CC BY
3BULLETIN OF OSH STATE UNIVERSITY MATHEMATICS. PHYSICS. TECHNOLOGY. 2024, № 1(4)
УДК 517.956
DOI: https://doi.org/10.52754/16948645 2024 1(4) 47
BLOW-UP OF SMOOTH SOLUTIONS OF THE PROBLEM FOR THE KORTEWEG-DE VRIES-BURGERS EQUATION WITH THE HILFER FRACTIONAL
DIFFERENTIAL OPERATOR
Mamanazarov Azizbek (PhD) mamanazarovaz1992@gmail. com Mukhtorov Diyorbek, researcher diyorbekmuxtorov81@,gmail. com Fergana State University Fergana, Uzbekistan
Abstract. This work is devoted to studying the non-existence of the global-in-time solutions for the Korteweg-de Vries-Burgers equation including Hilfer time fractional differential operator which in particular cases of the parameters follows the classical and other time-fractional Korteweg-de Vries-Burgers equation. Applying the method of nonlinear capacity which was suggested by S.I. Pokhozhaev for some initial-boundary value problems, it has been obtained sufficient conditions for the non-existence of global solutions.
Keywords: Hilfer derivative, the method of nonlinear capacity, non-existence of the solution.
РАЗРУШЕНИЕ ГЛАДКИХ РЕШЕНИЙ ЗАДАЧИ ДЛЯ УРАВНЕНИЯ КОРТЕВЕГА-ДЕ ВРИСА-БЮРГЕРСА С ОПЕРАТОРОМ ДРОБНОГО ДИФФЕРЕНЦИАЛА
ХИЛЬФЕРА
Маманазаров Азизбек, (PhD) mamanazarovaz1992@gmail. com Мухторов Диёрбек, студент diyorbekmuxtorov81@,gmail. com Ферганский государственный университет
Фергана, Узбекистан
Аннотация. Настоящая работа посвящена изучению отсутствия глобальных по времени решений уравнения Кортевега-де Фриза-Бюргерса, включающего дробно-дифференциальный оператор Гильфера по времени, который в частных случаях параметров следует классическому и другим дробным по времени уравнениям Кортевега-Бюргерса. уравнение де Фриза-Бюргерса. Применяя метод нелинейной емкости, предложенный С.И. Похожаевым для некоторых начально-краевых задач, получены достаточные условия отсутствия глобальных решений.
Ключевые слова: производная Гильфера, метод нелинейной емкости, отсутствие решения.
1. PRELIMINARIES. In this section, we give some basic concepts of fractional calculus.
Definition 1.1. [1] Let f e L ([a, b]) . The following integrals
and
= -f{s)ds (11) 1 b
11[т = ТЩ\{8 ~*Г f{S)ds (12)
are called the left-sided and the right-sided Riemann-Liouville integrals of the fractional order a > 0, respectively, where r( z) denotes the Euler's gamma function.
Definition 1.2. The Riemann-Liouville left-sided fractional derivative Da+ f of order a (0<a< 1) is defined by
D" [f №)=^ [f №)=^(ib) 7 j(t - s r f (>) 7 - (13)
Definition 1.3. The Riemann-Liouville right-sided fractional derivative D"_ f of order a(0 < a < 1) is defined by
[ f ](,)=- T;1-"[ f ](,)=- r(I"-a) 7 f(s-,)a f (s) 7 - (14)
ition
defined by
Definition 1.4. The Hilfer derivative D"f f of order 0 < a < 1 and type 0 < /3 < 1 is
d"3 [ f ](t)=iTa) dft™-" [ f ](,) (1-5)
where , <J > 0 is the Riemann-Liouville fractional integral.
The Hilfer derivative was introduced in [2], [3]. These references provide information about the applications of this derivative and how it arises. It is easy to see that this derivative
interpolates the Riemann-Liouville fractional derivative (/= 0) and the Caputo fractional
derivative (/ = 1) (see [1]).
The fractional integration by parts is defined as follows.
Lemma 1.1. Le a > 0, p > 1, q > 1 and — + — < 1 + a (p ± 1 and q * 1 in the
a
p q
1 1
case —i— = 1 + a). If (p e Lp (a, b) and ^e (a, b), then
p q
b b
fp( t) ia+ ty]( t) dt=jV( t) it [p] dt. (1.6)
a a
2. NON-EXISTENCE OF THE SOLUTION OF TIME-FRACTIONAL KORTEWEG-DE-VRIES-BURGERS EQUATION
Let denote by nab a rectangular domain of R2 , i.enab = t,X)e R2 : 0 < t < T,
a < x < b j. In the domain na b, we consider the time-fractional Korteweg-de Vries-Burgers equation
D"+Ptu (t, x ) + u (t, x ) Ux (t, x ) + Uxxx (t, x ) = vuxx (t, x ) (2.1)
with the following initial condition
Il~\u(0, x) = u0 (x), x e [a, b], (2.2)
where D"f is the Hilfer derivative of order 0 < a < 1 and type 0 < / < 1 with respect to t, v> 0 and u0 (x) is a given function.
If /3 = 1 then the equation (2.1) takes the form which studied in [4]. And when /3 = 1 and a= 1 it becomes the classical Korteweg-de Vries-Burgers equation [5]. We should note the
Korteweg-de Vries-Burgers equation can be applied as the mathematical model for many reallife processes [5].
Our aim is to investigate blow-up solutions of the problem (2.1)-(2.2). To do this we apply the method of nonlinear capacity. This concept for analyzing blow-up of solutions nonlinear equations was suggested by Pokhozhaev in [6].
We consider a class 0(nab) of test functions p(t,x), defined on the domain nab with arbitrary parameters T > 0, a, b e R, have the following properties:
0) P (x ( e C ( na,b ) ;
(ii) px > 0 in na,b ;
(iii) i/(1-")p(x,t) = 0 at t = T and x e (a,b);
(iv) C (najb) = jj (Lp)- dtdx < ,
na,b px
U T* 7-(1-/)(1-a) 1-/(1-")
where L p = A )Dr- )p + ( - Pxx .
Suppose that there exists an T > 0 for which weak solution of the problem (2.1)-(2.2) satisfying ux,D"£u e C([a,b] x [0,t]) .
By multiplying the equation (2.1) by a test function pe 0(Uab) and then integrating
over na b obtained equality, we get
jj p(t,x) D"ftu (t,x) dtdx + jfj p(t,x) u (t,x) ux (t,x) dtdx +
n n
+ U P(^x) Uxxx (^x) dtdx = V JJ p(^x) Uxx (^x) dtdx . (2 3)
Applying the rule of integration by parts, it is easy to obtain the following equalities
jj p (t, x) u (t, x) u (t, x) dtdx =
1 t 1
= 1 u-t,x)p(t,x)Ibdt--jj u2(t,x)px(t,x)dtdx , (2.4)
2 0 2 n
jjp(t,x) ^xx (t,x) dtdx =
na ,b
t b
= j[ux (t,x)p(t,x)-u(t, x)px (t, x)] dt - jj u(t,x)pxx (t, x)dtdx, (2.5)
0 a nab
jj P (^ x) ^xxx (t, x) dtdx =
na b
t b
j[(uxx -Pxux +Pxxu ] dt -jj Pxxx (t,x)u (t,x)dtdx . (26)
0 a na,b
Using Definition 1.4 and applying Lemma 1.1, we have
dt
jj p(t,x)D"ftu(t,x)dtdx = jj p(t,x)i/(1-")-IJ1/*"(t,x)dtdx =
jj I^1-^t,x)/)(1-a)u(t,x)dtdx .
dt
na ,b
Hence, applying the rule of integration by parts and using Lemma 2.1, we obtain
n
b
Jj>(i, x ) DZPtu (t, X ) dtdx = J{/0+—f)( 1—a)u(t, x ) If——"a)p(t, x )}
T dx —
JJ t, x) I^(1—a)u (t, x) dtdx :
dt
J{4——fX——a)u(t,x)If——a)p(t,x)} Tdx — JJ u(t,x)I™-'a) f/fl^Ht,x)dtdx .
d
dt
Taking this and equalizes (2.4), (2.5) into account and also using Definition 2.3, from (2.3) we drive
— JJ u2 (t, x) H (t, x) dtdx J — JJ u (t, x) (L*p) (t, x) -
J ^u (t, x ) I^a)p(t, x dx + Jb (u (t ,x ) ,p(t,x ))
dt,
(2.7)
a
where
B (u (t, x ) ,p(t, x )) j — u 2 (t, x )p( t, x) —vux (t, x )p(t, x ) + vu (t, x )px (t, x )-
P(t, x) uxx (t, x) — H (t, x) ux (t, x) + Hxx (t, x) u (t, x) .
Taking (2.2) and (iii) property of test functions, from the last we get
1 T
— JJ u2 (t, x)h (t, x) dtdx J — JJ u (t, x)(Lp)(t, x) dtdx + J B (u (t, x) ,p(t, x))
dt —
—Ju0 (x) If\ap(x, t) dx .
a
By applying Holder and Young's inequalities, it is easy to see that JJ u (t, x) (L*p) (t, x ) dtdx
(2.8)
Яl-(L p) (t, x)
u (t, x ( x, t) , =- dtdx
^Px ( x't)
<
V'2 r ((,,x))2 v'2
JJ u2 (t, x) px (t, x) dtdx
vn,
JJ
( X, t)
dtdx
<
1 rr 1 rr ((L H)(t,x))
— II u2 (t,x)h(t,x)dtdx + — II ---—dtdx.
2 nat 2 nb (x (x,t)
Taking this inequality and (iv) property of test functions into account from (2.8), we have
1 T b b 0 < — C(na,b) + \B(u(t,*),p(t,*)) ^dt — Ju0(x)IPT^a)p(x,t)|t=0dx . (2.9)
a
The following theorem is valid:
Theorem 3.1. Suppose that the boundary conditions and the initial function u0 (x) e L [a,b] satisfy the following assumption: there exists a function p(x, t) e O(nab) such
b
that B (u (t, x) ,p( t, x))
e
a
L [0, T ] and the following inequality
1 T b b
— C(na_b) + {B(u(t,x),p(t,x)) dt — jU0 (x)If———a)p(x,t)|tj0dx < 0 . (2.10)
0
n
a
0
a
b
0
a
0
a
Then problem (2.1)-(2.2) does not admit a global-in-time solution in ПаЪ with these
initial and boundary conditions.
Proof. Let us assume the opposite i.e. the problem (2.1)-(2.2) admits a global-in-time
solution in Па b . Then we arrived at contradiction by virtue of inequalities (2.9) and (2.10).
Now, we consider the fractional Korteweg-de Vries-Burgers equation (2.1) with v = 1 in
the rectangular domain ПаЬ = {(t,x)e R2 : 0 < t < T,0 < x < l| with the initial condition (2.2)
and the following boundary conditions
u (t,0) = т1 (t), ux (t,0) = t2 (t), 0 < t < T, (2.11)
where T- and T2 are given functions such that t1, t2 e L[0, T ].
Multiply the time-fractional Korteweg-de Vries-Burgers equation (2.1) by a test function p e Ф (Пг, ), after some calculations and simplifications we obtain
1 t l i
0 < - C(П ,1 ) + J Я (u (t, x), p (t, x)) 1 dt - J u (x) I^p (x, t ) 11=0dx .
2 0 0 0
We take a test function satisfying the following boundary conditions:
p(t,l) = 0, px (t,l) = 0, 0 < t < T.
(2.12)
Then,
B (u,p)
-Ti2 (t) T2 (t )
p( t ,0)-T- (t )(x ( t ,0)
In this case, the following theorem is valid:
Theorem 2.2. Let the initial-boundary problem (2.1), (2.2), (2.11) be such that there exists a test function ^e 0(no^ satisfying the boundary conditions (2.12) and also the following inequality
7 П0,-)< J
- T-2 (t) p (t, 0) - T2 (t) p (t, 0) + T- (t) px (t, 0)
dt +
+
-J«0 (*)t)\t=0dx. (2.13)
0
Then the problem (2.1), (2.2), (2.11) does not admit a global-in-time solution in n0l.
References
1. Kilbas A.A., Srivastava H.M. and Trujillo J.J. Theory and Applications of Fractional Differential Equations, Elsevier, North-Holland, 2006.
2. Hilfer R. Applications of Fractional Calculus in Physics. World Scientific, Singapore, 200, p.87 and p.429.
3. Hilfer R. Experimental evidence for fractional time evolution in glass materials, Chem. Physics. 284 (2002), 399408.
4. Ahmed Alsaedi, Mokhtar Kirane and Berikbol T.Torebek (2020) Blow-up smooth solutions of the time-fractional Burger equation, Questiones Mathematical, 43:2, 185-192, D0I:10.2989/16073606.2018.1544596.
5. Burger J.M. A Mathematical Model Illustrating the Theory of Turbulence, Adv.in Appl. Mech. I, pp.171-199, Academic Pres, New York, 1948.
6. Pokhozhaev S.I. Essentially nonlinear capacities induced by differential operators. Dokl.Ros. Akad. Nauk. 357(5) (1997), 592-594.
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