to x. Using the Leibniz formula for the k-th derivative of the product of two functions, we obtain the equation in general form

dk T T dk T T dk T A T dk-j+1 T

dxkut = 90 dxkux + g 1dxku +^ gj dxk-j+1'

j=2

xut = g0 dlkux + g 1 dlku +7 .9

where

g0 = 3b{t- 3,x,u (t- 3,x) ^ - 3)) ,

gT=3Ck Kt - 3 ,x,uT(t - 3 ^^- D).

Writing the solution in explicit form, we obtain the following estimate

dxku (t,x)

/ \k-1 \ < e^{u{0))(uUl + CPyi (U(0)) J ]T UJ(№) <

( ) k- ( )

^ eCTPYi{u{0))( Ul (3) + CtP1i (U(0))eCTPYi{u{0)^ UJ (3) J <

V j=1 )

k

^ eCTPYi{u {0))f £ UJ (3))(1 + ctpyi (U (0))) <

\ j=1 J

< eCTPYi{u{0))( J2 UJ (3)) , k = 3,... ,p + 2, 3 <t < 23.

1j= 1

Now we take from the left and right parts of the resulting inequality sup for x G E and obtain

Ul(t) < eCTPYi{u{0))( £ UJ (3) ) , 3 <t < |. (16)

j= 3 3 3

C

k

Combining inequalities (13), (14), (15) and (16), we obtain estimates for the solution at the second fractional step

UT(t) < UT (3) eCTPYi(U(0)), 3 <t < (17)

At the third fractional step <t <t^ we integrate equation (9) with respect to variable

t

(2T \ rt

T(t x)= uT I ,„,„ ,„ ,„

3 ) /2tV 3 V 3 J V 3/

uT(t,x) = u (y^) +3 J2 f (p - 3,x,uT (n - 3,x) (n - 3)) dn-

Condition (2) implies that

UT(t) < UT (f) + CtPy/2 (V (Ç))

Differentiating equation (9) k times with respect to x,k = 1,... ,p + 2 and using condition (2), we obtain

UT(t) < UT (y) + CrPl2 (uT

3 ) ,2\ \3 ) Combining the obtained inequalities, we have

, 2, ,, Y2

TT/j-\ ^ TTT I 2 ' \ I D I TTT I 21 \ \ ^ 1 I TTT I 21 \ I I 1 I TTT ' 2 '

UTt) < UT (f ) + CTP, (UT (*)) s , + UT (*) + ) (, + Ut (£))" -, < < 0 + 'U (*)) (1+ O (1 + UT (I))~) - ! <

^ + UT eCT(i+uT(¥))Y2-1 - I. (18)

Using estimates (11), (17), (18), the following inequality holds at the zero time step t € [0, t]

^(t) < (l + U(0)eCTPY1 (u(0)^ eC'T[1+u<0))f21 - 1 ^

< (1 + '(0))eCTPY1 (U(0)) + CT(l + U(0))Y2-le<<2-1)CPY1<U<0)) - 1 ^

< (1 + '(0))eCTK (U(0)) + (1 + U(0))Y2-1]e(Y2-1CP->i(U<0)) - L Let us choose = max{71; j2 - 1} then

^(t) < (1 + U(0))eC'TPY3(1+U(0)W3CtPy3<-u(0)) - h Let t be such that inequality

eY3CTPY^ (U(0)) ^ 2

is satisfied then

UT(t) < (1 + U(0))e2C'TPY3 (1+U(0)) - 1, t € [0, t]. Using the same line of reasoning, at the first time step (t <t < 2t), we obtain the estimate

UT (t) ^(1 + UT (t))e2C'TPY3 (1+UT (T)) - 1 <

^ ^ + (1 + ' (0))e2CTPY3 (l + U (0)) __ ^ e2CTPY3 [1+(1+U (0))e2CTP-y3<i + U <0)) -i] __ 1 ^ < (1 + ' (0))e2CTPY3 (l + U (0))+2CtPy3 (l+U (0))e2CT->3P->3<1 + U<°) ^

< (1 + ' (0))e2CTP->3 (l+U (0))[l+e2CrY3 PY3 (1 + U<0))] - ^

Let be such that inequality

e2Crj3PJ3 {1+u{0)) < 2

is true, then

UT(t) < (1 + U(0))e6CTPY3{1+u{0)) - 1, t G [0, 2t]. At the second time step (2t <t < 3t) we obtain the estimate

UT(t) ^{1 + UT(2r))e2CTPY3{1+uT {2t)) - 1 <

< ^ + (1 + U(0))eCP-l3 {1+u{0)) __ ^ e2CTPY3 [ 1+{1+u+ 1 ] __ 1 < < (1 + U(0))e6CTPY3 {1+u{0))+2CTPy3 {1 + u{0))eGCrY3PY3 (i + U(0)) <

< (1 + U (0))e2CTPY3 (1 + u m[3+e6CTY3PY3(i + U (0))] - ^

Let be such that inequality

e6Crl3P13 (1+u{0)) < 2

is satisfied, then

UT(t) < (1 + U(0))e10CTPY3{1+u{0)) - 1, t G [0, 3r]. Continuing given above argument, at the i-th time step we obtain the estimate UT (t) < (1 + U (0))e{4i+2)CT PY3 {1+u {0)) - 1, t G [0,'ir]. Let t* (0 <t* < T) be such that

t*C'Y3PJ3 {1 + u{0)) < 2

e

i*

Then for all i ^ 0 such that (4i + 2)r < t* the following estimate holds

UT(t) < (1 + U(0))e{4i+2)CTPY3 {1+u{0)) - 1 < (1 + U(0))eeCPY3 {1+u{0)) - 1. Since t*,C,Y3 and U(0) depend on the input data but do not depend on r we obtain

d k

dXk u (t,x)

C UT(t) C (1 + U(0))ettCPY3(1+U(0)) - 1 = K, t G [0,t*]. (19)

This implies that function u (t, x) and its derivatives with respect to x are bounded uniformly in terms of variable r up to order p + 2 inclusive in the strip C[0,t,].

By virtue of equations (7)-(9) it also follows that derivatives are bounded uniformly in terms of variable

d dkuT , , N

k = 0,...,p. (20)

dt dxk

"uT

d dkz,

Taking into account the boundedness of derivatives ———k, k = 1,... ,p, it guarantees equicon-

dx dxk

( dkut ]

tinuity in = {(t,x)|0 C t C t*, \x\ C N} of sets of functions < k > ,k = 0,.. .p for any

fixed constant N.

By virtue of the Arzela theorem some subsequence uTk (t,x) of the sequence uT (t,x) of solutions of split problem (7)-(10) converges together with derivatives with respect to x up to order p inclusive to function u(t,x) G Cl'£ (G[0jt*]). By virtue of the convergence theorem for the weak approximation method [1], u(t,x) is a solution of original problem (1), (2). Moreover, for (t, x) G G[0jt»] the following estimate is satisfied

E

dku(t, x)

dxk

C C.

k=0

Thus, u(t, x) G Zp([0, t*]). The theorem is proved. □

2. Example

Let us consider an example of application of Theorem (1) to the proof of the solvability of one inverse coefficient problem for a parabolic type equation. Let us consider the Cauchy problem

ut(t, x) = a2Uxx(t, x) + (u(t, x) + X1(t))ux(t, x) + \2(t)f (t, x), u(0, x) = uo(x).

(21) (22)

that is posed in the domain G[0,T] = {(t,x) \0 < t < T,x G E1}. Functions X1(t), X2(t) are to be determined simultaneously with solution u(t,x) of problem (21), (22) satisfying redefinition conditions

u(t,a) = pi(t), Ux(t, a) = ^2(t)

(23)

(24)

and conditions of agreement

u(0,a) = ^1(0), Ux(0, a) = ^2(0).

(25)

(26)

Regarding functions (t),p2(t),u0(x), f (t,x), we assume that they are sufficiently smooth, and they have all continuous derivatives that satisfy the following inequality for all (t, x) € G^0<T]

\Mt)\ + H (t)| + \v2(t)\ + \v2(t)\ +

dk

dxkuo(x)

+

dk

dxkf (t,x)

< C, k = 0,..., 5. (27)

Let us also assume that for all t € [0, T] the following inequality is satisfied

d2

P2(t)fx(t, a) - f (t, a)dx2 u0(a) > S > 0. The original problem is reduced to the auxiliary direct problem

(28)

ut = a uxx +

where

u+

(^i(t) - a2u,xx(t, a))fx(t, a) V2 fx (t, a) - uxx(t, a)f (t, a)

(Mt)

- a2 uxxx

(t, a) - ^1Uxx(t, a))f (t, a)

¥2fx(t, a) - uxx (t, a)f (t, a)

(^2 (t) - a2u xxx (t,a) - ^iuxx(t,a))^2 V2fx(t, a) - uxx(t, a)f (t, a)

ux +

+

(^i(t) - a2u xx (t, a))u xx (t, a)

V2fx(t, a) - uxx(t, a)f (t, a) _

u(0, x) = uo(x),

Mt) = v'i(t) - Mt)Mt), Mt) = v>2(t) - ^i(t)-

f (t,x), (29) (30)

In order to guarantee that denominator of expression (29) does not vanish we introduce the cut-off function Sg (y). It is differentiable as many times as needed and has the following properties

Ss (y) > 3 > 0,

Vy G Ei,

(31)

Ss (y)

y,

6

3 '

6

y > 2; 6

y < - •

y ^ 3

(32)

Let us substitute the cut-off function into the denominator of fractional expressions

(^i(t) - a?uxx(t, a))fx(t, a)

Ut

2 + a uxx +

U +

Ss(w2fx(t, a) - Uxx(t, a)f (t, a))

(02(t) - a2Uxxx(t, a) - wiUxx(t, a))f (t, a)

+

Ss(W2fx(t, a) - Uxx(t, a)f (t, a))

(02 (t) - a2U xxx (t,a) - WiUxx(t,a))W2

Ux +

Ss(W2fx(t, a) - Uxx(t, a)f (t, a))

(0i(t) - a2U xx (t, a))U xx (t, a)

Ss(w2fx(t,a) - Uxx(t,a)f (t,a))_

f (t,x), (33)

u(0,x) = uo(x). (34)

The resulting direct problem (33), (34) is a problem of form (1), (2). Let us check the conditions of Theorem (1) for p = 3,

b = u +

(0i(t) - a Uxx (t, a))fx(t, a) - (02 (t) - a Uxxx(t, a) - WiUxx(t, a))f (t, a)

Ss(W2fx(t, a) - Uxx(t, a)f (t, a))

f

(-02 (t) - a2Uxxx(t, a) - WlUxx(t, a))w2 - (0l (t) - a2Uxx (t, a))Uxx(t, a)

Ss(W2fx(t, a) - Uxx(t, a)f (t, a))

Condition (1) is satisfied due to assumption (27), and condition (2) becomes

5

E

k=0 5

E

k=0

d k

dXk b(t,X,U(t,X),^(t)) dk

dxxk f (t,X,U(t,x),M(t))

< Pi(U(t)),

< P2(U(t))•

Thus, all conditions of Theorem (1) are satisfied for p = 3,71 = = 2. Therefore,

there exists a constant t* : 0 <t* < T depending on the constants that constrain the input data such that classical solution u(t,x) of problem (33), (34) exists in the class ZX(G[0,t»j).

Note that at this point, the existence of a solution of direct problem is proved but not the existence of a solution of inverse problem. After that, we need to remove the cut-off function from the denominators of fractional expressions. In order to guarantee that conditions imposed on cut-off function (31)-(32) are satisfied it is necessary to use inequality (28).

Then, using the agreement conditions and redefinition conditions, one can show that solution of the inverse problem also exists, and function u(t,x) which is the solution of direct problem (33), (34) is also the solution of inverse problem (21)-(22). Parameters X1(t),X2(t) are defined as follows

(t)--

Mt) =

(0i(t) -a2Uxx(t, a))fx(t, a)-(02(t)-a2Uxxx(t, a)-WiUxx(t,a))f (t, a) W2fx(t, a) - Uxx (t, a)f (t, a) '

(02 (t)-a2Uxxx(t, a)-<piUxx(t, a))w2 - (0i(t)- a2Ux:x(t, a))Ux:x(t, a) W2fx(t, a) - Uxx (t, a)f (t, a) '

This work is supported by the Krasnoyarsk Mathematical Center and financed by the Ministry of Science and Higher Education of the Russian Federation (Agreement no. 075-02-2023-936).

References

[1] Yu.Ya.Belov, K.V.Korshun, Weak approximation method, Krasnoyarsk State University, Krasnoyarsk, 1999 (in Russian).

[2] I.V.Frolenkov, M.A.Darzhaa, On the existence of solution of some problems for nonlinear loaded parabolic equations with Cauchy data, Journal of Siberian Federal University. Mathematics & Physics, 7(2014), 173-185.

[3] E.Kamke, The directory on the differential equations in partial derivatives of the first order, Nauka, Moskva, 1966 (in Russian).

[4] Yu.Ya.Belov, I.V.Frolenkov, Some identification problems of the coefficients in semilin-earparabolic equations, Doklady Mathematics, 404(2005), no. 5, 583-585 (in Russian).

[5] Yu.Ya.Belov, K.V.Korshun, An identification problem of the source function for the Burgers equation, Journal of Siberian Federal University. Mathematics & Physics, 5(2012), no. 4, 497-506 (in Russian).

О разрешимости уравнения типа Бюргерса с нелинейностью специального вида

Игорь В. Фроленков Роман В. Сорокин Иван Е. Зубров

Сибирский федеральный университет Красноярск, Российская Федерация

Аннотация. В данной работе рассматривается одномерное параболическое уравнение Бюргерса специального вида с данными Коши. При доказательстве теоремы о разрешимости этой задачи используется метод слабой аппроксимации, разработанный Ю. Я. Беловым. Результаты, полученные в данной работе, усиливают результаты, полученные в [2].

Ключевые слова: обратная задача, параболическое уравнение, уравнение типа Бюргерса, задача Коши, метод слабой аппроксимации.