THE CAUCHY PROBLEM FOR A NONLINEAR DEGENERATE PARABOLIC SYSTEM IN NON-DIVERGENCE FORM Текст научной статьи по специальности «Математика»

Математические заметки СВФУ Июль—сентябрь, 2020. Том 27, № 3

UDC 517.95

THE CAUCHY PROBLEM FOR A NONLINEAR

DEGENERATE PARABOLIC SYSTEM

IN NON-DIVERGENCE FORM

M. Aripov, A. S. Matyakubov, and B. Kh. Imomnazarov

Abstract: We deal with degenerate quasilinear parabolic systems in the non-divergence form under positive initial conditions. An asymptotic behavior of self-similar solutions in the case of slow diffusion is established. Depending on values of the numerical parameters and the initial value, the existence of the global solutions of the Cauchy problem is proved. In addition, the asymptotic representation of the solution is obtained.

DOI: 10.25587/SVFU.2020.93.40.003 Keywords: a nonlinear degenerate parabolic system, non-divergence form, Cauchy problem.

In this paper, we consider the following Cauchy problem to solving a quasilinear degenerate parabolic system in the non-divergence form:

^ =vmiV{\Vuk\p~\un), ^=um*V(\Vvkr\un), t>0,xGRN, (1)

u|t=o = uo(x) > 0, v|t=o = vo(x) > 0, x e Rn, (2)

where p, mi, m2, k, n are positive constants, N > 1.

As compared to the classical equations of the divergence form equations (1), (2) in some cases are closer to the actual circumstances. For example, for the biological species, the diffusion of divergence form implies that a species is able to move to all the locations within its environment with equal probability. However, if we consider this problem under the objective conditions, the population density will affect the rate of diffusion, so a kind of a "biased" diffusion equation will be more realistic, for the non-divergence form diffusion, the diffusion rate being regulated by the population density that is increasing for large populations and decreasing for small populations.

System (1) has been proposed as a mathematical model for a variety of physical problems (see [1-11]). For instance, this system can be used to describe the development of multiple groups in the dynamics of biological populations, where (u,v) are the densities (see [1]).

This work is supported by the Russian—Uzbek project MRT—0T—81—2018 and the Russian Foundation for Basic Research (grants 18-51-41002 and 18-31-00120).

© 2020 M. Aripov, A. S. Matyakubov, and B. Kh. Imomnazarov

In [1], Gao et al. consider the following degenerate quasilinear parabolic systems in the non-divergence form

uit = upi (Auj + alul+x), i = 1, 2,..., m, um+1 = ui, 0) = Uio(x), i = 1, 2, .. ., m, x G O, ui(x, t) = 0, i = 1, 2,..., m, x G dO, t > 0,

with zero Dirichlet boundary conditions and positive initial conditions. The local existence and uniqueness of the classical solution are proved. Moreover, it is proved that all solutions exist globally with the homogeneous Dirichlet boundary condition.

In [5], Zhou and Zheng investigate the positive solutions to the degenerate parabolic equation in the non-divergence form

ut = upAu + uq|Vu|2 - ur,

with zero Dirichlet boundary condition. The existence of global solutions and the large time needed for solving them is studied. The main effort goes into obtaining uniform asymptotic profiles for the decay solutions, under various dominations of the nonlinear diffusion or absorption. It is shown that the large time property of the solution behaves just like (1 + (r — 1)t)(-1/r — 1) if the decay is governed by the nonlinear absorption with 1 < r < p +1.

In [6], Lu deals with positive solutions of some degenerate quasilinear parabolic systems in the non-divergence form

uit = fi(ui+i)(Aui + aiui), i = 1, 2,..., n — 1,

unt = fn(ui)(Au„ + a„u„), x G O, t > 0.

The local existence and uniqueness of the classical solution have been proved. In [3], Chunhua and Jingxue study the self-similar solutions in the form

u(t,x) = (t + 1)-af ((t +1)" |x|2).

The existence and uniqueness of solutions with compact supports are proved, which implies that the self-similar solution is shrink. The convergence rates of these solutions on the boundaries of the supports are established. The convergence rates of the solutions and comparisons with the Dirac function at t ^ to are considered.

In [8], Raimbekov studies the asymptotic behavior of solutions in the case of the single equation (1). The properties of solutions in slow and fast diffusion cases are investigated. The results of the numerical experiments are discussed.

Porous medium equations with local sources or with non-local sources subject to non-local boundary conditions were studied (see [12-15]). The conditions of existence and blow-up are discussed.

In [9], Sun et al. study the non-existence and long-time behavior of a weak solution for the degenerate parabolic equation

dtun = um div(|Vum|p-2Vum) + 7|Vum|p + ^un

with zero boundary condition. The blow-up time is detected when the blow-up does occur.

Zhou et al. [10] investigate the positive solutions to the doubly degenerate parabolic equation in the non-divergence form with the gradient term

ut = um div(|Vu|p—2Vu) + 7ur|Vu|p + ,0uq

and with zero Dirichlet boundary condition. Zhou et al. first established the local existence of weak solutions to the problem, and then determined in what way the gradient term affects the behavior of the solutions. The conditions for global and non-global solutions were obtained with the critical exponent rc = . The authors have represented a certain precise technique based on the "concavity method" for a complicated non-divergence form of the model.

Wang [11] studies the non-existence and the behavior of the non-negative solution to the nonlinear degenerate parabolic equation with the Dirichlet boundary condition and the initial condition

ut = um div( |Vu |p—2 Vu) + Yum-1|Vu|p

where m £ [1; 2),p > 1, 7 > 0. The above equation, as a typical divergent equation, is known as a fast diffusive one for 0 < n < 1, where n = -r^—, m < 0 and n > 1,

' 1—m' '

which corresponds to 0 < m < 1. When 1 < m < 2, the above equation is a non-divergent one, which has been proposed as a mathematical model of physical problems in many fields such as the resistive diffusion of a force-free magnetic field, dynamics of biological populations, etc. (see [16,17] and references therein). Many authors studied the above equation with p = 2 (see, for example, [18] and references therein). Recently, the authors [17] have discussed the case p> 2, m = 1, 7 = 0 and have constructed different solutions. In [13], the non-existence and the extended time behavior of the solutions for m =1 are studied. In the general case, the authors [17] have discussed the existence of weak solutions to the initial boundary value problems. However they did not study the non-existence and the extended time behavior of the solution.

In [19-22], the authors study the properties of self-similar solutions of the system in a two-component medium of the Cauchy problem with a variable density and a source of the following form:

\x\n^ = ^ivil xfu^^Vu) + \x\nu^, dt

\x\n^ = um2V dxfv^^Vv) + \x\nv^,

In particular, solutions of the Zeldovich-Barenblatt type equations for cross-diffusion systems of a non-divergent type were constructed, and slow and fast diffusion cases were investigated.

The global solvability of the Cauchy problem for system (1) in the divergence case with a source and the asymptotic self-similar solution are studied in [23,24].

This paper is organized as follows. In Section 1, using the comparison principle and the standard equation method, we prove the global solvability of solutions and prove Theorem 1. In Section 2, using the standard equation method and the method of nonlinear splitting, we construct an asymptotic representation of the solution and prove Theorem 2 and Theorem 3.

1. The Fujita type global solvability of solutions

Let us begin with the local existence of solutions to problem (1), (2). One of the particular features of problem (1), (2) is that the system is degenerate at the points where u = 0,v = 0 or Vu = 0, Vv = 0. Hence, there is, in general, no classical solution, and we introduce the following definition of a weak solution (see [2-4, 8, 25,26]).

Definition 1. The non-negative measurable functions u(x, t), v(x, t) defined in RN x (0, T) are called a weak solution of the Cauchy problem (1), (2) if for every bounded open set Q with a smooth boundary dQ, uk, un, um2 G Lfoc(0, T; W 1,p(Q)), u G Cioc(Q X (0,T)), v G Cioc(Q X (0,T)), vk,vn,vmi G Lfoc(0,T; W 1lP(Q)), and

t

J J ( —u^t + |Vuk|P VuV(vmi dxdt + | u^dx = j uo(to, x)<^(to, x) dx, to n n n

t

J J (—v^t + |Vvk |P-2VvV(um2dxdt + J v^dx = J vo(to, x)^(to, x) dx to n n n

for all 0 < to < t < T and all the test functions ^ G Co1(Q x (0, T)). Moreover,

u(x,t)n(x) dx = I u(x,to)n(x)

—mo J u(x,t)n(x) dx = j u(x,to)n(x) dx

t-

n

for any n(x) G Cq(Q). Let

1 — a1(k(p — 2) + n — 1) — a2m1 > 0, a1 > 0, a2 > 0,

and

a1(k(p — 2) + n — 1) + a2m1 = a2(k(p — 2) + n — 1) + a1m2. Then we introduce the notations

hi =

1 — ai(k(p — 2) + n — 1) — a3_iTOi'

_ (p - l)(fc(p - 2) + n - 1 - to,) _ ^ (A;(p — 2) + n — l)2 — TOITO2 ' '

Applying the method of comparison of solutions [26] and the method of standard equations [25], we obtain the estimates of the solution of problem (1), (2).

Theorem 1. Let k(p — 2)+ n > m + 1, m = max{mi,m2}, Yi > miYz-i,

N

P < 0, W+(0,x) > u0(x), v+(0,x) > v0(x), X G R, i

problem (1), (2) has a global solution with the estimate

1, 2. Then

u(t, x) < u+(t, x), v(t,x) < v+ (t,x),

where

u+ (t, x) = A(T + t)~

v+(t,x) = B(T + t)~

ai

a —

a —

|x| !

1 — ai ( k(p— 2) + n — 1) — «2 mi

(T +t)

|x|i

1 — ai (k(p — 2)+n — 1) — a 2 m 1

(T +1)

(3)

Ak(p-2) + n-1 Bm1 =

2)+n-1 Am2 =

p — 1

p2n(Yi — miY2) p — 1

p — 1

p2n(Y2 — m2Yi)

pkYi p — 1

pkY2

p-2

p-2

(b) + = max(0, b).

Proof. We prove the properties of the global solvability of weak solutions of system (1) using the comparison principle (see [26]). To this end, we construct a new system of equations using the standard equation method as in [25]:

u(t,x) = (T + t)-a1 /(0, v(t,x) = (T + t)-a ¿(0,

where

-i .L 1 — q ]_ ( fc (p — 2 ) + n — 1) — q 2 ml

C = paf |x|(T + i) 5 ,

p-2

Ak(p-2) + n-1 Bm1

p — 1

p2n(Yi — miY2)

p — 1

p2n(Y2 - TO27l)

pkYi p — 1

pkY2

p-2

It is easy to check that

Lu+ (t, x) Lv+(t, x) = ( —

N Yi

+ h

p Yi — miY2 P 72- ™27i

+h

lJ /(0, i) ¿(C).

From this expression it follows that the fulfilment of the inequalities Lu+ < 0,

Lv+ < 0 is in the satisfaction of the conditions —

N

+ hi < 0, i = 1, 2.

p ji-mij3 — i

By the conditions of Theorem 2.1 the latter inequalities are satisfied. Then, by the theorem of comparison of solutions to problem (1), (2), there is a global solution in Q and for it, the following estimation u+(t, x) > u(t, x), v+ (t, x) > v(t, x), x G RN holds.

Theorem 1 is proved.

p

h

i

«1

p

h

a

i

i

+

Y

2. Asymptotic behavior of solutions to problem (1.1), (1.2)

In Theorem 1, it is proved that the upper solution of problem (1), (2) is of the form (3). We now show that the functions u+(t, x),v+ (t,x) are of the asymptotic form among all the solutions of the problem (1), (2).

First, we assume that the solution is radially symmetric, that is, depending only on r = |x| at a given time t > 0.

Therefore, we study the following problem:

du ~dt

vmi r1-N

d_

dr

,N-1

9V = um-2rl-N_^_ ! rN-1

dt

dr

duk p-2dun

dr dr

dr dr

u|t=o = uo(r) > 0, v|t=o = vo(r) > 0, r G (0, +to).

(4)

2.1. Case 1: 1 - ai(k(p - 2) + n - 1) - a2mi = 0.

It is easy to establish that system (4) has, approximately, a self-similar solution of the form

u(t,x) = (T + t)-ai f (C), v(t,x) = (T + t)-a 0(C),

(5)

where

^ = hJafr(T + ty

1 — ai (k(p — 2) + n — 1) — «2 mi v

is a self-similar variable and the functions f (£), satisfy the approximate self-similar system of equations

jni2 ^1 — N

A (V-1 dfk p-2d.r^

dC d4 y

d [Cn-1 d(f>k p dC

dï, di; y

(6)

+ M = 0.

According to the statement of the original problem, we will consider non-trivial, nonnegative solutions to the system of equations (6) satisfying the following conditions:

f (0) = Ml > 0, 0(0) = M2 > 0, f (di)= 0(d2) = 0,

0 < d1 < TO, 0 < d2 < ro.

2 — p j 2 — p 1 — p

= 1,2.

(7)

We introduce the notations ci1 Let us have the equality

pn(7i-mi73_i)

a1(k(p — 2) + n — 1) + a2m1 = a2(k(p — 2) + n — 1) + a1m2. Hence the following theorem is valid:

Theorem 2. Let Yi > miY3-i, 1 — a1(k(p — 2) + n — 1) — a2m1 = 0, i =1, 2.

p-i

Then the compactly supported solution of problem (6), (7) at £ —>■ a p ¿as tie following asymptotics:

(p-l)(k(p-2) + n.-l—m.1 )

/(C) = A(a - «^H—1)2—!^ (! + o(l)),

(p — l)(fc(p — 2) + n — 1 — m2)

(8)

0(0 = B(a - ^ ) «P-2)+„-i)2-mim2 (! +

where the coefficients A and B are the roots of the systems of the nonlinear algebraic equations

Ak(p-2)+n-1Bm1 = ci1, Bk(P-2)+n-1Am2 = c21.

Proof. For convenience we transform system (6), (7). For such auxiliary systems of equations, we apply the following transformations:

m = №yi№, № = №v2(v), 7? = -in(o-^), (9)

p 'Yi p T2

where /(£) = (a — C^1) , <^>(£) = (a — C^1) , a > 0, yi(r/), y2(??) are the new functions.

Now let us study the asymptotic behavior of the solutions to problem (6), (7)

p-i

at £ —>■ a p .

After carrying out transformation (9), system (6) will take the following form: d

Here

+ an{ri)y™-i{Lyi) + ai2{r}) + ai0{r})y^j + at3(17)^ = 0,. (10)

+ aio(n)y

T (k-1)(p-2)+n-1

= yi

+ a-ioiVWi

p-2

/dyi

dn

(p — 1)(k(p — 2)+ n — 1 — mi) aio(i?) = —^——^->

(k(p — 2) + n — 1) — m1m2 N (p — 1)e-n

a*i(v) = —r-br +P " 1 + (*(P " 2) + n)aio(v),

p(a — e n)

k2-pY1-p hie-n

CLrtiw) =-, a,jr,(n) = —----;--, 2 = 1,2.

pn ' nkP-2jP(a- e-v)'

p- 1

Here we assume that £ G [£o, £1), 0 < £0 < £1, £1 = a p . Therefore, the function n(C) has the properties

n(0 > 0 at C G [Co,Ci), no = n(Co) > 0, lim n(C) =

€—€1

Hence follows that an auxiliary system of equations (10) with the following limits: lim aij (n) = aij-, i = 1, 2, j = 0,1, 2, 3,

exist, is finite and nonzero, that is, 0 < Iaij I < +œ. In the case when n ^ +œ, the

1 j 1 '

solutions to systems (10) hold, each of which in a certain neighborhood satisfies the inequalities

yi(n) > ° y'i + ai0(n)yi = 0 i = 1, 2.

Assuming in system (10) that

i \ (k-1)(p-2)+n-1

vi(n) = Vi we obtain the identity

dyi , , + aMVWi

^+aio(v)yi), * = (11)

(rj) = -ani^Viirj) - al3(rj)yiy3 - al2(r])yl p 1 y31 (77), i = 1,2.

(12)

Now consider the functions

i }z i

9i(Xi,r,) = —an(ri)Xi - a^y^ - 0^(77)% "-1 y^K" (v), (13)

where Ai € R, i = 1, 2. Assume that

1 - ai(k(p - 2)+ n - 1) - a2mi = 0, Yi > m_iY3-i, i = 1, 2.

Then the functions gi(Ai, n), i =1, 2, are fixed-sign ones on some interval [ni, C [no, for every fixed value Ai, i = 1, 2, different from the values satisfying the system

- ^(^^"^(W^AT1 - =0, < = 1,2.

Then

k2-p71-p

lim al2(??) =-, lim £^3(77) = 0, ¿=1,2,

n^+TO pn

lim aniv) =P- 1 - {k[P ~ f + " - 1 - ,

(k(p - 2) + n - 1) m1m2

and the functions gi(Ai,n), i = 1, 2, are fixed-sign ones on some interval [n1, C

[no, for every fixed value Ai, i = 1, 2. Thus, the functions gi(Ai, n), i = 1, 2, for

all n £ [n1, satisfy one of the inequalities

gi(Ai,n) > 0 or gi(Ai,n) < 0, i = 1, 2. (14)

Now assume that for the functions vi(n) i = 1, 2, the limit at n ^ does not exist. Now consider the case, where one of inequalities (14) is satisfied. Because of the oscillations of the functions vi(n), i = 1, 2, the straight lines t = Ai, i = 1, 2, their graphs intersect an infinite number of points on the interval [ni, i = 1, 2.

But this is impossible, since the interval [ni, i = 1, 2, is just one of inequalities

(14), and therefore from (13) it follows that the graphs of the functions vi(n), i = 1, 2, intersect the straight line ti = Ai, i = 1, 2, only once on the interval [ni, i = 1, 2. Therefore, for the function vi(n), i = 1, 2, there exists the limit at n ^

We assume that the functions vi(n), i = 1, 2, are defined according to (11) and have the limit at n ^ Therefore y'i(n), i = 1, 2, have the limit at n ^

which is equal to zero. Then

vi(n)

dy* r \

-r- + a-ioiVW dn

P 2 / dVi . / x A (k-1)(p-2)+n-1

+ai0{r))yi m

0 IP-2 0 / 0\k(P-2)+n I \

a°0)| a0o( Vi0) +o(1)

at n ^ and by (12), the derivatives of the functions v,(n), i = 1, 2, have the limit at n ^ which is obviously equal to zero. Consequently, it is necessary that

1 h 71 — fc I

lim (aii(vH(v) + at2(v)yt P_1 Vz'^¡^i'1 M)

+ lim a,3(n)^-™' = 0, i = 1, 2.

It is easy to see that system (10) has the solution (yi(n),y2(n)) with a finite non-zero limit as n ^ that is necessary to match the conditions of Theorem 2. Then the solution with a compact supporter of problem (6), (7) has the asymptotics of the form of (8) at £ —>■ a p . Theorem 2 is proved.

Corollary 1. A weak solution of problem (1), (2) has the following asymptotics

p 7l \ p- '

uA(t, x) = A(T + tyai (a - (¡xIt-^p )p-1) (1 + 0(1)),

, , _£_ 72

vA(t,x) =B(T + t)-a2(a-(\x\T-1/P)*-1) (l + o(l)) at \x\ —>■ a~p~T1/p , where the constants 71, 72, A, B are defined above and t = hia-1(T + i)1-»i(k(p-2)+"-1)-»2™>.

2.2. Case 2: 1 - a1(k(p - 2) + n - 1) - a2m1 = 0.

It is easy to establish that system (1) has an approximate self-similar solution of the form of (5), where

N

C = c ln(T + t) - ^ji,

i=1

is a self-similar variable, c > 0, y, > 0, i =1, 2, and the functions f (C), ¿(C) satisfy the self-similar system of equations

df k

dC

d/

dC

^ df M f n p-2 , (15)

^^ ^ n

p

N ^

where « = y \j=1

Let us introduce the notations: &2 = —-r^--—r-—, i = 1,2.

Let us have the inequality

a1(k(p - 2) + n - 1) + a2m1 = a2(k(p - 2) + n - 1) + a1m2.

Then the following theorem is valid:

Theorem 3. Let toito2 > 1, 1 — ai(k(p — 2) + n — 1) — a2mi = 0. Then

p-1

tie solution with a compact supporter of problem (15), (7) at £ —>■ a p has the following asymptotics:

(p-l)(fc(p-2) + n-l-rr11 )

/(£) = A(a - ^ ) «P-2)+„-i)2-mim2 (! +

(p — l)(fc(p — 2) + n — 1 — m2)

(16)

0(C) = B(a - ^) (1 + 0(1)),

where the coefficients A and B are the solutions of systems of the nonlinear algebraic equations

Ak(p-2)+n-1Bmi = c Bk(P-2)+n-1Am2 = c22-Theorem 3 is proved in a similar manner as Theorem 2.

Corollary 2. A weak solution of problem (1), (2) has the following asympto-

tics:

v 71 N \ p-i\

UA(t, x) « A(T + t)-ai a — cln(T + t) — ^

i=1

P 72

JV \ p-I \

VA(t, x) « B(T + t)-a2 a — cln(T + t) —

at

N

p-i

^^ HiXi —>■ cln(T + i) — a p

i=1

where the constants y1, y2, A, B are defined above.

3. Conclusion

An asymptotic behavior of the self-similar solutions of degenerate quasilinear parabolic systems in the non-divergence form with positive initial conditions is studied. Depending on the value of the numerical parameters and on an initial value, the existence of global solutions of the Cauchy problem is proved. In addition, the asymptotic representation of the solution at

(/Ha-V-^T + i)1-"1^-2^™-1'-"2"11^

and

N

x - p=1

i=1

l^iXi —>■ cln(T + t) — a p

is obtained.

The results of the computational experiments show that the self-similar solutions are more appropriate and the iterative method based on the Picard method is efficient for the solution of nonlinear problems and brings about the nonlinear effects if we use as initial approximation the solutions of the self-similar equations constructed by the method of nonlinear splitting and by the method of standard equation (9).

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22. Matyakubov A. S., "Finite speed of the perturbation distribution and asymptotic behavior of the solutions of a parabolic system not in divergence form," Univ. J. Comput. Math., 5, № 3, 57-67 (2017).

23. Aripov M. and Matyakubov A. S., "To the qualitative properties of solution of system equations not in divergence form of polytrophic filtration in variable density," Nanosystems: Phys., Chem., Math., 8, № 3, 317-322 (2017).

24. Aripov M. and Sadullaeva Sh. A., "Qualitative properties of solutions of a doubly nonlinear reaction-diffusion system with a source," J. Appl. Math. Phys., 3, 1090-1099 (2015).

25. Aripov M., Method of the standard equation for the solution of the nonlinear value problem, Fan, Tashkent (1988).

26. Samarskii A. A., Galaktionov V. A., Kurdyumov S. P., and Mikhailov A. P., Blow-Up in Quasilinear Parabolic Equations, De Gruyter, Berlin (2011) (De Gruyter Expo. Math., vol. 19).

Submitted September 29, 2019 Revized June 30, 2020 Accepted August 30, 2020

Mersaid Aripov

National University of Uzbekistan University street 4, Tashkent 100174 mirsaidaripov@mail.ru

Alisher S. Matyakubov National University of Uzbekistan University street 4, Tashkent 100174 almasaSlist.ru

Bunyod Kh. Imomnazarov Novosibirsk State University, 1 Pirogov Street, Novosibirsk 630090, Russia buned11998.07Smail.ru

, Uzbekistan

, Uzbekistan