Differential Equations, Vol. 39, No. 2, 2003, pp. 267–274. Translated from Differentsial’nye Uravneniya, Vol. 39, No. 2, 2003, pp. 246–252. c 2003 by Khei. Original Russian Text Copyright
ORDINARY DIFFERENTIAL EQUATIONS
Necessary Conditions for the Existence of Global Solutions of Systems of Higher-Order Nonlinear Ordinary Differential Equations and Inequalities Dzh. Khei Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia Received June 14, 2002
INTRODUCTION In the present paper, we consider systems of nonlinear differential equations and inequalities of the general form dk1 x/dtk1 ≥ a(t)|y|r |x|s , dk2 y/dtk2 ≥ b(t)|y|p |x|q . (1) Recent results on the corresponding equations and systems can be found in [1–7] (see also the bibliography therein). A detailed review of the publications until 1990 is given in [8]. As to the initial data, in the present paper, we only assume that dk1 −1 x/dtk1 −1 (t0 ) + dk2 −1 y/dtk2 −1 (t0 ) > 0. Note that our technique permits one to consider problems with various sets of initial data depending on the choice of test functions. Recently, Mitidieri and Pokhozhaev [9] suggested a new approach to obtaining necessary conditions for the existence of global solutions of nonlinear differential inequalities, which also permits one to obtain a priori estimates for the blow-up time of the solution. A number of results in this direction was obtained in [10], but the problem on the corresponding results for higher-order inequalities and systems with general right-hand side remains open. Estimates for the existence time of solutions of equations and inequalities with arbitrarily many terms on the right-hand side were obtained in [11]. In the present paper, we generalize these results to the case of systems of equations and inequalities. 1. A MODEL PROBLEM Consider the simplest case s, p = 0 of system (1) of two differential inequalities with variable coefficients. For example, consider the system dk1 x/dtk ≥ a(t)|y|r ,
dk2 y/dtk ≥ b(t)|x|q ,
(1.1)
where q, r > 1, t ≥ 1, a(t) > Atα , b(t) > Btβ , and A, B > 0, with the initial conditions dk1 −1 x(1)/dtk1 −1 = xk1 −1 (1) > 0,
dk2 −1 y(1)/dtk2 −1 = yk2 −1 (1) > 0.
(1.2)
Remark 1.1. We do not impose any conditions on the initial data (in particular, on the values of lower derivatives) other than (1.2). Remark 1.2. Instead of (1.2), one can consider the less restrictive condition xk1 −1 (1) + yk2 −1 (1) > 0. The following assertion is the main result of this section. c 2003 MAIK “Nauka/Interperiodica” 0012-2661/03/3902-0267$25.00
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Theorem 1.1. Let
or
Q ≡ rq (k1 − 1) + r (k2 + β) + (α + 1) ≥ 0 P ≡ rq (k2 − 1) + q (k1 + α) + (β + 1) ≥ 0.
(1.3)
Then the system of nonlinear ordinary differential inequalities (1.1), (1.2) has no global solutions i.e., solutions defined on the interval t ∈ [1, ∞) . Proof. Using the technique introduced in [9, 12] and applied in [11], we introduce a test function ϕ(t) ∈ C ki [1, T ], ki = max (k1 , k2 ), by setting ϕ(t) =
1 for t ∈ [1, T /2] 0 for t ∈ [t ≥ T ],
0 ≤ ϕ(t) ≤ 1 for
t ∈ [1, T ].
(1.4)
The test function ϕ(t) must also satisfy an additional condition given below [see (1.15)]. By multiplying the differential inequalities (1.1) by ϕ(t) and by integrating the resulting relation from 1 to T , we obtain ZT
dk1 x ϕ(t) k1 dt ≥ dt
1
ZT
ZT r
ϕ(t)a(t)|y| dt, 1
dk2 y ϕ(t) k2 dt ≥ dt
1
ZT ϕ(t)b(t)|x|q dt.
(1.5)
1
Let us integrate by parts on the left-hand side in (1.5): ZT 1
ZT k1 kX 1 −1 i (k1 −1−i) T x dk1 x d ϕ(t) i dϕ (t) d k1 ϕ(t) k1 dt = (−1) + (−1) x dt, i (k1 −1−i) dt dt dt dtk1 1 i=0 1
ZT
k2
ϕ(t)
d y dt = dtk2
1
(1.6)
ZT k2 i (k2 −1−i) T y d ϕ(t) i dϕ (t) d k2 (−1) + (−1) y dt. dti dt(k2 −1−i) dtk2
kX 2 −1 i=0
1
1
Since ϕ(1) = 1, ϕ(T ) = 0, and di ϕ(t)/dti = 0, i ∈ [1, . . . , k], for t = 1 and t = T by the definition of the function ϕ(t), it follows from (1.6) that ZT
ZT
dk1 x ϕ(t) k1 dt = −xk1 −1 (1) + (−1)k1 dt
x
1
1
ZT
ZT
dk2 y ϕ(t) k2 dt = −yk2 −1 (1) + (−1)k2 dt
1
dk1 ϕ(t) dt, dtk1 (1.7)
dk2 ϕ(t) y dt. dtk2
1
By substituting these expressions into (1.5), we obtain ZT −xk1 −1 (1) + (−1)
k1
dk1 ϕ(t) x dt ≥ dtk1
1
ZT −yk2 −1 (1) + (−1)k2 1
ZT ϕ(t)a(t)|y|r dt, 1
dk2 ϕ(t) y dt ≥ dtk2
(1.8)
ZT ϕ(t)b(t)|x|q dt. 1
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We introduce the following notation: ZT
ZT q
X=
ϕ(t)b(t)|x| dt,
ϕ(t)a(t)|y|r dt,
Y =
1
1
0 ZT k2 d ϕ(t) r |ϕ(t)a(t)|1−r0 dt, A(T ) = k dt 2
(1.9)
1
0 ZT k1 d ϕ(t) q |ϕ(t)b(t)|1−q0 dt. B(T ) = k 1 dt 1
By applying to the H¨ older inequality to the second terms on the left-hand sides in (1.8), we obtain 0
Y ≤ X 1/q B 1/q − xk1 −1 (1),
0
X ≤ Y 1/r A1/r − yk2 −1 (1),
whence it follows that 1/q 0 0 0 0 Y ≤ Y 1/r A1/r B 1/q − xk1 −1 (1) = Y 1/(qr) A1/(qr ) B 1/q − xk1 −1 (1), 1/r 0 0 0 0 X ≤ X 1/q B 1/q A1/r − yk2 −1 (1) = X 1/(rq) B 1/(rq ) A1/r − yk2 −1 (1).
(1.10)
(1.11)
By applying the parametric Young inequality to the first terms on the right-hand sides of the relations (1.11), we obtain the estimates 0
0
0
0
Y 1/(qr) A1/(qr ) B 1/q − xk1 −1 (1) ≤ Y /(rq) + Ap/(qr ) B p/q /p − xk1 −1 (1), 0
0
0
0
X 1/(rq) B 1/(rq ) A1/r − yk2 −1 (1) ≤ X/(rq) + B p/(rq ) Ap/r /p − yk2 −1 (1),
(1.12)
where 1/(rq) + 1/p = 1, rq, p > 1; thus, inequalities (1.11) acquire the form 0
0
Y ≤ Ap/(qr ) B p/q − pxk1 −1 (1), α
0
0
X ≤ B p/(rq ) Ap/r − pyk2 −1 (1).
(1.13)
β
By assumption, a(t) > At and b(t) > Bt ; therefore, 0 ZT k2 d ϕ(t) r 0 |ϕ(t)Atα |1−r dt, A(T ) ≤ dtk2
0 ZT k1 d ϕ(t) q 0 ϕ(t)Btβ 1−q dt. B(T ) ≤ dtk1
1
(1.14)
1
To estimate these integrals, we introduce a function ϕ0 (τ ) such that ϕ(t) = ϕ0 (τ ), where τ = t/T , ϕ0 (τ ) = 1 for 1/T ≤ τ ≤ 1/2, and ϕ0 (τ ) = 0 for τ ≥ 1, ϕ0 (τ ) ∈ C ki [1/T, 1], ki = max (k1 , k2 ). One can readily show that there exists a test function of this form satisfying the conditions 0 Z1 k2 d ϕ0 (τ ) r 0 α 1−r C0 ≡ dτ < ∞, dτ k2 |ϕ0 (τ )Aτ | 1/2
0 Z1 k1 d ϕ0 (τ ) q 0 ϕ0 (τ )Bτ β 1−q dτ < ∞. D0 ≡ dτ k1
(1.15)
1/2
By performing a change of variable in (1.14) with regard to the relations t = T τ , dt = T dτ , dϕ/dt = (dϕ0 /dτ ) dτ /dt = T −1 dϕ0 /dτ , and dki ϕ/dtki = T −ki dki ϕ0 /dτ ki , we rewrite inequalities (1.14) in the form C0 D0 , B(T ) ≤ q0 (k1 +β)−(β+1) . (1.16) T r0 (k2 +α)−(α+1) T Now the values of the integrals in A(T ) and B(T ) depend only on the function ϕ(t) = ϕ0 (τ ). A(T ) ≤
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Then inequalities (1.13) can be rewritten in the form
ZT ϕ(t)a(t)|y| dt ≤ r
p/(qr0 )
C0 T r0 (k2 +α)−(α+1)
D0
p/q0 − pxk1 −1 (1),
T q0 (k1 +β)−(β+1)
1
ZT ϕ(t)b(t)|x|q dt ≤
p/(rq0 )
D0 T q0 (k1 +β)−(β+1)
C0 T r0 (k2 +α)−(α+1)
(1.17)
p/r0 − pyk2 −1 (1)
1
or
ZT
ϕ(t)a(t)|y|r dt ≤ C0(r−1)/(rq−1) D0r(q−1)/(rq−1) T −Q/(rq−1) − pxk1 −1 (1),
1
(1.18)
ZT q(r−1)/(rq−1)
ϕ(t)b(t)|x|q dt ≤ C0
(q−1)/(rq−1)
D0
T −P/(rq−1) − pyk2 −1 (1).
1
If Q > 0, then, passing to the limit in the first inequality in (1.18) as T → ∞, we obtain ZT ϕ(t)a(t)|y|r dt ≤ −pxk1 −1 (1).
lim
T →∞
(1.19)
1
Since the left-hand side of (1.19) is positive and the right-hand side is negative, we arrive at a contradiction, which shows that the condition Q > 0 is sufficient for the nonexistence of a global solution of system (1.1), (1.2). In a similar way, if P > 0, then for the second inequality in (1.18), we have ZT ϕ(t)b(t)|x|q dt ≤ −pyk2 −1 (1),
lim
T →∞ 1
and so the condition P > 0 is also sufficient for the nonexistence of a global solution of system (1.1), (1.2). Remark 1.3. The simultaneous validity of the inequalities Q < 0 and P < 0 is a necessary condition for the existence of a global solution of system (1.1), (1.2). Remark 1.4. By using the argument in [9], one can show that, in the limit case Q = 0, P = 0, the problem also has no global solutions. 2. ESTIMATES FOR THE SOLUTION EXISTENCE TIME AND SHARPNESS OF RESULTS We use Theorem 1.1 to estimate the solution existence time for the system of nonlinear ordinary differential inequalities (1.1), (1.2). Following [13], we refer to the supremum of T for which there exists a solution of the differential inequalities (1.1), (1.2) as the maximum solution existence time Tmax . If Tmax = +∞, then problem (1.1), (1.2) has a global solution . If Tmax < +∞, then the solution of problem (1.1), (1.2) blows up in finite time. The following assertion is the main result of this section. Theorem 2.1. If the system of nonlinear ordinary differential inequalities (1.1), (1.2) satisfies condition (1.3), then Tmax < Test = min (T1 , T2 ) , where (rq−1)/Q −1 (r−1)/(rq−1) r(q−1)/(rq−1) T1 = C0 D0 (pxk1 −1 (1)) , (2.1) (rq−1)/P −1 T2 = C0(q−1)/(rq−1) D0q(r−1)/(rq−1) (pyk2 −1 (1)) . DIFFERENTIAL EQUATIONS
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Proof. We again consider inequalities (1.18). Since the left-hand sides of these inequalities are positive, it obviously follows that system (1.1), (1.2) has no global solution if the right-hand sides of inequalities (1.18) are negative, which obviously implies the desired assertion. Let us show by examples that the results of Section 1 are sharp, i.e., if Q < 0 and P < 0, then the system may have a global solution. For example, if k1 , k2 = 1, then the conditions Q < 0 and P < 0 acquire the form r(1 + β) + (α + 1) < 0,
q(1 + α) + (β + 1) < 0.
(2.2)
y(t) = C2 tλ2 ,
(2.3)
We seek a solution in the form x(t) = C1 tλ1 ,
where λ1 , λ2 > 0, C1 , C2 > 0, and t ∈ [1, ∞). Note that xk−1 (1), yk−1 (1) > 0. By substituting (2.3) into system (1.1), we obtain r q dx/dt = λ1 C1 tλ1 −1 = Atα C2 tλ2 , dy/dt = λ2 C2 tλ2 −1 = Btβ C1 tλ1 . (2.4) By equating the exponents λ1 − 1 = α + λ2 r and λ2 − 1 = β + λ1 q, we obtain λ1 = (r(β + 1) + (1 + α))/(1 − rq),
λ2 = ((β + 1) + q(α + 1))/(1 − rq).
(2.5)
Note that the numerators [by (2.2)] and the denominators in (2.5) (r, q > 1) are negative, i.e., λ1 , λ2 > 0, as desired. Therefore, the results of Section 1 are sharp for first-order problems. In a similar way, one can show that the results are sharp for systems of equations of arbitrary orders k1 and k2 . To this end, we match the exponents in the expressions " k −1 # 1 Y dk1 x λ1 −k1 α λ2 r C = (λ = At , 1 − i) C1 t 2t k dt 1 i=0 (2.6) " k −1 # 2 Y dk2 y λ2 −k2 β λ1 q = (λ2 − i) C2 t = Bt C1 t , dtk2 i=0 i.e., λ1 − k1 = α + λ2 r and λ2 − k2 = β + λ1 q, whence it follows that λ1 = (r (k2 + β) + k1 + α)/(1 − rq),
λ2 = (q (k1 + α) + k2 + β)/(1 − rq).
(2.7)
To verify the inequalities λ1 , λ2 > 0, it suffices to note that the conditions Q < 0 and P < 0 imply that r (k2 + β)+k1 +α ≤ Q < 0 and q (k1 + α)+k2 +β ≤ P < 0, which ensures the negativity of the numerators in (2.7). The denominators are also negative by virtue of the condition rq > 1. Qkj −1 The validity of the estimates i=0 (λj − i) > 0, j = 1, 2, follows from the expressions for λ1 − i and λ2 − i and the inequalities r (k2 + β) + k1 + α − i(1 − rq) ≤ Q < 0 for i = 1, . . . , k1 − 1 and q (k1 + α) + k2 + β − i(1 − rq) ≤ P < 0 for i = 1, . . . , k2 − 1, which imply that λ1 − i > 0, i = 1, . . . , k1 − 1, and λ2 − i > 0, i = 1, . . . , k2 − 1. 3. A FIRST-ORDER SYSTEM WITH MIXED RIGHT-HAND SIDE Consider a system of two equations in which the right-hand side contains the product of some powers of x and y. We first consider the system of first-order inequalities dx/dt ≥ a(t)|y|r |x|s ,
dy/dt ≥ b(t)|y|p |x|q ,
(3.1)
where t ≥ 1, a(t) > Atα , b(t) > Btβ , A, B > 0, s, p < 1, r/(1 − p), q/(1 − s) > 1, and xk−1 (1) = x(1) > 0, DIFFERENTIAL EQUATIONS
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(3.2)
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Theorem 3.1. Let (r/(1 − p))(1 + β) + (α + 1) ≥ 0
(q/(1 − s))(1 + α) + (β + 1) ≥ 0.
or
(3.3)
Then system (3.1), (3.2) has no global nontrivial nonnegative solutions. Proof. By multiplying the first and second inequalities in (3.1) by θxθ−1 and λy λ−1 , respectively, and by performing the change of variables u = xθ , x = u1/θ , v = y λ , y = v 1/λ , we reduce system (3.1) to the form du/dt ≥ θa(t)|y|r |x|s+θ−1 = θa(t)v r/λ u(s+θ−1)/θ , (3.4) dv/dt ≥ λb(t)|y|p+λ−1 |x|q+λ−1 = λb(t)v (p+λ−1)/λ uq/θ . We take θ and λ such that s + θ − 1 = 0 and p + λ − 1 = 0 (the negative values of θ and λ are not admitted by the assumption that s, p < 1), i.e., θ = 1 − s and λ = 1 − p. Then we obtain du/dt ≥ θa(t)v r/(1−p) ,
dv/dt ≥ λb(t)uq/(1−s) ,
(3.5)
i.e., system (3.1), (3.2) is reduced to the form considered in Section 1, and the use of Theorem 1.1 completes the proof of the theorem. One can show that this result is sharp. 4. A HIGHER-ORDER SYSTEM WITH MIXED RIGHT-HAND SIDE Let us generalize the results of Section 3 to higher-order systems. More precisely, we consider system (1), where t ≥ 1, a(t) > Atα , b(t) > Btβ , and A, B > 0, with the initial conditions (1.2). Theorem 4.1. Let 0<
p −r −q s p−r s−q , , , , , <1 sp − rq sp − rq sp − rq sp − rq sp − rq sp − rq
(4.1)
and let one of the following inequalities be satisfied : (sp − rq − s) (k1 − 1) − r (k2 + β) + (p − 1)(α + 1) ≥ 0, (sp − rq − p) (k2 − 1) − q (k1 + α) + (s − 1)(β + 1) ≤ 0.
(4.2)
Then system (1), (1.2) has no global nontrivial nonnegative solutions. Proof. By analogy with Section 1, we introduce a test function ϕ, multiply by it the original system, and integrate by parts in the resulting relations, thus obtaining ZT −xk−1 (1) + (−1)
k1
dk1 ϕ x dt ≥ dtk1
ZT ϕa(t)|y|r |x|s dt,
1
1
ZT
ZT
−yk−1 (1) + (−1)k2
k2
d ϕ y dt ≥ dtk2
1
(4.3) ϕb(t)|y|p |x|q dt.
1
We introduce the change of variables u = y r xs ,
v = y p xq ;
(4.4)
y = u%1 v %2 ,
(4.5)
the inverse change of variables has the form x = uθ1 v θ2 ,
where θ1 = p/(sp − rq), θ2 = −r/(sp − rq), %1 = q/(rq − sp), and %2 = −s/(rq − sp). DIFFERENTIAL EQUATIONS
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In the new variables, inequalities (4.3) acquire the form ZT −xk−1 (1) + (−1)
k1
dk1 ϕ θ1 θ2 u v dt ≥ dtk1
ZT ϕa(t)|u|dt,
1
1
ZT
ZT
−yk−1 (1) + (−1)k2
k2
d ϕ %1 %2 u v dt ≥ dtk2
1
(4.6) ϕb(t)|v|dt.
1
We introduce the coefficients 1/θ1
= d1
1/θ2
= d2
c1 c2
1/%1
= ϕa(t),
c1 = ϕθ1 a(t)θ1 ,
d1 = ϕ%1 a(t)%1 ,
1/%2
= ϕb(t),
c2 = ϕθ2 b(t)θ2 ,
d2 = ϕ%2 b(t)%2
and set
ZT
ZT 1/θ c1 1 |u|dt
U= 1
1
ZT
ZT 1/θ c2 2 |v|dt
V = 1
ZT C0 =
ZT 1/% d1 1 |u|dt
=
=
ϕa(t)|u|dt, 1
ZT 1/% d2 2 |v|dt
=
(4.7)
1
=
ϕb(t)|v|dt, 1
k1 1/(1−θ1 −θ2 ) d ϕ −1/(1−θ1 −θ2 ) (c1 c2 ) dt, dtk1
(4.8)
1
ZT k2 1/(1−%1 −%2 ) d ϕ −1/(1−%1 −%2 ) D0 = k2 (d1 d2 ) dt. dt 1
We require that 0 < θ1 , θ2 , %1 , %2 < 1, 0 < θ1 + θ2 < 1, and 0 < %1 + %2 < 1. These conditions impose constraints on the original powers of x and y in system (1). In particular, 0 < p/(sp − rq), −r/(sp − rq), −q/(sp − rq), s/(sp − rq) < 1,
(4.9)
which is valid by virtue of assumption (4.1). Note that if sp − rq > 0, then relation (4.9) implies that s, p > 0 and r, q < 0; if sp − rq < 0, then s, p < 0 and r, q > 0. By assumption, sp − rq 6= 0. By applying the parametric Young inequality to (4.6), we obtain n n0 θ (1−%1 −%2 )/(1−%2 ) −xk−1 (1) + (1/n) C01−θ1 −θ2 D02 + (1/n0 ) U (θ2 %1 +θ1 (1−%2 ))/(1−%2 ) ≥ U, (4.10) m m0 −yk−1 (1) + (1/m) D01−%1 −%2 C0%1 (1−θ1 −θ2 )/(1−θ1 ) + (1/m0 ) V (%1 θ2 +%2 (1−θ1 ))/(1−θ1 ) ≥ V, where 1/n + 1/n0 = 1/m + 1/m0 = 1 and n0 , m0 > 1. Note that we require the validity of the constraints 0 < (θ2 %1 + θ1 (1 − %2 )) / (1 − %2 ) , (%1 θ2 + %2 (1 − θ1 )) / (1 − θ1 ) < 1.
(4.11)
However, one can readily show that the change of variables (4.4) can be modified so as to ensure the validity of (4.11). In particular, if we set uδ = y r xs and v δ = y p xq , then δ, 0 < δ < 1, can be chosen so as to satisfy condition (4.11). Further, by using the relations 1/n = 1 − 1/n0 and 1/m = 1 − 1/m0 , we obtain n θ (1−%1 −%2 )/(1−%2 ) −nxk−1 (1) + C01−θ1 −θ2 D02 ≥ U, m % (1−θ1 −θ2 )/(1−θ1 ) −myk−1 (1) + D01−%1 −%2 C0 1 ≥ V. DIFFERENTIAL EQUATIONS
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By using the inequalities a(t) > Atα and b(t) > Btβ , by choosing a test function ϕ0 (τ ) as in Section 1, and by performing an appropriate change of variables and a number of arithmetic computations, we obtain the desired assertion by analogy with the proof of Theorem 1.1. Remark 4.1. For rq = 0, we obtain the following condition for the nonexistence of solutions: α ≥ −s (k1 − 1) − 1, β ≥ −p (k2 − 1) − 1, which coincides with the result in [11]. ACKNOWLEDGMENTS The author is grateful to S.I. Pokhozhaev for the statement of the problem and useful discussion of results. REFERENCES 1. Kon’kov, A.A., Izv. RAN. Ser. Mat., 2001, vol. 65, no. 2, pp. 81–125. 2. Kiguradze, I.T., Nachal’nye i kraevye zadachi dlya sistem obyknovennykh differentsial’nykh uravnenii. I (Initial and Boundary Value Problems for Systems of Ordinary Differential Equations. I), Tbilisi, 1997. 3. Rabtsevich, V.A., Differents. Uravn., 2000, vol. 36, no. 1, pp. 85–93. 4. Rabtsevich, V.A., Differents. Uravn., 2000, vol. 36, no. 12, pp. 1642–1652. 5. Rabtsevich, V.A., Mem. Diff. Equations Math. Phys., 2000, vol. 20, pp. 149–153. 6. Rabtsevich, V.A., Dokl. NAN Belarusi, 2001, vol. 45, no. 1, pp. 13–16. 7. Rabtsevich, V.A., Dokl. NAN Belarusi, 2002, vol. 46, no. 1, pp. 15–17. 8. Kiguradze, I.T. and Chanturiya, T.A., Asimptoticheskie svoistva reshenii neavtonomnykh obyknovennykh differentsial’nykh uravnenii (Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations), Moscow, 1990. 9. Mitidieri, E. and Pokhozhaev, S.I., Dokl. RAN , 1998, vol. 359, no. 4, pp. 456–460. 10. Mitidieri, E. and Pokhozhaev, S.I., Apriornye otsenki i otsutstvie reshenii nelineinykh differentsial’nykh uravnenii i neravenstv v chastnykh proizvodnykh (A Priori Estimates and Absence of Solutions for Nonlinear Partial Differential Equations and Inequalities), Tr. Mat. Inst. im. Steklova, Moscow, 2001, vol. 234. 11. Khei, Dzh., Differents. Uravn., 2002, vol. 38, no. 3, pp. 344–350. 12. Mitidieri, E. and Pokhozhaev, S.I., Tr. Mat. Inst. im. Steklova, 1999, vol. 227, pp. 192–222. 13. Levine, H.A., SIAM Reviews, 1990, vol. 32, pp. 371–386.
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