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On the behavior of the solution of a nonlinear polytropic filtration problem with a source and multiple nonlinearities
Z. R. Rakhmonov, A. I. Tillaev
National University of Uzbekistan, Applied Mathematics and Computer Analysis, Universitet, 4, Tashkent, 100174, Uzbekistan zraxmonov@inbox.ru, tillayev1@mail.ru
PACS 02.30.Jr, 02.30.Mv, 02.60.-x, 11.10.Jj, 11.10.Lm DOI 10.17586/2220-8054-2018-9-3-323-329
In this paper, we study the global solvability and unsolvability conditions of a nonlinear filtration problem with nonlinear boundary flux. We establish the critical global existence exponent and critical Fujita exponent of nonlinear filtration problem in inhomogeneous medium. An asymptotic representation of the solution with a compact support is obtained, which made it possible to carry out a numerical experiment. Keywords: filtration, global solutions, blow-up, critical curve, asymptotic behavior, numerical analysis.
Received: 23 April 2018
1. Introduction
Consider the following parabolic equation
du
du d
P (x) ât = dx
p-2 du
with nonlinear boundary flux
and initial value condition
dx
du dx
dX| + P (x) u^, (x,t) G R+ x (0, , (1)
P-2 du
— (0,t)= uq (0,t), t> 0, (2)
dx
u (x, 0) = m0 (x) > 0, x G R+, (3)
where p > 2, p, q > 0, p(x) = x-n, n g R, u0(x) - is a bounded, continuous, nonnegative and nontrivial initial data.
Equations (1) occur in various fields of natural science [1,3-5]. For example, equation (1) arises in the mathematical modeling of thermal conductivity of nanofluids, in the study of problems of fluid flow through porous media, in problems of the dynamics of biological populations, polytropic filtration, and the formation of structures in synergetics and in nanotechnologies, and in a number of other areas [1,4].
Equation (1) is called a parabolic equation with inhomogeneous density [1] and the case p > 2 corresponds to the slow diffusion equation [2]. The problem (1) - (3) has been intensively studied by many authors (see [2,6-17] and references therein) in different values of numerical parameters.
The authors in [2] considered the Cauchy problem for the equation (1) at n = 0, proving that if 0 < p < 1, then all solutions of (1) are global in time, while for p > 1 there are solutions with finite time blow-up. Also in this paper the following statements were proved:
- If 1 < p < 2p - 1, then every solutions of Cauchy problem (1), (3) blow up in finite time;
- If p > 2p — 1, then the Cauchy problem (1), (3) admits nontrivial global solutions with small initial data. The value p = 2p — 1 is Fujita type critical exponent.
Some properties of the solutions of (1)-(3) at p(x) = 1 were studied in [9] by Zhongping Li, Chunlai Mu and Li Xie. They obtained the critical global existence exponent and the critical Fujita exponent by constructing sub and supersolutions.
In [6,7], the authors studied the unboundedness of the solution for the following reaction-diffusion model with nonlocal nonlinearities
ut = Aum + u^, (x,t) G Q x (0,T),
du
— = uq, (x,t) G dQ x (0,T), dn
u (x, 0) = uo (x), x G Q,
where Q e RN is a bounded domain. Authors obtained that all positive solutions exist globally if and only if p, q < 1 when m > 1 and if and only if p < 1, q < 2m/(m + 1) when m < 1.
The authors of the work [15] have studied problem (1)-(3) at n = 0. They obtained the principal terms for the asymptotic of self-similar solutions.
As it is well known, degenerate equations need not possess classical solutions. Therefore, its solution is understood in the generalized sense.
Definition 1. The function u (x, t) is said to be the weak solution to problem (1)-(3) in Q = {R+ x (0, T)},
if 0 < u(x,t) e C (Q),
du dx
p-2
du
— e C (Q) and if it satisfies (1)-(3) in the sense of distribution in Q, where
T > 0 is the maximal existence time. 2. Main results
In this section, we show the critical global existence curve and critical Fujita curve, and characterize when all solutions to the problem (1)-(3) are global in time or they blow up.
(2 — n) (p — 1)
Theorem 1. If 0 < p < 1 and 0 < q <- then each solution of problem (1)-(3) is global in time.
p — n
Proof. We look for a globally defined in time supersolution of the following self-similar form
x(x,t) = eBt (D + e-LÇ) , £ = xe
-Jt
where D > ||u0|U, №->/. — e-L = D, B = "" " — "+ 'D + ^, J = BÇP-=>.
D p — n
After computation, we have:
p (x) = (fieBt(D + e-L«) + LJe(B-J)t(1 + x)) e-"Jt£-n > BDe(B-nJ)t,
/ = e^Bt(D + e-L«fe-nJt£-n < e(^B-nJD + e
d
dx
du dx
du
dx
p-2 du
p-2
dx> = LP (p—1) e
du
(B(p-1)-Jp)te-L(p-1)Ç
^ (0,t) = LP-1e(B-J)(P-1)te-B(P-1). dx
Now, we shall show that the function u (x,t) is a supersolution of problem (1)-(3). According to the comparison principle it must satisfy the following inequality:
du d
p (x) at > dx
du p-2du^
dx dx y
du p-2 du
dx dx
(x,t) e R+ x (0, +to) ,
(4)
(5)
(2 — n) (p — 1)
It is not difficult to verify that if 0 < P < 1 and 0 < q <-, by definition B, L, D, J the inequalities
p — n
(4) and (5) are valid. Hence, u (x, 0) > u0 (x) and u (0,0) > u0 (0). Thus, by comparison principle, Theorem 1 is proved.
Remark 1. Theorem 1 shows that the critical global existence exponent of the problem (l)-(3) is
P =1, 0 < q < (2 — n)(P — 1) 1 U Rq =(2 — n)(P — 1) , P < ll.
p — n
p — n
(2 — n) (p — 1)
Theorem 2. If p < 1 and q >- then every solution of problem (1)-(3) blows up in time.
p — n
Proof. We shall seek a blow up subsolution of the self-similar form:
where a
1 — P
p — 1 — p
(1 — p) (p — n)
u- (x,t)= tCT(n) , n = xt 7
and ^ (n) is the solution of the following problem
d dn
dn
P-2
d^J + Yn1-n ? — an-> + n-V = 0,
dn dn
(6)
q
1
Y
P-2.
't (0)
(0).
In order u- (x, t) to be a subsolution of problem (1)-(3) function ^ (n) should satisfy following:
P-2
d dn
dn
+ Yn1-n dr - + n-V > 0,
d^ / dn
dn
P-2
^ (0) < ^ (0). dn
We set:
^ (n) = E ( a — n p-1
p-i p-n \ p-2
325 (8)
(9) (10) (11)
where E, a are constants to be determined. It is easy to see that
p-i
p — nF i—if p—^p-2
V (n) =--En p-1 a — n p-1
p — 2
d dn
dn
P-2
dn
^ = - EP-
1 / p — n p — 2
p-1
p-i p-n ^ p-2 a — n p-i
dn
P-2
| (0) = 0.
By taking
EP-
2/ P — n
P-1
(p-i)Q-i)
> 7, a p-2
P- 1
> Ep-2-^+ E1-^
p — 2 p — 2
it is easy to check that if fi < 1, then (9) and (10) are valid. Thus u- (x, t) is a subsolution of the problem (1)-(3) with every nontrivial initial data.
(2 — n) (p — 1)
Theorem 3. If fi > 2p — 1 and q < - then every solution of problem (1)-(3) blows up in time.
p — n
Proof. In this case we prove that the flux condition makes the solution large enough to be in the set of initial data for which the reaction term alone is enough to cause blow up. We consider the self-similar subsolution of the problem (1)-(3) without a source:
ub (x,t)= tag (0 , e = xt-Y (12)
p — 1 p — 1 — q , .
where a = --——-------, 7 = --——-------. Let us show that function ub (x,t)
(p — 1) (2 — n) — q (p — n) (p — 1) (2 — n) — q (p — n)
defined by (12) is a subsolution. Then, according to the comparison principle, function g (£) should satisfy the
following inequalities:
d
de
dg
de
P-2
dg \ , ¿1-n dg /îw n
^ + 7e -jT — ag (e) > 0,
de
dg
de
P-2
de
de (0) < gq (0).
(13)
(14)
p-i \ p-2
Let us set
g(e) = b (b — e)+
where b and B are positive constants to be determined. After some computations, we have
g' (e) = —b^—1 (b — e)+^,
p — 2
dgr P-2 dg
de de
( dg P-2
'I de
p- nP-1 p-i p—1 <b -«>+"
\ 1A p — 1
— = BP dd
p — 2
P
p-2
(b — e)+
+
n
1
Noting that
7ei-„ — a; (e) > —(b — e)p-2 b — aB (b — e)p-2 b =
—Bb (b — o
1
p-2
p — 1
7P—2 + a
By taking:
-2/'p — n > + «
p — 2
p — 2
-i-q
p — 1 p — 2
p-i
, (q i)(p 1) < b p-2
(2 — n) (p — 1)
and q <-, it is easy to check that (13) and (14) are valid. It follows from the comparison principle
p — n
that the problem (1)-(3) exists a solution blowing up in a finite time.
Theorem 4. If P > 1 and q < (2 — n) (p — 1), then every solution of problem (1)-(3) blows up in time.
Remark 2. Theorem 3 and Theorem 4 shows that the critical Fujita curve of the problem (1)-(3) is {P = 2p — 1, q > 2 (p — 1)}U {P > 2p — 1, q = 2 (p — 1)}.
(p_n) q_p —+ 1
Theorem 5. If P >--- and q > (2 — n) (p — 1) then each solution of problem (1)-(3) is global in
time.
p — 1
Theorems 4 and 5 can be proved in the same manner as it was done in [13,16]. Let us show the asymptotics of self-similar solutions.
„ (2 - n) (p — 1) The case 1 /(p — 1) < fl < 1, q > -. Consider the following self-similar solution of problem
(1)-(3):
where
1 — P
p - n
ui (x,t) = (o, e = xt-Y,
p — 1 — P
Y = 7-—-^ and f (e) is solution of the following problem
(p — n)(1 — P)
d de
df
de
p-2
^1 + Ye1-n § — «e->+e-V
df de
(15)
(16) (17)
. - , (18)
p — n +
where a > 0, (y) + = max(0, y). We show that the function (18) is the asymptotics of the solutions of problem (16), (17).
Theorem 6. The compactly supported solution of problem (16), (17) has the asymptotic
df p- 2 df
de
de
5=0
fq (0)
Consider the function
N P=1
I p — 2 -L rp-^\p-2
f (e) = a--y p-1 ep-1
f (e) = f(e)(1 + 0(1)),
when e ^
a ( p — n)
(p-i)/(p-n)
> — 2) y 1/(p-1),
Proof. We seek a solution of equation (16) in the following form
f = f(e) w(t ^
with t = — ln (a — be, b = ^—^y 1/(p-1), where t ^ at e ^ (a/b) Substituting (19) into equation (16) with regard to (18), it takes the form:
(19)
(p-1)/(p-n)
dT(L1w)p-1 + (t) — (L1w)p-1 + k2L1W — (T) — (T) = 0, (20)
w w/ ,, . 8(p-1)-1 I .
¿1 (t) = e-T, ^2 (t) = e-T/ (a — e-T), ^3 (t) = e--p--t (a — e-T),
where L1w = -----
p — 2 p — 1
p — 1 Y (p — 1)
k
b ( p — n)
k2
(b (p — n))
p-1
k3
a(p — 1) bp-1(p — n)p
k4
p — 1
(b (p — n))p
1
We note that the study of the solutions of the last equation is equivalent to the study of those solutions of equation (1), each of which satisfies the inequalities in some interval [t0 , +œ):
/ \ w (t) w' (t) , w (t) > 0, —^--^ =0.
p — 2 p — 1
We verify that the solution w (t) of the equation (20) has a finite limit wo or not at t ^ Let v (t)
(Liw)p-1. Then, for the derivative of the function v (t), we have:
v' = — ( ki^i (t) — p—1 ) v — k2^iw + (t) + k4w^^3 (t) .
V p — 2/
We introduce the auxiliary function to analyze the solutions of the last equation:
p — 1N
e (t,m) = — k1 ¿1 (t ) —
p — 2
^ — k2^1W + &3W^2 (t) + ¿3 (t)
(21)
where ^ G R. Hence, it is easy to see that for each value of ^ the function 6 (t, keep the sign on some interval [t1, c [t0, and for all t g [t1, holds one of the inequalities v'(t) > 0, v'(t) < 0. Then, analyzing (21) with the help of Bohl's theorem [13], we conclude that there for the function v (t) exists a limit at t g [t1, It is easy to see that
lim (t) ^ 0, lim t2 (n) ^ 0, lim t3 (n) ^ 0
(P-1)/(P-n)
at e ^ (a/b)-P algebraic equation for w
Then taking into account the last limit and w' = 0 from (20), we obtain following
(p - 1)
p — 2
P-1
— k2W = 0.
From this equation, we get that w = 1, thus we have f (e) = f (e) (1 + °(1)).
Th O t / (2 - n)(p — 1)
The case p > 2p — 1, q < -.
(p - n)
Theorem 7. The compactly supported solution of problem (1)-(3) has the asymptotic:
fp — 2 \ 1/(P-2) p — 2
where C = ( -r 67
u (x,t) = cta 3(e) (1 + 0(1)),
g is above defined function.
p — 1"7 B (p — 1)'
The critical case (p — n) q = (2 — n) (p — 1). In this case, we consider following exponential form solution of problem (1)-(3):
u4 (x, t) = ea(t-TV (n), n = xe-Y(i-T>,
p_n p_2
where a = --7-, 7 = --- and t (n) is the solution of the problem
2p - 1
2p - 1
d dn
d¿ dn
P-2
+ 7^1-n ^ — = 0,
dn
d¿ P- 2 d¿
dn dn n=o
dn
= ¿q (0)
(22) (23)
We take:
¿g (n) = ^ ( p 1
P-1
P \ P-2
p—V _ nJ + ,
where D > 0.
Theorem 8. The compactly supported solution of problem (22)-(23) has the asymptotic
t (n) = C<Mn)(1 + o(1)),
at n ^ DP
p — 1
where C =
p — 1
7
1/(P-2)
vp — 27 Vp _ 2
Theorem 7 and Theorem 8 are proved similarly to the proofs of Theorems 6. Numerical experiments. It is known that choosing a suitable initial approximation preserving nonlinear properties is very important in numerical analysis. For this purpose, a computer experiment was conducted on the basis of the above qualitative properties of solutions for the case of global solvability. Since equation (1) approximated the
w
P
2
second order of accuracy respect to x and first order of accuracy respect to t. The iterative process for numerical modeling was constructed, in the inner steps of iteration the node values are calculated by the Thomas algorithm. It is well known that, the iteration methods required a good initial approximation, which quickly converge to the exact solution and retain the qualitative properties of the studied nonlinear processes. This is the difficulty of the numerical analysis of the nonlinear problem. It depends on the values of the numerical parameters of the equation and can be overcome by successful choice initial approximations. We present some results of numerical experiments.
Figures 1-3 show that filtration process dependents on the density of the medium. Numerical experiments show rapid convergence of iterations to the exact solution. It is due to the choice of a suitable initial approximation. The number of iterations is not more than 5 for various values of numerical parameters.
Fig. 1. Numerical solution of (1)-(3): P = 2.1, p = 2.65, n = 1.2, q = 2.5
Fig. 2. Numerical solution of (1)-(3): P = 2.1, p = 2.35, n = 3.5, q = 3
Fig. 3. Numerical solution of (1)-(3): P = 1.6, p = 2.25, n = 0.8, q = 2.5
References
[1] Kalashnikov A.S. Some problems of the qualitative theory of nonlinear degenerate second-order parabolic equations. Russian. Math. Surveys, 1987, 42 (2), P. 169-222.
[2] Galaktionov V.A. On global nonexistence and localization of solutions to the Cauchy problem for some class of nonlinear parabolic equations. Zh. Vychisl. Mat. Mat. Fiz., 1983, 23, P. 1341-1354. English transl.: Comput. Math. Math. Phys. 1983, 23, P. 36-44.
[3] M.Aripov. Standard Equation's Methods for Solutions to Nonlinear problems, Tashkent, FAN, 1988, 138 p.
[4] Galaktionov V.A., Vazquez J.L. The problem of blow-up in nonlinear parabolic equations. Discrete and continuous dynamical systems, 2002, 8 (2), P. 399-433.
[5] Galaktionov V.A., Levine H.A. On critical Fujita exponents for heat equations with nonlinear flux boundary condition on the boundary. Israel J. Math., 1996, 94, P. 125-146.
[6] Pablo A.D., Quiros F., Rossi J.D. Asymptotic simplification for a reaction-diffusion problem with a nonlinear boundary condition. IMA J. Appl. Math., 2002, 67, P. 69-98.
[7] Song X., Zheng S. Blow-up and blow-up rate for a reaction-diffusion model with multiple nonlinearities. Nonlinear Anal., 2003, 54, P. 279-289.
[8] Li Z., Mu Ch. Critical exponents for a fast diffusive polytrophic filtration equation with nonlinear boundary flux. J. Math. Anal. Appl.,
2008, 346, P. 55-64.
[9] Zhongping Li, Chunlai Mu, Li Xie. Critical curves for a degenerate parabolic equation with multiple nonlinearities. J. Math. Anal. Appl.,
2009, 359, P. 39-47.
[10] Wanjuan Du, Zhongping Li. Critical exponents for heat conduction equation with a nonlinear Boundary condition. Int. Jour. of Math. Anal., 2013, 7 (11), P. 517-524.
[11] Mersaid Aripov, Shakhlo A. Sadullaeva. To properties of solutions to reaction-diffusion equation with double nonlinearity with distributed parameters. Jour. Sib. Fed. Univ. Math. Phys., 2013, 6 (2), P. 157-167.
[12] Aripov M., Rakhmonov Z. On the asymptotic of solutions of a nonlinear heat conduction problem with gradient nonlinearity. Uzbek Mathematical Journal, 2013, 3, P. 19-27.
[13] Rakhmonov Z. On the properties of solutions of multidimensional nonlinear filtration problem with variable density and nonlocal boundary condition in the case of fast diffusion. Journal of Siberian Federal University. Mathematics & Physics, 2016, 9 (2), P. 236-245.
[14] Aripov M., Rakhmonov Z. Estimates and Asymptotic of Self-similar Solutions to a Nonlinear Filtration Problem with Variable Density and Nonlocal Boundary Conditions. Universal Journal of Computational Mathematics, 2016, 4, P. 1-5.
[15] Aripov M.M., Rakhmonov Z.R. On the asymptotics of solutions of the heat conduction problem with a source and a nonlinear boundary condition. Computational technologies, 2015, 20 (2), P. 216-223.
[16] Aripov M.M., Matyakubov A.S. To the qualitative properties of solution of system equations not in divergence form of polytrophic filtration in variable density. Nanosystems: Physics, Chemistry, Mathematics, 2017, 8 (3), P. 317-322.
[17] Aripov M.M., Matyakubov A.S. Self-similar solutions of a cross-diffusion parabolic system with variable density: explicit estimates and asymptotic behavior. Nanosystems: Physics, Chemistry, Mathematics, 2017, 8 (1), P. 5-12.
