On a problem of cross-diffusion with Nonlocal boundary conditions Текст научной статьи по специальности «Математика»

Journal of Siberian Federal University. Mathematics & Physics 2019, 12(5), 614—620

УДК 517.957

On a Problem of Cross-Diffusion with Nonlocal Boundary Conditions

Zafar R. Rakhmonov*

National University of Uzbekistan University, 4, Tashkent, 100174 Uzbekistan

Jasur E. UrunbayeV

Samarkand State University University Blvd., 15, Samarkand, 140104

Uzbekistan

Received 01.04.2019, received in revised form 15.06.2019, accepted 20.08.2019 Condition of global existence of solution of a non-linear system of cross-diffusion with non-linear boundary conditions is studied in the paper. Critical exponents of Fujita type and critical exponents of global existence of solution are established.

Keywords: nonlinear parabolic system, cross-diffusion, critical Fujita curve, blow-up. DOI: 10.17516/1997-1397-2019-12-5-614-620.

1. Introduction and preliminaries

Let us consider the following nonlinear system of cross-diffusion equations with nonlinear boundary conditions

du dt

д

дх

(mi-l M ) ^ = A (um2-l ^

\ дх j dt дх \ дх J

— I , x G R+, t > 0

ди ди

-umi-1 — (0,t)= uqi (0,t), -um2-1 — (0,t) = uq2 (0,t), t> 0, дх дх

и (x, 0) = uo (x), u (x, 0) = uq (x), x G R+,

(1)

(2) (3)

where mi > 1, q > 0 (i = 1, 2), u0 and (x) are non-negative continuous functions with compact support in R+.

The process of cross-diffusion means that the spatial displacement of one object, described by one of the variables, occurs due to diffusion of another object described by another variable.

Cross-diffusion models are used in various fields of natural science. For example, in physical systems (plasma physics) [1-3], in chemical systems (dynamics of electrolytic solutions), in biological systems (cross-diffusion transport, dynamics of population), in ecology (dynamics of the forest structure), in seismology — the Burridge-Knopoff model which describes the interaction of tectonic plates [4-7]. In recent years, mathematical models with cross-diffusion have been actively used to study biological populations and the motion of tectonic plates [4, 5].

* zraxmonov@inbox.ru turin1987@rambler.ru © Siberian Federal University. All rights reserved

It is known that a system of degenerate equations may not have a classical solution in the region where u, u = 0. In this case, the generalized solution of system (1) is studied in a class where

du du

u (x,t), u (x,t) > 0, um1-1 — um2-1 — € C (R x (0, +œ)),

dx dx

and system (1) is satisfied from the point of distribution [1, 3].

In recent years, the condition for global existence of solutions and the condition for the emergence of a blow-up regime have been intensively studied (see [5-16]). The following system of equations was considered [8, 9]

ut = uxx, ut = Uxx, x> 0, 0 <T < 0, (4)

-ux (0,t)= uaup, —ux (0,t)= uq u3, 0 <t<T, (5)

u (x, 0) = uo (x), u (x, 0) = uo (x), x > 0. (6)

The conditions for global solvability in terms of time were studied and the estimation of the solution near explosion time was stated. It was proved that if pq < (1 — a) (1 — ¡3) then each solution of problem (4)-(6) is global.

The following problem was studied [10]

ut = (un)xx, ut = (uk)xx, x € R+, t> 0, (7)

— (un)x (0,t)= up (0,t), — (uk)x (0,t) = uq (0,t), t> 0, (8)

u (x, 0) = uo (x), u (x, 0) = uo (x), x € R+. (9)

It was shown that solution of problem (7)-(9) is global if pq < (n + 1) (k + 1)/4. Conditions were obtained for the numerical parameters of system (7)-(9) wherein the solution of the problem blows-up in a finite time.

System (7) was considered also with the following boundary conditions [11]

— (un)x (0,t) = uaup (0,t), — (uk)x (0,t) = uqu3 (0,t), t> 0.

It was shown that min{y1 — r1,y2 — r2} = 0, where

2p + k + 1 — 23 2p + n + 1 — 23 ri = --71—,-——;-, r2

4pq - (k +1 - 2a) (n +1 - ' 2 4pq - (k + 1 - 2a) (n + 1 - 2^)'

1 - ri (n - 1) 1 - r2 (k - 1)

yi =-2-' y2 =-2-'

are critical exponents of Fujita type.

Definition 1 ([1]). The solution u (x,t), — (x,t) is called lower solution (upper solution) in QT = {(x,t) : x e R+, 0 <t < if u (x,t) = 0 and - (x,t) = 0 in QT\Dt, Dt = = {|x| < p(t)} x

(0, +^),u(x,t),v(x,t) e Ct'X (D) n C(Qt), u 1 u 2 d- e C (Qt) and it satisfies

' dx dx

du d I „mi-i du\ du d I „m2-1 dv\

at ^ (>) eï{u ox), m ^ (>) dxl^ ox),

„mi— 1 du . . . „qi . „m2 — 1 du . . . „ 02 /

—u dx (0,t) ^ (»u (0,t), — u 2 — (0,t) < (»u (0,t).

u (x, 0) ^ uo (x), v (x, 0) ^ uo (x), x € R.

The purpose of this paper is to find conditions for existence of solutions of problem (1)-(3) in time on the basis of self-similar analysis. Various self-similar solutions of problem (1)-(3) were constructed, the assessment and asymptotic solutions were obtained, critical exponents of Fujita type and critical exponents of global existence of the solution were established.

2. Main results

Theorem 2.1. If qi ^ 1, q2 ^ 1 then each solution of problem (1)-(3) is global.

Proof. Sufficient conditions for the solvability in time of problem (1)-(3) are determined by constructing its bounded upper solutions. The bounded upper solutions of the problem are taken in the following self-similar form

(x,t) = eLit (K + e-M^) , £i =

u (x,t)

Lt

-M2&

-J2t

x > 0, t > 0, x > 0, t > 0,

(10)

where K > max{|K||TO, ||uoHTO>, Mi = (K +1)mi Mi(K + 1)mi-2 (MiK + M2 (mi - 1)) M2(K +1)

_ -, L2 = -

qj__, s i2

, M2 = (K +1) m2-1 , Li =

m2-2 (M2K + Mi (m2 - 1)) j

-, Ji =

K + 1 K + 1

= L2 (mi - 1), J2 = Li (m2 - 1).

Let us show that functions (10) are upper solutions of problem (1)-(3). According to the comparison principle of solutions [1, pp. 21-22], they must satisfy the following system of inequalities

du ^ d lu mi-i du

dt ^ dx I dx

du ^ d I um2-i du dt ^ dx \ dx

x > 0, t> 0, x > 0, t > 0,

(11)

m1-i du dx

,m2-i du dx

^u 1 (0,t), t> 0,

(12)

^v 2 (0,t), t> 0.

x=0

Taking into account that

LieLit (K + e-^1) + MiJixe(Li-Ji)t > LieLit (K +1),

-,mi — i.

Mie(L2(mi-i)+Li-Ji)t(K + e-M2&)mi-ie-Mi£i,

-,mi —i

> mi —2

Ux) = Mie(L2(mi-i)+Li-Ji)t(K + e-M2^2 y

/ X

x (Mie-(Mi^i+Jit) (K + e-M2^y + (mi - 1) M2e-J2^ < < Mie(L2(mi-i)+Li-Ji)t(K + 1)mi-2 (Mi (K + 1) + (mi - 1) M2), ut =L2eL2t (K + e-M2^2) + M2J2xe(L2-J2)t > L2eL2t (K +1),

m2 — i.-

= M2e(Li(m2-i)+L2-J2)t( K + e-Miti) m2-ie-

M2&

xe

x

t

x

x

( um2-1u = M2e(L! (m2-D+L2-j2)t{K K + e-Mli^ m2-2x

x (M2e-(M2?2+J2t) K + e-Mi+ (m2 - 1) Mie-Ji^ <

< M2e(Ll(m2-1)+L2-j2)t(K + 1)m2-2 (M2 (K + 1) + (to2 - 1) Mi) and considering definitions of Mi, Ji, Li (i = 1,2) and K, it is easy to prove that at q1 < 1, q2 < 1 systems of inequalities (11) and (12) are always true. □

Theorem 2.2. Let qt > 1, q2 > 1. Then each solution of problem (1)-(3) is unbounded at sufficiently large initial data.

Proof. To prove Theorem 2.2 the solution of problem (1)-(3) is taken in the following self-

similar form

where T > 0, ß

i u (x,t) = (T - t)-a1 v (0 , \ U (x,t) = (T - t)-a24 (e), e = x(T - t)-ß, qi - m2 q2 - mi 1

(13)

1

ai

2q1 — m2 — 1 2q2 — m,1 — 1 ' 2q1 — m2 — 1 '

Functions (^ (£) ,4 (£)) are solutions of the following problems

a2

2q2 - mi - 1

d (*"-1 V - ßede - aiV

d

de

V

•m2-id4) - ße^ - a2

de

dv ^

d4 ■de

= 0, 0,

-4m 11 dde (0) = v1 (0),

Vm2-id4 (0) = 4q2 (0),

(14)

(15)

which are obtained after substituting (13) into (1)-(3) and some simplifications. Conditions can be attained wherein solution (13) is an unbounded lower solution of problem (1)-(3). The following functions are compared

v(e) = Ai(a - e) m2—1 ■

~ 1

4(e) = A2(a - e) m1-1,

(16)

where Ai > 0 (i = 1, 2). Then in order to use the comparison theorem the following inequalities should be satisfied

d

l1-i dv

de d

de

de

dvp

1? 4m1-i^ - ße^7 - aiv > 0

Vp

m2-i- ße^ - °-24 > 0,

de

de

d4p

de

-4mi-i < vqi (0), -<pm2-i^(0) < 4q2 (0).

This is equivalent to stating that the following inequalities

Ami-i + ( + Pi

---2 - aia + ai +---

(TOt - 1)2 ( TO2 - 1

AT2- + ( + P2

---2 - a2a + [a2 +---

(to2 - 1)2 V toi - 1

e > 0,

e > 0,

J A1Amjl1-1a m— < Af a m— ,

{ A<2A™2-1a ^^ < Af a

are true. It is clear that the first systems of inequalities are true if the following sufficient conditions

Ami-i Am2-i

tin <|

a < min ^ ——-2, —2-2}, qi > 1, > 1

a2(mi - 1) ai(m2 - 1)

are satisfied.

From the second systems of inequalities the following restrictions on a are obtained

A q2-1 a qi-1

a > max ' 1 2

(mi - 1) A™1-1 (m2 - 1) qi-1 A™2 -

Then, choosing values of A1, A2, a, we can always achieve that the last system of inequalities are satisfied at q1 > 1, q2 > 1. According to the comparison principle of solutions, the lower self-similar solutions (13), (16) proved conditions for the initial data:

u0 (x) > T-aiA1 (a - x2T-2?,

1

v0 (x) > T a2 A2 (a - x2T-2?)m1-1 .

Thus, by comparison we conclude that

j u (x,t) > (T - t)-ai <£(0) —>■ tt, t — T, \ v (x,t) > (T - t)-a24>(0) — tt, t — T,

at q1 > 1 and q2 > 1. It implies that u and v are unbounded. The theorem is proved.

Theorem 2.3. Suppose that q1 > m2 + 1, q2 > m1 + 1 and initial data are sufficiently small. Then each solution of problem (1)-(3) is global.

Proof. Conditions for the solvability in time of problem (1)-(3) are determined by constructing bounded upper solutions. They are taken in the following self-similar form:

(x,t) = (T + t)-ai f (0 (x, t) = (T + t)-a2g (0 , £ = x(T + t)

-, (17)

where T > 0. Taking into account the comparison theorem of solutions, functions (f (£) ,g (£)) should satisfy the system of inequalities

d ( gmi-l df\ , cJf

H ) + «5 + aif < 0 (18)

d ( fm.2-1 od

fTe) + ^ + -g <0,

-gmi-1 f (0) > fqi (0),

t (19)

-fm2-1 dg (0) > gq2 (0) • de

Let us consider the following functions

f(e) = Ai(a - (e+hi)2)

^ ^ (20) g(e) = A2[a - (e+h2)2)m1—,

i i where HU h2 G (0, y/à), a> 0, ¿1 = [¡3 m — , A2 = {¡3 ^ — ^. Systems

of inequalities (18), (19) are solvable with respect to unknowns a, h1 and h2 at q1 > m2 + 1, q2 > m1 + 1. Then, substituting functions (20) in (18) and (19), we obtain

(«1 — 3) (a — (£ + h1)2) < 0, (a2 — 3) (a — (£ + h2)2) < 0.

Then necessary conditions q1 > m2 + 1, q2 > m1 + 1 and conditions for numerical parameters a,

h1 , h2

a < m,»{ h\ + {¿m'-1A-> m2h ) ^. h2 + (A-r-KA— ) (21)

are obtained. Thus, if q1 > m2 + 1, q2 > m1 + 1, and initial functions u0 (x), (x) satisfy conditions

u0 (x) < T-a1 A^a - (xT-ß + h^2)

2 \ m2 — 1

uo (x) < T a2 A^a - (xT-ß + h2f) m1—1

where a, h1, h2 are chosen from condition (21), then the solution of problem (1)-(3) is global. Remark 1. Theorem 2.1 shows that critical exponents of global existence of solution are

qio = 1, q20 = 1.

Remark 2. Theorem 2.3 shows that critical exponents of Fujita type are q1c = to2 + 1,

q2c = toi + 1.

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Об одной задаче кросс-диффузии с нелокальными граничными условиями

Зафар Р. Рахмонов

Национальный университет Узбекистана Университет, 4, 100174 Узбекистан

Жасур Э.Урунбаев

Самаркандский государственный университет Университетский бул., 15, Самарканд, 140104

Узбекистан

В 'работе изучается условие глобального существования и несуществования решения нелинейной системы кросс-диффузии с нелинейными граничными условиями. Установлены критические экспоненты типа Фуджита и критические экспоненты для глобального существования решения.

Ключевые слова: нелинейная параболическая система, кросс-диффузия, критические экспоненты типа Фуджита, режим с обострением.