On the solvability of a system of two multidimensional loaded parabolic equations with the Cauchy data Текст научной статьи по специальности «Математика»

УДК 517.9

On the Solvability of a System of Two Multidimensional Loaded Parabolic Equations with the Cauchy Data

Galina V. Romanenko* Igor V. FrolenkoV

Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041

Russia

Received 17.01.2016, received in revised form 03.04.2016, accepted 15.06.2016 We study a multidimensional system of two loaded parabolic equations of a special kind with the Cauchy data. Sufficient conditions for the existence of a solution in the class of smooth bounded functions are obtained. The splitting method at differential level (the method of weak approximation) is used in the proof.

Keywords: inverse problem, direct problem, loaded equation, parabolic equation, weak approximation method, system of partial differential equations, Cauchy problem. DOI: 10.17516/1997-1397-2016-9-3-364-373.

Introduction

A study of inverse problems for systems of parabolic equations can be a time-consuming process, therefore, in [1] there was proposed an algorithm to the study of one-dimensional direct problems for systems of parabolic equations, to which inverse problems for loaded systems of a special kind can be reduced. Increasing dimension of a problem leads to estimations obtained in the process become more complex, which significantly slows down the study.

The article presents a generalization of the algorithm proposed in [1] to the multidimensional case. In the present article we propose and investigate the following model: a system of two multidimensional loaded parabolic equations connected by the lower terms with the Cauchy data. Our motivation is that inverse problems for systems of linear or semilinear parabolic equations with unknown coefficients depending only on time variable can be reduced to such a model. The obtained result can be used as a sufficient condition for existence of a solution to auxiliary direct problems. To prove existence of a solution we use the weak approximation method, which is the splitting method at the differential level [2,3].

Another method of the study of systems of multidimensional parabolic equations in a special form is presented in [4]. Similar algorithms for the study of inverse problems for parabolic equations have been previously presented in [5,6]. The Cauchy problem for a loaded Burgers-type system has been investigated in [7].

* galina.romanencko@yandex.ru 1 igor@frolenkov.ru © Siberian Federal University. All rights reserved

1. Statement of the problem

In the space En of variables x1,... ,xn choose ri different points a\ , (hi = 1,ri) for each variable xi (i = 1,n). In the strip

G[o,t ] = {(t, x)|0 < t < T,x = (x1,...,xn) £ En}

consider the Cauchy problem for the system of loaded nonclassical parabolic equations

nn

Ut(t,x) = a1 (t,1fu(t),1fv (t))uXiXi +Y1 b1(t,¥u(t),<fv (t))uXi + f1(t,x,u,v,¥u(t),1Pv (t)),

1=1 l=1 (1) nn

vt(t,x) = J2 a2 (t,^u(t),Vv (t))vXiXi +J2 h2(t,Vu(t),Vv (t))vXi + f2 (t,x,U,V,¥u(t),1Pv (t)), i=1 i=1

u(0,x) = uo(x), v(0,x) = vo(x), x £ En, (2)

here

Vu(t) = (u(t,ah,...,ann^ D'Xu(t,al1, ...,arkn^ ,

¥v(t) = (v(t,aki,...,annDXv(t,aki,...,ann))

k1 kn X k1 kn

s = (si,..., Sn), Si = 0,1,...,pi; hi = 1,...,ri; i = 1,...,n;

DX ¿(x) = n(X1iX2|-'iXn) C(x) = dX£(xi,x2, . . . ,xn)

Dx^(x) D(xi,x2,...,xn)^x) dxli 3x2? ...dxnr '

where s = (s1; s2,..., sn) is a multiindex, sr > 0 is integer, r = 1, 2,...,n, |s| = s1 + ... + sn.

Henceforth, choose and fix the constants pi = max{p>i, 2}, i = 1,... ,n. Also, introduce the following definitions.

Definition 1. Denote by ([0, t*]) the set of the functions u(t,x), v(t,x) defined in G[0,t*]

belonging to the class Cl'^''.'..'p(G[0,t*]) where

^ £ C(G[0't*]),

dt

s = (s1,... ,sn), si = 0,1,...,pi, i = 1(3) and bounded in G[0,t*] together with their derivatives occurring in (1),

£ (Dxu(t,x)l + Dxv(t,x)l) < C. (4)

X=(Xi,...,Xn), Si =0'1'..;Pi '

i=1,...,n

Definition 2. By a classical solution to problem (1), (2) in G[0,t*] we mean a pair of functions u(t,x),v(t,x) £ Z^;^([0,t*]), satisfying (1), (2) in G^t*]-

Here 0 <t* < T is a fixed constant. If t* depends on the constants bounding the initial data and t* < T then we say that the functions u(t,x),v(t,x) are a solution (1), (2) 'in the small'. If t* is fixed and t* = T for every collection of initial data satisfying the sufficient solvability conditions then we say that u(t,x),v(t,x) are a solution (1), (2) 'in the whole' (or use the term global solvability).

2. The main result

Suppose that the following conditions hold: Condition 1. The functions a\, ai2, b\, b\ (i = T,n) are real-valued, defined, and continuous for all values of their arguments; a\, a\ satisfy the conditions a\ > a0 > 0, al2 > a0 > 0, a0 is a constant, (i = T~n). For all t1 € (0,T\, q(t,x), w(t,x) € Zp\+2xfn+2([0,t{\) the relation holds

n (

(t),ëw(t)) \ + \al2(t,vq (t),Tpw(t)) \ +

+ \b\(t,?q (t),Pw(t))\ + \bi2 (t,Pq (t),Pw (t)}\) < Pyi (Sq,w (t)) . (5) Remark 1. In Conditions 1 and 3, by Y1, 72 ^ 0 we mean some fixed real numbers,

Sq,w(t)= SUP SUP DXq(£,x)\ + sup sup \DXw(£,x)\ ,

s \0<^tx£En 0<$^tx€En )

s = (si,...,sn ), Si = 0,1,...,Pi + 2, i=1

q(t, x), w(t, x) € Zp+2:xfn+2(l0,ti\),

P((y) = C(1 + y + ■ ■ ■ + yz), C > 1 is a constant independent of q(t,x), w(t,x) and their derivatives.

Condition 2. The functions u0(x), v0(x) are real-valued and have all continuous derivatives occurring in the following relation and satisfying it:

£ (Du0(x)\ + \D"Xv0(x)\) < C.

s = (si,...,sn), Xi = 0,1,...,pi + 2, i=1,...,n

Condition 3. The functions f1, f2 are real-valued, defined, and continuous for all values of their arguments. For all t1 G (0, T], q(t,x), w(t,x) G Zpii+2',;xn,Pn+2([0, t1]) as functions of the variables (t,x) G G[0,tl], these functions are continuous and have continuous derivatives occurring in the relation

E (\DX ilit,x,q,w,Pq (t),Pw (t)) \ + s = (si,...,sn),

Si=0,1,...,pi+2, i=1,...,n

+ \DX f2(t,x,q,W,Pq (t),Pw (t))\) < Py2 (Sq,w (t)) . (6) Theorem. Suppose that Conditions 1-3 hold. Then

[a] If Conditions 1-3 hold for ^ 0, y2 = 0 or Y2 = T, then a classical solution {u(t,x),v(t,x)} to problem (1), (2) exists in the class Zp^'-'X™ ([0,T]).

[b] If Conditions 1-3 hold for Y1 ^ 0, y2 > 1 then there exists a constant t*, 0 < t* ^ T, depending on the constant C of (5), (6), such that a classical solution {u(t,x), v(t,x)} to problem (1), (2) exists in the class ZXl.fXi([0,t*]).

Proof. The proof is conducted by using the splitting method on the differential level. We use the spitting of the initial problem into two fractional steps with time shift by (t — in the traces of the unknown functions and nonlinear terms:

-(t,x) = 2J2 a\(t,<pU4 - 2) t - 2)) < x, + 2J2 bi^Ut - 2 WAt - 2)) ,

i=1 i=1 n n

-(t,x) = a^Ut - 2) ,vAt - 2)) vTxixi+ bifaFAt - 2) ,^At - 2)) , (7)

i=i

i=1

nT < t ^ ^n +2) T,

l(t,x) = 2 fi (t - 2 ,x,uT (t - 2 ,x),vT (t - 2 ,x),ruT (t - 2),^ (t - 2) ),

-(t,x)=2 fi (t - 2 ,x,uT (t - 2 (t - 2 (t - 2) Fvt (t - 2))

2 j

(n + ^t < t < (n + 1)t,

uT(t,x)\t^o = uo(x), vT(t,x)\t^o = vo(x), x € En here n = 0,...,N - 1, Nt = T,

Fu

FV

(t- 2)= {V'T(t- 2),DXuT{t- 2,aki'■■■'aL))

t T

2

vTi t — T, al

DXvT{t --,aki,

(8)

(9)

s = (si,. . . ,sn), Si = 0, 1,...,Pi, ki = 1,...,ri, i = 1,...,n.

Now, establish a priori estimates that guarantee the compactness of the family of solutions {uT(t,x),vT(t,x)} to (7),(8) in classes Ct'^'^'XZ(G[o,t*]) for some constant 0 <t* < T.

Refer as the n-th integer time step to the half-interval (nr, (n + 1)t], n = 0,1,... ,N — 1. Introduce the notation:

Suv (0) =

T-A

s = (si,...,sn ), Bi=0,1,..,pi+2, i=1,..,n

sup

xeEn

D'X uo(x)

+ sup

xeEn

DX vo(x)

sut ,vt (t) =

T (

Bl,...,Bn ), Y

B=(Bl,...,Bn ),

Bi = 0,1,...,pi + 2, i=1,

sup sup

nT<£^.t^.(n+1)T xeEn

DX uT (Ç,x)

+ sup sup

nT<Ç^.t^.(n+1)T xeEn

Dx vT (ç,x)

nT <t < (n + 1)t. (10)

The following hold:

1.

DXuT(£,x) + DXvT(tx) < Sut^ (t), e e (nr,t],

t e (nr, (n + 1)t], s = (si, .. ., sn), si = 0, l,...,pi + 2, i = l,n; (11)

2. the functions Sut,Vt (t) are nonnegative and nondecreasing on each time interval (nr, (n + 1)t] .

n

n

a

.,a

k

k

)

[a]. At the first fractional step t € (0, 2], for the solution uT, vT to (7) with initial data (9), by Conditions 1-3 and the maximum principle, we obtain the estimate

\uT (£,x)\ ^ sup \u0(x)\,

xeEn

\vT(£, x)\ < sup \vo(x)\,

xEEn

Summing up the obtained estimates, we have

0<««t, 0<t«^•

\uT («,x)\ + \vT («, x)\ « sup \uo(x)\ + sup \vo(x)\, 0 < « « t, 0 <t « -.

x£En

x£En

2

(12)

We can apply the operation of differentiation Dsx to problem (7), (9) s = (s1,..., sn), si = 0, T,... ,pi + 2, i = T,... ,n. By Condition 1, 2, obtain:

\DSxuT(«,x)\ « sup \DSxuo(x)\,

x£En

\DSxvT(«,x)\ « sup \Dxvo(x)\,

0 < « « t,t € (0, 2], s = (s1,. .. ,sn), si =0, T,...,pi + 2, i = T,...,n.

Apply sup to both sides of the inequalities (12), (13), then sup , and sum the results up, xeEn 0<ç^t

taking into account the notation (10), we get

(13)

SuT,VT (t) « Suv(0), 0 <t « 2•

(14)

Consider the second fractional step of the zero integer step t G (2 . Integrating the system of equations (8) with respect to the time variable over the interval (2, £],£ G (2,t] and T> <t < t, we obtain

T /*> / t T T T T \

(«,x) = uT{-,x) + 2 h(e - -,x,uT(e - -,x,vT(e - -,x,pUt(e - -),prvT(e - -)J de,

T f^ / T T T T T \

(«,x) = vT(-,x) + 2 f2{e - -,x,u(e - -,x)v(e - -(e - -),prvT(e - -)) de,

« € ( 2 ,

2 <t « T,

\uT(e,x)\ « uT(T,x)

+

f^ / T T T T T \

+2JT f1 (e- 2,x,uT(e- 2,x,vT(e- 2,x),pUt(e- 2),prvT(e- 2)J

\vT(«x « vT{T,x)

+

f^ / T T T T T \

+2JT f2 (e- 2,x,uT(e- 2,x,vT(e- 2,x),pUt(e- 2),pVt(e- 2)J

de,

de,

« €(

€ .§,«

The last inequality, Condition 3 (inequality (6)) and the conditions of case [a] imply

\uT («,x)\ « \uT ( - ,x) \ + 2 JT C(i + SuT vvt (e - -) yde,

\vT («,x)\ « \vT (-,x)\ + 2 I c(t + SuT vvt (e - -)J ,de,

« €(

, - << « T,

T

whence, using the properties Sut ,vt (t) in (10) and summing up the inequalities, we infer

\uT («,x)\ + \vT («,x)\ « \uT( - x \ + \vT( - x \ + C(T + SuT ,vt( 2)) T,

« € (-,t], - <t « T• (15)

We apply the operation of of differentiation Dsx, s = (s1,..., sn), si =0, T,... ,pi + 2, i = T, . . . , n, to the equation (8) and then integrate with respect to time variable. By condition 3, we obtain

\DxuT(«,x)\ + \DxvT(«,x)\ « DxuT(-,x) + DxvT(-,x) + C(t + sutvvt(-)) T,

2

£ G (2 ,t], 2 <t ^ T, s = (s1,---,sn), Si = 0, l,...,Pi + 2,i = 1,...,n. (16)

Apply sup and sup to both sides of the inequalities (15), (16) and sum them. By (10), we

xeEn o<£^t

see that

T ~ / T \ T

Sut vt (t) < Sut ,vt( C(1 + Sut ^ ( ^ T, 2 <t ^ T.

Taking into account (14), from this inequality at the zero integer time step we obtain:

Sut ,vt (t) < Su,v (0) + C(l+ Su,v (0))t, 0 <t < T,

here and below we assume that C > l are some (generally speaking, different) constants bounding the input data in Conditions 1-3 and independent of the splitting parameter t .

Sut,vt (t) < Su,v(0)+1-1+C(1+Su,v(0))t < (1+Su,v(0))(1+Ct)-1 < (1+Su,v(0))eCT-l. (17)

At the first integer time step t G (t, 2t], arguing similarly to the zero integer step, we have

Sut,vt (t) < (1 + Sut,vt (t))e - 1 < (1 + Su,v (0))e2CT - 1. In finitely many steps, on the interval ((N - 1)t, Nt] we get

Sut,vt (t) < (1 + Suv (0))eNCT - 1= (1 + Su,v(0)) ■ eCT - 1 < C, mt G ((N - 1)t, Nt] As a result, on [0, T] we obtain

Sut,vt (t) < (1 + Su,v(0)) ■ eCT - 1 < C, mt G [0,T]. Thus we have proved the following estimates, which are uniform in t:

(t, x)

+

(t, x)

« C,

s = (s1,..., sn), si = 0, T,...,pi + 2, i = T,... ,n, (t,x) € G[o,t ]• (18)

Estimates (18) imply that the right-hand sides of (7), (8) are bounded uniformly in t on each time step, and hence the left-hand sides of the equations are bounded uniformly in t:

\uT(t,x) \ + \vT(t, x)\ « C, (t,x,z) € G[0;T].

We apply operation of differentiation Dx for s = (s1,..., sn ), si = 0, T,... ,pi to (7), (8), by (18), we obtain

Dx uT (t, x) + Dx vT (t, x) « C, s = (s1,..., sn), si = 0, T,... ,pi, i = T,....

(19)

x

x

Put GM0t] = {(t,x)\0 < t < T,x = (x1,..,xn) <M}. Estimates (18), (19) guarantee the fulfillment of the conditions of the Arzela Compactness Theorem. By this theorem, some subsequences {uTk(t,x),vTk(t,x)} of the sequences {uT(t,x),vT(t,x)} of solutions to (7), (9) converge together with all relevant derivatives to the functions u(t,x) G C0];;;%Z (G[o, T]), v(t,x) G C0,'Z'XZ (G[o0tT]) respectively, which, by the theorem of convergence of the weak approximation method, are the solution to (1), (2) (by virtue of the arbitrariness of M); moreover, u(t,x) G Cl%;;;;;£ G,t), v(t,x) g C1 %;;;;£ G,t] ) . Case [a] is proved.

For case [b], repeating analogous arguments at the first fractional step, we obtain some estimate similar to (14).

At the second fractional step, by the hypotheses of the theorem, we infer the estimate

\DsxuT(Ç,x)\ + \DSXvT(£,x)\ < DsxuT{2,x) + DsxvT{2,x)

" 2

+

+ 2 J*Py2(Sut, „t(e — 2})de, eG (2T

, 2 <t < T, j = 1,...,p + 2.

Summing the last inequality and using the notation (10) at the second fractional step, we obtain

T i T T

Sut , „ t (t) < Sut , „ t (-) + C Pl2 (Sut „T (e — -)) de, 2 <t < T,

J 2

which, since Sut,„t is nondecreasing in (2,t] , implies

T (t T

Sut „T (t) < Sut „T (2) + C Pl2 (Sut „T (e)) de, 2 <t < T, (20)

ft

*2' ' ~ J - <.->>-■> 2

Consider the Cauchy problem for the ordinary differential equation

dw(t)

dt CP12 (w(t)), w(0) = SUtV (0). (21)

Recall that PY2 (y) = C(1 + y + ■ ■ ■ + y12), yi > 1 is an integer constant, and the constants C, C are independent of the splitting parameter t. By the Cauchy theorem, there is a constant 0 < t* < T such that a solution to this problem w G C*[0,t*], where t* depends on C, C and initial condition SUvv (0). Obviously, w(t) is a strictly increasing function. We have

Lemma 1. It follows from (20), (21) that if Sutvt (t0) < w(t0) for some t0 G (0,t*), then Sut ' vt (t) < w(t), t G [to,t*].

The proof of lemma is in [1].

Thus, we have proved the following estimates uniform over :

Dsx uT (t,x) + Dax vT (t,x)

^ C,

S = (Si,..., s„), Si =0,1,...,pi + 2, i = 1,...,n, (t,x) G G[0 , *.

Repeating the arguments similar to the arguments in [a], we infer that some subsequences {uTk (t,x),vTk (t,x)} of the sequences {uT(t,x),vT(t,x)} of solutions to (7)-(9) converge together with all relevant derivatives to the functions u(t,x) G C0'!''.'! (G[0,t*]), v(t,x) G C°£l(G[0,t*]) respectively, which, by the theorem of the convergence of the weak approximation method, are a solution to (1), (2); moreover, u(t,x) G Ct!'"''X!^ (G[0,t*]), v(t,x) g ClG'..

Thus, we have proved the existence of a solution "in the small"; i.e., case [b] is proved. The proof of Theorem is complete. □

3. An example of the application of theorem

We examine the following Cauchy problem for a system of two-dimensional parabolic equations.

Consider in the domain G[0,T] = {(t, x)\0 < t < T, (x, z) G E2} the system

ut(t, x, z) = uxx + uzz + u + v + A(t)Fi(t, x, z), v(t,x,z) = (cos(t) + 3)vxx + vzz + vx + vz + u + v + F2(t,x, z),

u(0,x,z)= uo(x,z), v(0,x,z) = vo(x,z), (x, z) € E2•

(22) (23)

The function A(t) is to be determined simultaneously with the solution u(t, x, z), v(t, x, z) of the problem.

The overdetermination condition is given by

u(t, a, 3) = ^(t), (24)

(a, ft are constants), the compatibility condition holds

uo(a, 3) = V(0).

About the input data we assume that they are known functions, sufficiently smooth, have all continuous derivatives occurring in the relations and satisfy them

F1(t, a, 3) ^ 5 > 0, S is a constant.

№(t)\ + \^'(t)\ +

dkl+k2 u0 dkl+k2 v0 dkl + k2 F1 dki+k2 f2

dxkl dzk2 + dxkl dzk2 + dxkl dzk2 + \\ dxkl dzk2

« C,

k1,k2 =0, T,..., 4. (25)

By using the overdetermination conditions (24), the problem (22), (23) is reduced to the auxiliary direct problem:

I \ . . . ^'(t)-uxx (t,a,P)-uzz (t,a,ft)-^ - v(t,a,ft)

\Ut(t,x,z)= Uxx+ Uzz+ u + v +-----—-F 1(t, x, z),

F1(t,a,P) (26)

Kvt(t, x, z) = (cos(t) + 3)vxx + vzz + vx + vz + u + v + F2(t, x, z),

u(0,x,z)= U0(x,z), v(0,x,z) = v0(x,z), (x, z) G E2. The problem (26), (27) fits into the model proposed above. Here

a1(t,1fu(t),'fv (t)) = aKt,1fu (t),1fv (t)) = a1^{t,Pu(t"),1fv (t)) =

4( t,Vu(t),Vv (t)) = cos(t) + 3

(27)

= b1( t,Pu(t),Pv (t))= b2( t,Pu(t),Pv (t)) =T,

f1(t,x,u,v,Pu(t),Pv (t)) = u + v +

, ^'(t) - uxx(t,a,3) - uzz(t, a, 3) - ^ - v(t,a,3)

+--r\-f1(t x z),

F1(t,a,3)

f2(t, x, u, v, <fiu(t),<fiv(t)) = u + v + F2(t,x,z).

The constants p>i = {2,1}, pi = max{p>i, 2} = 2. We verify the conditions of the theorem. Condition 1.

^(\a\(t,¥u(t),¥v (t))\ + \4(t,¥u(t),¥v (t))| + i=i^

+ \b\ (t,ëu(t),Vv (t))\ + \b2(t,lfu(t),^v (t))\) =5 + cos(t) + 3 Thus, the condition is satisfied when yi =0.

Condition 2 is satisfied by the assumptions on the input data (25). Condition 3.

(\DX fl(t,x,u,v,1Pu (t),1fv (t)) \ + \DX h(t,x,u,v,1fu(t),1fv (t))\) =

< 9.

s = (s 1' s 2) ' si=01 '... ' 4 ' i=l ' 2

E

s=(s 1' S2) ' Si = 0 '1 '... ' 4 ' i=1 ' 2

d Is I

dxs 1 dzs2

(2(u + v)+

+ (t) — uxx(t,<*,P) — u*z (t,g,P) — * — v(t,a,p) Fit x, z) + F2t x, z)

Fi(t,a,p)

<

<

C

s = (s 1' s2 ) ' si=0 '1 '... ' 4 ' i=1 ' 2

dxS1 dzs2 ((1 + u + v + uxx(t, a, P) + uzz(t, a, P) + v(t, a, ft)))

<

< Pi (Su,v (t)) .

Consequently, condition 3 is satisfied when y2 = 1. Hence, the classical solution u(t,x,z), v(t,x,z) to the auxiliary direct problem (26), (27) exists, for example, in the class Zx'l([0,T]).

The research for this paper was carried out in Siberian Federal University within the framework of the project "Multidimensional Complex Analysis and Differential Equations" funded by the grant of the Russian Federation Government to support scientific research under supervision of a leading scientist, no. 14-Y26.S1.0006.

s

References

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[2] Yu.Ya.Belov, Inverse Problems for Partial Differential Equations, Utrecht, VSP, 2002

[3] N.N.Yanenko, Fractional Steps for Solving Multidimensional Problems of Mathematical Physics, Nauka, Novosibirsk, 1967 (in Russian).

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О разрешимости системы двух многомерных нагруженных параболических уравнений с данными Коши

Галина В. Романенко Игорь В. Фроленков

Институт математики и фундаментальной информатики Сибирский федеральный университет Свободный, 79, Красноярск, 660041

Россия

Исследована многомерная система двух параболических нагруженных уравнений специального вида в случае данных Коши. Получены достаточные условия существования решения в классе гладких ограниченных функций. Для доказательства используется метод расщепления на дифференциальном уровне (метод слабой аппроксимации).

Ключевые слова: обратная задача, прямая задача, нагруженное уравнение, параболическое уравнение, метод слабой аппроксимации, системы уравнений в частных производных, задача Коши.