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Blow up of solutions to the initial boundary value problem for quasilinear strongly damped wave equations
J. Math. Anal. Appl. 403 (2013) 89–94
Contents lists available at SciVerse ScienceDirect
Journal of Mathematical Analysis and Applications journal homepage: www.elsevier.com/locate/jmaa
Blow up of solutions to the initial boundary value problem for quasilinear strongly damped wave equations Bilgesu A. Bilgin, Varga K. Kalantarov ∗ Department of Mathematics, Koç University, Rumelifeneri Yolu, Sariyer, Istanbul, Turkey
article
abstract
info
Article history: Received 31 August 2012 Available online 4 February 2013 Submitted by Kenji Nishihara
We obtain sufficient conditions on initial functions for which the initial boundary value problem for second order quasilinear strongly damped wave equations blow up in a finite time. © 2013 Elsevier Inc. All rights reserved.
Keywords: Strongly damped wave equation Blow up Concavity method Initial boundary value problem
1. Introduction We consider the following initial boundary value problem: utt − ∇
a0 + a|∇ u|m−2 ∇ u − b1ut = g (x, t , u, ∇ u) + |u|p−2 u,
u(x, t ) = 0,
x ∈ Ω , t > 0,
x ∈ ∂ Ω , t > 0,
u(x, 0) = u0 (x),
ut (x, 0) = u1 (x),
(1.1) (1.2)
x ∈ Ω,
(1.3)
where p > m ≥ 2,
a0 ≥ 0,
a, b > 0
(1.4)
are given numbers, and Ω ⊂ R is a bounded (or unbounded) domain with a sufficiently smooth boundary ∂ Ω . We assume that n
|g (x, t , u, p)| ≤ M1 (|p|m/2 + |u|p/2 ), where M1 > 0 is some positive number.
∀(x, t ) ∈ Ω × R+ , u ∈ R, p ∈ Rn ,
(1.5)
As far as we know the first result about blow up of solutions for a strongly damped equation is the result of H. Levine [7], where by using the concavity method introduced in [8] he got sufficient conditions for blow up of solutions to the Cauchy problem for a nonlinear differential-operator equation in a Hilbert space of the form Putt + Qut + Au = F (u), where P , A are positive symmetric operators and F (·) is a nonlinear operator that is the gradient of some functional G(·) such that (F (u), u) − kG(u) ≥ 0, ∀u ∈ D(F ) for some k > 2. This result was developed in [5,3] (see also [4]) for a more general class of differential-operator equations of the form Putt + Au + Qut = B(u) + F (u), where B(·) is a nonlinear operator subordinated to A.
∗
Corresponding author. E-mail address: [email protected] (V.K. Kalantarov).
0022-247X/$ – see front matter © 2013 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2013.01.056
90
B.A. Bilgin, V.K. Kalantarov / J. Math. Anal. Appl. 403 (2013) 89–94
Later on results about blow up of solutions to the strongly damped Kirchhoff equation utt − 1ut − (α + β∥∇ u∥2γ )1u − |u|q−2 u = 0 were obtained in [11,10] under some restrictions on α ≥ 0, β > 0, γ > 0 and q > 2. Our result obtained for solutions to the problem (1.1)–(1.3) can be considered as a development of results on the blow up of solutions to initial boundary value problems for quasilinear hyperbolic equations obtained in [9]. In what follows we will use the following notations: (·, ·), ∥ · ∥ will denote the inner product and norm of L2 (Ω ) and ∥ · ∥r will denote the standard norm of Lr (Ω ). The proof of blow up of solutions to the problem (1.1)–(1.3) is based on the following lemma. Lemma 1.1 ([5]). Suppose that a positive, twice-differentiable function Ψ (t ) satisfies for t > 0 the inequality
2 Ψ ′′ (t )Ψ (t ) − (1 + α) Ψ ′ (t ) ≥ −2C1 Ψ ′ (t )Ψ (t ) − C2 [Ψ (t )]2 ,
(1.6)
where α > 0, C1 , C2 ≥ 0. If Ψ (0) > 0, Ψ ′ (0) > −γ2 α −1 Ψ (0), and C1 + C2 > 0, then Ψ (t ) → +∞ as t → t1 ≤
1
ln
2 C12 + α C2
where γ1 = −C1 +
γ1 Ψ (0) + α Ψ ′ (0) , γ2 Ψ (0) + α Ψ ′ (0)
C12 + α C2 , γ2 = −C1 −
C12 + α C2 .
2. Blow up theorems By using a modification of the concavity method suggested in [5] first we prove the following theorem Theorem 2.1. Suppose that (1) the conditions (1.4) and (1.5) are satisfied, (2) the initial functions satisfy the conditions E0 :=
1 2
∥ u1 − λ u0 ∥ 2 + b
(u1 , u0 ) >
2
a0 + λ b 2
p+m+4 p+m−4
∥∇ u0 ∥2 +
a m
∥∇ u0 ∥m m +
− 1 ∥∇ u0 ∥ + (λ + 1) 2
λ2 2
1
∥u0 ∥2 − ∥u0 ∥pp ≤ 0,
p+m+4 p+m−4
p
∥ u0 ∥ 2 ,
(2.1)
(2.2)
where
λ := M1
2(m + ap) a(p − m)(p + m − 4)
.
(2.3)
Then there exists T0 ≤
2
(p + m)2 − 16
ln
(u1 , u0 ) +
b 2
p+m+4 p+m−4
(u1 , u0 ) −
b 2
p+m+4 p+m−4
m+4 − 1 ∥∇ u0 ∥2 + (λ + 1) pp+ ∥ u0 ∥ 2 +m−4 m+4 − 1 ∥∇ u0 ∥2 − (λ + 1) pp+ ∥ u0 ∥ 2 +m−4
such that
∥∇ u(t )∥2 → ∞ as t → T0− . Proof. For λ > 0 we make the change of variables u(x, t ) = v(x, t )eλt ,
x ∈ Ω , t ≥ 0,
in (1.1) and obtain the differential equation
vtt − ∇
a0 + λb + aeλ(m−2)t |∇v|m−2 ∇v + λ2 v + 2λvt − b1vt
= e−λt g (x, t , eλt v, eλt ∇v) + eλ(p−2)t |v|p−2 v
(2.4)
under the initial and boundary conditions
v(x, 0) = u0 (x),
vt (x, 0) = −λu0 (x) + u1 (x),
x ∈ Ω ; v|Ω = 0, t > 0.
(2.5)
B.A. Bilgin, V.K. Kalantarov / J. Math. Anal. Appl. 403 (2013) 89–94
91
Multiplying the Eq. (2.4) by vt in L2 (Ω ) we get
d dt
1 2
a0 + λ b
2
∥vt ∥ +
2
a λ2 1 ∥∇v∥ + eλ(m−2)t ∥∇v∥m + ∥v∥2 − eλ(p−2)t ∥v∥pp m m 2 p
2
+ 2λ∥vt ∥2 + b∥∇vt ∥2 −
aλ(m − 2) λ(m−2)t λ(p − 2) λ(p−2)t e ∥∇v∥m e ∥v∥pp m + m p
= e−λt g (x, t , eλt v, eλt ∇v), vt .
(2.6)
Using the assumption (1.5) and the Cauchy’s inequality with ε we estimate the last term above in the following way. e
−λt
λt
λt
|(g (x, t , e v, e ∇v), vt )| ≤ ε1 e
λ(m−2)t
λ(p−2)t
∥∇v∥ + ε0 e m m
p p
∥v∥ +
M12 4
1
ε1
+
1
ε0
∥vt ∥2 .
We define H (t ) :=
1 λ(p−2)t 1 a0 + λb a λ2 e ∥v∥pp − ∥vt ∥2 − ∥∇v∥2 − eλ(m−2)t ∥∇v∥m ∥v∥2 , m − p 2 2 m 2
(2.7)
and note that H (0) = −E0 (E0 is defined by (2.1)). Employing the last inequality, and using this notation, we obtain from (2.6) the following inequality d dt
H (t ) ≥ [λ(p − 2) − ε0 p] H (t ) +
+ +
λ2 2 a m
λ(p−m) 2p
1 2
[λ(p − 2) − ε0 p] + 2λ ∥vt ∥2
[λ(p − 2) − ε0 p]∥v∥2 + b∥∇vt ∥2 +
1
[λ(p − 2) − ε0 p] (a0 + bλ)∥∇v∥2
2
λ(p−2)t [λ(p − 2) − ε0 p − λ(m − 2)] eλ(m−2)t ∥∇v∥m ∥v∥pp m + ε0 e
− ε0 e Choosing ε0 =
λ(p−2)t
∥v∥ − ε1 e p p
aλ(p−m) 2m
and ε1 =
λ
λ(m−2)t
m m
∥∇v∥ −
M12
1
ε0
4
+
1
ε1
∥vt ∥2 .
we obtain from (2.8) the inequality
(m + ap)M12 H (t ) ≥ (p + m − 4)H (t ) + (p + m − 4) + 2λ − ∥vt ∥2 + b∥∇vt ∥2 . dt 2 4 2aλ(p − m) 2(m+ap) According to (2.3) λ = M1 a(p−m)(p+m−4) . Therefore we obtain from (2.9) d
d dt
H (t ) ≥
λ 2
(2.8)
λ
(p + m − 4)H (t ) + 2λ∥vt ∥2 + b∥∇vt ∥2 .
(2.9)
(2.10) λ
Since by (2.1) H (0) ≥ 0, we obtain from (2.10) that H (t ) ≥ e 2 (p+m−4)t H (0) ≥ 0. Therefore integrating (2.10) we get H (t ) ≥ 2λ
t
∥vτ (τ )∥2 dτ + b 0
t
∥∇vτ (τ )∥2 dτ .
(2.11)
0
Let us consider the function
Ψ (t ) = ∥v(t )∥ + 2λ 2
t
∥v(τ )∥ dτ + b 2
0
t
∥∇v(τ )∥2 dτ + C0 , 0
where v is a local solution of the problem (2.4)–(2.5) and C0 is a positive parameter to be chosen below. It is easy to see that
Ψ ′ (t ) = 2(v, vt ) + 4λ
t
(vτ , v)dτ + 2b 0
t
(∇vτ , ∇v)dτ + 2λ∥u0 ∥2 + b∥∇ u0 ∥2
(2.12)
0
and Ψ ′′ (t ) = 2∥vt ∥2 + 2(v, vtt + 2λvt − b1vt ). By using the Eq. (2.4) we obtain from the last equality
Ψ ′′ (t ) = 2∥vt ∥2 + 2 v, ∇ a0 + λb + aeλ(m−2)t |∇v|m−2 ∇v − 2λ2 ∥v∥2 + 2e−λt v, g (x, t , eλt v, eλt ∇v) + 2eλ(p−2)t ∥v∥pp .
(2.13)
92
B.A. Bilgin, V.K. Kalantarov / J. Math. Anal. Appl. 403 (2013) 89–94
Due to the condition (1.5) we have 2e
−λt
λt
λt
|(v, g (x, t , e v, e ∇v))| ≤ ν1 e
λ(m−2)t
λ(p−2)t
∥∇v∥ + ν0 e m m
∥ v∥ + p p
M12
1
ν1
+
1
ν0
∥v∥2 ,
where ν0 and ν1 are positive parameters we are going to choose below. By using this inequality we infer from (2.13) the following estimate from below for Ψ ′′ (t )
1 1 Ψ ′′ (t ) ≥ 2∥vt ∥2 − 2λ2 + M12 + ∥v∥2 ν1 ν0 − 2(a0 + λb)∥∇v∥2 + (2 − ν0 )eλ(p−2)t ∥v∥pp − (2a + ν1 )eλ(m−2)t ∥∇v∥m m. By choosing ν0 =
(p − m) we get M12 (m + ap) ′′ 2 2 ∥v∥2 − 2(a0 + λb)∥∇v∥2 Ψ (t ) ≥ 2∥vt ∥ − 2λ + a(p − m) p+m a(p + m) λ(m−2)t + eλ(p−2)t ∥v∥pp − e ∥∇v∥m m p m p+m (p + m)(a0 + λb) 2 = (p + m)H (t ) + 2 + ∥vt ∥ + − 2(a0 + λb) ∥∇v∥2 p−m p
and ν1 =
a m
2
2
M 2 (m + ap) p+m −2 − 1 ∥v∥2 . + λ2 2 a(p − m)
(2.14)
2M 2 (m+ap)
Since λ2 = a(p−m1 )(p+m−4) (see (2.3)) the coefficient of the last term in (2.14) is zero, and since p + m > 4 the coefficient of ∥∇v∥2 is positive. Thus (2.14) implies the estimate
p+m Ψ ′′ (t ) ≥ (p + m)H (t ) + 2 + ∥vt ∥2 . 2
By using the estimate (2.11) in the last inequality we get
t t 2 2 2 Ψ (t ) ≥ 4(1 + α) ∥vt ∥ + 2λ ∥vτ (τ )∥ dτ + b ∥∇vτ (τ )∥ dτ , ′′
0
where α =
p+m−4 . 8
(2.15)
0
As Ψ (t ) ≥ 0 it follows from (2.15) that
2 Ψ ′′ (t )Ψ (t ) − (1 + α) Ψ ′ (t ) t t ≥ −4(1 + α)C0 Ψ (t ) + 4(1 + α) ∥vt ∥2 + 2λ ∥vτ (τ )∥2 dτ + b ∥∇vτ (τ )∥2 dτ + C0 Ψ (t ) 0
− 4(1 + α) (v, vt ) + 2λ
t
(vτ , v)dτ + b 0
0
t
b
(∇vτ , ∇v)dτ + λ∥u0 ∥2 + ∥∇ u0 ∥2 0
2
2
.
We choose now C0 = λ∥u0 ∥2 + 2b ∥∇ u0 ∥2 . By the Cauchy–Schwarz inequality we deduce the following inequality
2 Ψ ′′ (t )Ψ (t ) − (1 + α) Ψ ′ (t ) ≥ −4(1 + α)C0 Ψ (t ) ≥ −4(1 + α)Ψ 2 (t ).
(2.16)
Thus, Ψ (t ) satisfies the condition (1.6) of Lemma 1.1 with C1 = 0, and C2 = 4(1 + α). The other conditions of the lemma are fulfilled thanks to the assumption (2.2). Thus the statement of Lemma 1.1 holds. Hence the statement of Theorem 2.1 also holds true. Finally we prove that there exist initial functions u0 and u1 with arbitrarily large initial energy E0 , for which the corresponding solutions to the problem (1.1)–(1.3) blow up in a finite time. A result about blow up of solutions to the initial boundary value problem for a semilinear strongly damped wave equation is obtained in [2]. Let us note that the concavity method and some modifications were used to establish blow up of solutions to nonlinear weakly damped wave equations with positive energy (see e.g. [6,12,1], and references therein). Theorem 2.2. Suppose that the number E0 defined (2.1) is a positive number and the initial functions u0 and u1 satisfy also the following condition
(u0 , u1 ) >
(λ + 1) 2
κ b ∥ u0 ∥ 2 + α 4
κ 1 κ − 2 ∥∇ u0 ∥2 + − d 0 E0 , α 2 α
(2.17)
B.A. Bilgin, V.K. Kalantarov / J. Math. Anal. Appl. 403 (2013) 89–94
where d0 := λ2 (p + m − 4), α = blows up in finite time.
p+m−4 8
93
and κ = max 4(1 + α), α d20 + p + m . Then the solution to the problem (1.1)–(1.3)
Proof. So, we assume that E0 = −H (0) > 0. For this case the inequality (2.10) derived in the previous section still holds. From (2.10) we deduce H (t ) ≥ H (0)ed0 t + 2λ
t
∥vτ (τ )∥2 ed0 (t −τ ) dτ + b
≥ −E0 ed0 t + 2λ
∥∇vτ (τ )∥2 ed0 (t −τ ) dτ
0
0 t
t
∥vτ (τ )∥2 dτ + b
t
∥∇vτ (τ )∥2 dτ .
(2.18)
0
0
This time let us consider the function
Ψ (t ) = ∥v(t )∥2 + 2λ
t
∥v(τ )∥2 dτ + b
t
∥∇v(τ )∥2 dτ + C0 + β ed0 t , 0
0
where β, C0 are positive constants which we will choose later. We have
Ψ ′ (t ) = 2(v, vt ) + 4λ
t
(vτ , v)dτ + 2b
t
(∇vτ , ∇v)dτ + 2λ∥u0 ∥2 + b∥∇ u0 ∥2 + d0 β ed0 t , 0
0
and
Ψ ′′ (t ) = 2∥vt ∥2 + 2(v, vtt + 2λvt − b1vt ) + d20 β ed0 t . Doing exactly the same calculations as in the previous section we arrive at
Ψ (t ) ≥ (p + m)H (t ) + 2 + ′′
p+m
2
∥vt ∥2 + d20 β ed0 t .
Utilizing (2.18) in this inequality we obtain
t t 2 2 2 Ψ (t ) ≥ 4(1 + α) ∥vt ∥ + 2λ ∥vτ (τ )∥ dτ + b ∥∇vτ (τ )∥ dτ + d20 β − E0 (p + m) ed0 t 0 0 t t d20 β d0 t 2 2 2 e = 4(1 + α) ∥vt ∥ + 2λ ∥vτ (τ )∥ dτ + b ∥∇vτ (τ )∥ dτ + C0 + ′′
0
− 4(1 + α)C0 − α d20 + where α =
p+m−4 . 8
4
0
E0 ( p + m )
β
β ed0 t ,
(2.19)
It follows from (2.19) that
Ψ (t )Ψ (t ) − (1 + α) Ψ ′ (t ) t t d20 β d0 t 2 2 2 ≥ 4(1 + α) ∥vt ∥ + 2λ ∥vτ (τ )∥ dτ + b ∥∇vτ (τ )∥ dτ + C0 + e Ψ (t ) ′′
0
4
0
t t − 4(1 + α) (v, vt ) + 2λ (vτ , v)dτ + b (∇vτ , ∇v)dτ + λ∥u0 ∥2 0
b
d0 β
2
2
+ ∥∇ u0 ∥2 +
ed0 t
2
0
E0 (p + m) − 4(1 + α)C0 + α d20 + β ed0 t Ψ (t ). β
We choose here C0 = λ∥u0 ∥2 + 2b ∥∇ u0 ∥2 , β = E0 and note that due to the Cauchy–Schwarz inequality the sum of the first two terms on the right of the last inequality is nonnegative. Thus we have
Ψ ′′ (t )Ψ (t ) − (1 + α) Ψ ′ (t ) ≥ −κ(C0 + β ed0 t )Ψ (t ) ≥ −κ Ψ 2 (t ). Remark 2.3. Let u0 be a smooth positive function on Ω and u1 =
p
2 2 u . For this choice of initial functions the initial energy p 0
has the form
p 2 λ a0 + λb a E0 = λ u0 , u02 + λ2 − ∥ u0 ∥ 2 + ∥∇ u0 ∥2 + ∥∇ u0 ∥m m. p
2
2
m
(2.20)
It is not difficult to see that there is a large class of positive smooth initial functions u0 for which the expression (2.20) takes an arbitrary positive value and the condition (2.17) holds true.
94
B.A. Bilgin, V.K. Kalantarov / J. Math. Anal. Appl. 403 (2013) 89–94
Acknowledgment The work of authors was supported in parts by The Scientific and Research Council of Turkey, grant no. 112T934. References [1] W. Chen, Y. Zhou, Global nonexistence for a semilinear Petrovsky equation, Nonlinear Anal. 70 (2009) 3203–3208. [2] F. Gazzola, M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2) (2006) 185–207. [3] V.K. Kalantarov, Global behavior of solutions of nonlinear equations of mathematical physics of classical and non-classical types, Post Doct. Thesis, St. Petersburg Department of Steklov Math. Inst., St. Petersburg, 1988, p. 213. [4] V.K. Kalantarov, Blow-up theorems for second order nonlinear evolutionary equations, in: O. Boratav, A. Eden, A. Erzan (Eds.), Turbulence Modeling and Vortex Dynamics, in: Lecture Not. in Physics, Springer Verlag, 1997, pp. 169–181. [5] V.K. Kalantarov, O.A. Ladyzhenskaya, The occurrence of collapse for quasilinear equations of parabolic and hyperbolic type, Zap. Nauchn. Sem. LOMI 69 (1977) 77–102. [6] M.O. Korpusov, On blowup of solutions to a Kirchhoff type dissipative wave equation with a source and positive energy, Sib. Math. J. 53 (2012) 702–717. [7] H.A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal. 5 (1974) 138–146. [8] H.A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form Putt = −Au + F (u), Trans. Amer. Math. Soc. 192 (1974) 1–21. [9] H.A. Levine, L.E. Payne, Nonexistence of global weak solutions for classes of nonlinear wave and parabolic equations, J. Math. Anal. Appl. 55 (1976) 329–334. [10] M. Ohta, Remarks on blowup of solutions for nonlinear evolution equations of second order, Adv. Math. Sci. Appl. 8 (2) (1998) 901–910. [11] K. Ono, On global existence, asymptotic stability and blowing up of solutions for some degenerate non-linear wave equations of Kirchhoff type with a strong dissipation, Math. Methods Appl. Sci. 20 (2) (1997) 151–177. [12] Y. Zhou, Global existence and nonexistence for a nonlinear wave equation with damping and source terms, Math. Nachr. 278 (11) (2005) 1341–1358.
Contents lists available at SciVerse ScienceDirect
Journal of Mathematical Analysis and Applications journal homepage: www.elsevier.com/locate/jmaa
Blow up of solutions to the initial boundary value problem for quasilinear strongly damped wave equations Bilgesu A. Bilgin, Varga K. Kalantarov ∗ Department of Mathematics, Koç University, Rumelifeneri Yolu, Sariyer, Istanbul, Turkey
article
abstract
info
Article history: Received 31 August 2012 Available online 4 February 2013 Submitted by Kenji Nishihara
We obtain sufficient conditions on initial functions for which the initial boundary value problem for second order quasilinear strongly damped wave equations blow up in a finite time. © 2013 Elsevier Inc. All rights reserved.
Keywords: Strongly damped wave equation Blow up Concavity method Initial boundary value problem
1. Introduction We consider the following initial boundary value problem: utt − ∇
a0 + a|∇ u|m−2 ∇ u − b1ut = g (x, t , u, ∇ u) + |u|p−2 u,
u(x, t ) = 0,
x ∈ Ω , t > 0,
x ∈ ∂ Ω , t > 0,
u(x, 0) = u0 (x),
ut (x, 0) = u1 (x),
(1.1) (1.2)
x ∈ Ω,
(1.3)
where p > m ≥ 2,
a0 ≥ 0,
a, b > 0
(1.4)
are given numbers, and Ω ⊂ R is a bounded (or unbounded) domain with a sufficiently smooth boundary ∂ Ω . We assume that n
|g (x, t , u, p)| ≤ M1 (|p|m/2 + |u|p/2 ), where M1 > 0 is some positive number.
∀(x, t ) ∈ Ω × R+ , u ∈ R, p ∈ Rn ,
(1.5)
As far as we know the first result about blow up of solutions for a strongly damped equation is the result of H. Levine [7], where by using the concavity method introduced in [8] he got sufficient conditions for blow up of solutions to the Cauchy problem for a nonlinear differential-operator equation in a Hilbert space of the form Putt + Qut + Au = F (u), where P , A are positive symmetric operators and F (·) is a nonlinear operator that is the gradient of some functional G(·) such that (F (u), u) − kG(u) ≥ 0, ∀u ∈ D(F ) for some k > 2. This result was developed in [5,3] (see also [4]) for a more general class of differential-operator equations of the form Putt + Au + Qut = B(u) + F (u), where B(·) is a nonlinear operator subordinated to A.
∗
Corresponding author. E-mail address: [email protected] (V.K. Kalantarov).
0022-247X/$ – see front matter © 2013 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2013.01.056
90
B.A. Bilgin, V.K. Kalantarov / J. Math. Anal. Appl. 403 (2013) 89–94
Later on results about blow up of solutions to the strongly damped Kirchhoff equation utt − 1ut − (α + β∥∇ u∥2γ )1u − |u|q−2 u = 0 were obtained in [11,10] under some restrictions on α ≥ 0, β > 0, γ > 0 and q > 2. Our result obtained for solutions to the problem (1.1)–(1.3) can be considered as a development of results on the blow up of solutions to initial boundary value problems for quasilinear hyperbolic equations obtained in [9]. In what follows we will use the following notations: (·, ·), ∥ · ∥ will denote the inner product and norm of L2 (Ω ) and ∥ · ∥r will denote the standard norm of Lr (Ω ). The proof of blow up of solutions to the problem (1.1)–(1.3) is based on the following lemma. Lemma 1.1 ([5]). Suppose that a positive, twice-differentiable function Ψ (t ) satisfies for t > 0 the inequality
2 Ψ ′′ (t )Ψ (t ) − (1 + α) Ψ ′ (t ) ≥ −2C1 Ψ ′ (t )Ψ (t ) − C2 [Ψ (t )]2 ,
(1.6)
where α > 0, C1 , C2 ≥ 0. If Ψ (0) > 0, Ψ ′ (0) > −γ2 α −1 Ψ (0), and C1 + C2 > 0, then Ψ (t ) → +∞ as t → t1 ≤
1
ln
2 C12 + α C2
where γ1 = −C1 +
γ1 Ψ (0) + α Ψ ′ (0) , γ2 Ψ (0) + α Ψ ′ (0)
C12 + α C2 , γ2 = −C1 −
C12 + α C2 .
2. Blow up theorems By using a modification of the concavity method suggested in [5] first we prove the following theorem Theorem 2.1. Suppose that (1) the conditions (1.4) and (1.5) are satisfied, (2) the initial functions satisfy the conditions E0 :=
1 2
∥ u1 − λ u0 ∥ 2 + b
(u1 , u0 ) >
2
a0 + λ b 2
p+m+4 p+m−4
∥∇ u0 ∥2 +
a m
∥∇ u0 ∥m m +
− 1 ∥∇ u0 ∥ + (λ + 1) 2
λ2 2
1
∥u0 ∥2 − ∥u0 ∥pp ≤ 0,
p+m+4 p+m−4
p
∥ u0 ∥ 2 ,
(2.1)
(2.2)
where
λ := M1
2(m + ap) a(p − m)(p + m − 4)
.
(2.3)
Then there exists T0 ≤
2
(p + m)2 − 16
ln
(u1 , u0 ) +
b 2
p+m+4 p+m−4
(u1 , u0 ) −
b 2
p+m+4 p+m−4
m+4 − 1 ∥∇ u0 ∥2 + (λ + 1) pp+ ∥ u0 ∥ 2 +m−4 m+4 − 1 ∥∇ u0 ∥2 − (λ + 1) pp+ ∥ u0 ∥ 2 +m−4
such that
∥∇ u(t )∥2 → ∞ as t → T0− . Proof. For λ > 0 we make the change of variables u(x, t ) = v(x, t )eλt ,
x ∈ Ω , t ≥ 0,
in (1.1) and obtain the differential equation
vtt − ∇
a0 + λb + aeλ(m−2)t |∇v|m−2 ∇v + λ2 v + 2λvt − b1vt
= e−λt g (x, t , eλt v, eλt ∇v) + eλ(p−2)t |v|p−2 v
(2.4)
under the initial and boundary conditions
v(x, 0) = u0 (x),
vt (x, 0) = −λu0 (x) + u1 (x),
x ∈ Ω ; v|Ω = 0, t > 0.
(2.5)
B.A. Bilgin, V.K. Kalantarov / J. Math. Anal. Appl. 403 (2013) 89–94
91
Multiplying the Eq. (2.4) by vt in L2 (Ω ) we get
d dt
1 2
a0 + λ b
2
∥vt ∥ +
2
a λ2 1 ∥∇v∥ + eλ(m−2)t ∥∇v∥m + ∥v∥2 − eλ(p−2)t ∥v∥pp m m 2 p
2
+ 2λ∥vt ∥2 + b∥∇vt ∥2 −
aλ(m − 2) λ(m−2)t λ(p − 2) λ(p−2)t e ∥∇v∥m e ∥v∥pp m + m p
= e−λt g (x, t , eλt v, eλt ∇v), vt .
(2.6)
Using the assumption (1.5) and the Cauchy’s inequality with ε we estimate the last term above in the following way. e
−λt
λt
λt
|(g (x, t , e v, e ∇v), vt )| ≤ ε1 e
λ(m−2)t
λ(p−2)t
∥∇v∥ + ε0 e m m
p p
∥v∥ +
M12 4
1
ε1
+
1
ε0
∥vt ∥2 .
We define H (t ) :=
1 λ(p−2)t 1 a0 + λb a λ2 e ∥v∥pp − ∥vt ∥2 − ∥∇v∥2 − eλ(m−2)t ∥∇v∥m ∥v∥2 , m − p 2 2 m 2
(2.7)
and note that H (0) = −E0 (E0 is defined by (2.1)). Employing the last inequality, and using this notation, we obtain from (2.6) the following inequality d dt
H (t ) ≥ [λ(p − 2) − ε0 p] H (t ) +
+ +
λ2 2 a m
λ(p−m) 2p
1 2
[λ(p − 2) − ε0 p] + 2λ ∥vt ∥2
[λ(p − 2) − ε0 p]∥v∥2 + b∥∇vt ∥2 +
1
[λ(p − 2) − ε0 p] (a0 + bλ)∥∇v∥2
2
λ(p−2)t [λ(p − 2) − ε0 p − λ(m − 2)] eλ(m−2)t ∥∇v∥m ∥v∥pp m + ε0 e
− ε0 e Choosing ε0 =
λ(p−2)t
∥v∥ − ε1 e p p
aλ(p−m) 2m
and ε1 =
λ
λ(m−2)t
m m
∥∇v∥ −
M12
1
ε0
4
+
1
ε1
∥vt ∥2 .
we obtain from (2.8) the inequality
(m + ap)M12 H (t ) ≥ (p + m − 4)H (t ) + (p + m − 4) + 2λ − ∥vt ∥2 + b∥∇vt ∥2 . dt 2 4 2aλ(p − m) 2(m+ap) According to (2.3) λ = M1 a(p−m)(p+m−4) . Therefore we obtain from (2.9) d
d dt
H (t ) ≥
λ 2
(2.8)
λ
(p + m − 4)H (t ) + 2λ∥vt ∥2 + b∥∇vt ∥2 .
(2.9)
(2.10) λ
Since by (2.1) H (0) ≥ 0, we obtain from (2.10) that H (t ) ≥ e 2 (p+m−4)t H (0) ≥ 0. Therefore integrating (2.10) we get H (t ) ≥ 2λ
t
∥vτ (τ )∥2 dτ + b 0
t
∥∇vτ (τ )∥2 dτ .
(2.11)
0
Let us consider the function
Ψ (t ) = ∥v(t )∥ + 2λ 2
t
∥v(τ )∥ dτ + b 2
0
t
∥∇v(τ )∥2 dτ + C0 , 0
where v is a local solution of the problem (2.4)–(2.5) and C0 is a positive parameter to be chosen below. It is easy to see that
Ψ ′ (t ) = 2(v, vt ) + 4λ
t
(vτ , v)dτ + 2b 0
t
(∇vτ , ∇v)dτ + 2λ∥u0 ∥2 + b∥∇ u0 ∥2
(2.12)
0
and Ψ ′′ (t ) = 2∥vt ∥2 + 2(v, vtt + 2λvt − b1vt ). By using the Eq. (2.4) we obtain from the last equality
Ψ ′′ (t ) = 2∥vt ∥2 + 2 v, ∇ a0 + λb + aeλ(m−2)t |∇v|m−2 ∇v − 2λ2 ∥v∥2 + 2e−λt v, g (x, t , eλt v, eλt ∇v) + 2eλ(p−2)t ∥v∥pp .
(2.13)
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B.A. Bilgin, V.K. Kalantarov / J. Math. Anal. Appl. 403 (2013) 89–94
Due to the condition (1.5) we have 2e
−λt
λt
λt
|(v, g (x, t , e v, e ∇v))| ≤ ν1 e
λ(m−2)t
λ(p−2)t
∥∇v∥ + ν0 e m m
∥ v∥ + p p
M12
1
ν1
+
1
ν0
∥v∥2 ,
where ν0 and ν1 are positive parameters we are going to choose below. By using this inequality we infer from (2.13) the following estimate from below for Ψ ′′ (t )
1 1 Ψ ′′ (t ) ≥ 2∥vt ∥2 − 2λ2 + M12 + ∥v∥2 ν1 ν0 − 2(a0 + λb)∥∇v∥2 + (2 − ν0 )eλ(p−2)t ∥v∥pp − (2a + ν1 )eλ(m−2)t ∥∇v∥m m. By choosing ν0 =
(p − m) we get M12 (m + ap) ′′ 2 2 ∥v∥2 − 2(a0 + λb)∥∇v∥2 Ψ (t ) ≥ 2∥vt ∥ − 2λ + a(p − m) p+m a(p + m) λ(m−2)t + eλ(p−2)t ∥v∥pp − e ∥∇v∥m m p m p+m (p + m)(a0 + λb) 2 = (p + m)H (t ) + 2 + ∥vt ∥ + − 2(a0 + λb) ∥∇v∥2 p−m p
and ν1 =
a m
2
2
M 2 (m + ap) p+m −2 − 1 ∥v∥2 . + λ2 2 a(p − m)
(2.14)
2M 2 (m+ap)
Since λ2 = a(p−m1 )(p+m−4) (see (2.3)) the coefficient of the last term in (2.14) is zero, and since p + m > 4 the coefficient of ∥∇v∥2 is positive. Thus (2.14) implies the estimate
p+m Ψ ′′ (t ) ≥ (p + m)H (t ) + 2 + ∥vt ∥2 . 2
By using the estimate (2.11) in the last inequality we get
t t 2 2 2 Ψ (t ) ≥ 4(1 + α) ∥vt ∥ + 2λ ∥vτ (τ )∥ dτ + b ∥∇vτ (τ )∥ dτ , ′′
0
where α =
p+m−4 . 8
(2.15)
0
As Ψ (t ) ≥ 0 it follows from (2.15) that
2 Ψ ′′ (t )Ψ (t ) − (1 + α) Ψ ′ (t ) t t ≥ −4(1 + α)C0 Ψ (t ) + 4(1 + α) ∥vt ∥2 + 2λ ∥vτ (τ )∥2 dτ + b ∥∇vτ (τ )∥2 dτ + C0 Ψ (t ) 0
− 4(1 + α) (v, vt ) + 2λ
t
(vτ , v)dτ + b 0
0
t
b
(∇vτ , ∇v)dτ + λ∥u0 ∥2 + ∥∇ u0 ∥2 0
2
2
.
We choose now C0 = λ∥u0 ∥2 + 2b ∥∇ u0 ∥2 . By the Cauchy–Schwarz inequality we deduce the following inequality
2 Ψ ′′ (t )Ψ (t ) − (1 + α) Ψ ′ (t ) ≥ −4(1 + α)C0 Ψ (t ) ≥ −4(1 + α)Ψ 2 (t ).
(2.16)
Thus, Ψ (t ) satisfies the condition (1.6) of Lemma 1.1 with C1 = 0, and C2 = 4(1 + α). The other conditions of the lemma are fulfilled thanks to the assumption (2.2). Thus the statement of Lemma 1.1 holds. Hence the statement of Theorem 2.1 also holds true. Finally we prove that there exist initial functions u0 and u1 with arbitrarily large initial energy E0 , for which the corresponding solutions to the problem (1.1)–(1.3) blow up in a finite time. A result about blow up of solutions to the initial boundary value problem for a semilinear strongly damped wave equation is obtained in [2]. Let us note that the concavity method and some modifications were used to establish blow up of solutions to nonlinear weakly damped wave equations with positive energy (see e.g. [6,12,1], and references therein). Theorem 2.2. Suppose that the number E0 defined (2.1) is a positive number and the initial functions u0 and u1 satisfy also the following condition
(u0 , u1 ) >
(λ + 1) 2
κ b ∥ u0 ∥ 2 + α 4
κ 1 κ − 2 ∥∇ u0 ∥2 + − d 0 E0 , α 2 α
(2.17)
B.A. Bilgin, V.K. Kalantarov / J. Math. Anal. Appl. 403 (2013) 89–94
where d0 := λ2 (p + m − 4), α = blows up in finite time.
p+m−4 8
93
and κ = max 4(1 + α), α d20 + p + m . Then the solution to the problem (1.1)–(1.3)
Proof. So, we assume that E0 = −H (0) > 0. For this case the inequality (2.10) derived in the previous section still holds. From (2.10) we deduce H (t ) ≥ H (0)ed0 t + 2λ
t
∥vτ (τ )∥2 ed0 (t −τ ) dτ + b
≥ −E0 ed0 t + 2λ
∥∇vτ (τ )∥2 ed0 (t −τ ) dτ
0
0 t
t
∥vτ (τ )∥2 dτ + b
t
∥∇vτ (τ )∥2 dτ .
(2.18)
0
0
This time let us consider the function
Ψ (t ) = ∥v(t )∥2 + 2λ
t
∥v(τ )∥2 dτ + b
t
∥∇v(τ )∥2 dτ + C0 + β ed0 t , 0
0
where β, C0 are positive constants which we will choose later. We have
Ψ ′ (t ) = 2(v, vt ) + 4λ
t
(vτ , v)dτ + 2b
t
(∇vτ , ∇v)dτ + 2λ∥u0 ∥2 + b∥∇ u0 ∥2 + d0 β ed0 t , 0
0
and
Ψ ′′ (t ) = 2∥vt ∥2 + 2(v, vtt + 2λvt − b1vt ) + d20 β ed0 t . Doing exactly the same calculations as in the previous section we arrive at
Ψ (t ) ≥ (p + m)H (t ) + 2 + ′′
p+m
2
∥vt ∥2 + d20 β ed0 t .
Utilizing (2.18) in this inequality we obtain
t t 2 2 2 Ψ (t ) ≥ 4(1 + α) ∥vt ∥ + 2λ ∥vτ (τ )∥ dτ + b ∥∇vτ (τ )∥ dτ + d20 β − E0 (p + m) ed0 t 0 0 t t d20 β d0 t 2 2 2 e = 4(1 + α) ∥vt ∥ + 2λ ∥vτ (τ )∥ dτ + b ∥∇vτ (τ )∥ dτ + C0 + ′′
0
− 4(1 + α)C0 − α d20 + where α =
p+m−4 . 8
4
0
E0 ( p + m )
β
β ed0 t ,
(2.19)
It follows from (2.19) that
Ψ (t )Ψ (t ) − (1 + α) Ψ ′ (t ) t t d20 β d0 t 2 2 2 ≥ 4(1 + α) ∥vt ∥ + 2λ ∥vτ (τ )∥ dτ + b ∥∇vτ (τ )∥ dτ + C0 + e Ψ (t ) ′′
0
4
0
t t − 4(1 + α) (v, vt ) + 2λ (vτ , v)dτ + b (∇vτ , ∇v)dτ + λ∥u0 ∥2 0
b
d0 β
2
2
+ ∥∇ u0 ∥2 +
ed0 t
2
0
E0 (p + m) − 4(1 + α)C0 + α d20 + β ed0 t Ψ (t ). β
We choose here C0 = λ∥u0 ∥2 + 2b ∥∇ u0 ∥2 , β = E0 and note that due to the Cauchy–Schwarz inequality the sum of the first two terms on the right of the last inequality is nonnegative. Thus we have
Ψ ′′ (t )Ψ (t ) − (1 + α) Ψ ′ (t ) ≥ −κ(C0 + β ed0 t )Ψ (t ) ≥ −κ Ψ 2 (t ). Remark 2.3. Let u0 be a smooth positive function on Ω and u1 =
p
2 2 u . For this choice of initial functions the initial energy p 0
has the form
p 2 λ a0 + λb a E0 = λ u0 , u02 + λ2 − ∥ u0 ∥ 2 + ∥∇ u0 ∥2 + ∥∇ u0 ∥m m. p
2
2
m
(2.20)
It is not difficult to see that there is a large class of positive smooth initial functions u0 for which the expression (2.20) takes an arbitrary positive value and the condition (2.17) holds true.
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B.A. Bilgin, V.K. Kalantarov / J. Math. Anal. Appl. 403 (2013) 89–94
Acknowledgment The work of authors was supported in parts by The Scientific and Research Council of Turkey, grant no. 112T934. References [1] W. Chen, Y. Zhou, Global nonexistence for a semilinear Petrovsky equation, Nonlinear Anal. 70 (2009) 3203–3208. [2] F. Gazzola, M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2) (2006) 185–207. [3] V.K. Kalantarov, Global behavior of solutions of nonlinear equations of mathematical physics of classical and non-classical types, Post Doct. Thesis, St. Petersburg Department of Steklov Math. Inst., St. Petersburg, 1988, p. 213. [4] V.K. Kalantarov, Blow-up theorems for second order nonlinear evolutionary equations, in: O. Boratav, A. Eden, A. Erzan (Eds.), Turbulence Modeling and Vortex Dynamics, in: Lecture Not. in Physics, Springer Verlag, 1997, pp. 169–181. [5] V.K. Kalantarov, O.A. Ladyzhenskaya, The occurrence of collapse for quasilinear equations of parabolic and hyperbolic type, Zap. Nauchn. Sem. LOMI 69 (1977) 77–102. [6] M.O. Korpusov, On blowup of solutions to a Kirchhoff type dissipative wave equation with a source and positive energy, Sib. Math. J. 53 (2012) 702–717. [7] H.A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal. 5 (1974) 138–146. [8] H.A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form Putt = −Au + F (u), Trans. Amer. Math. Soc. 192 (1974) 1–21. [9] H.A. Levine, L.E. Payne, Nonexistence of global weak solutions for classes of nonlinear wave and parabolic equations, J. Math. Anal. Appl. 55 (1976) 329–334. [10] M. Ohta, Remarks on blowup of solutions for nonlinear evolution equations of second order, Adv. Math. Sci. Appl. 8 (2) (1998) 901–910. [11] K. Ono, On global existence, asymptotic stability and blowing up of solutions for some degenerate non-linear wave equations of Kirchhoff type with a strong dissipation, Math. Methods Appl. Sci. 20 (2) (1997) 151–177. [12] Y. Zhou, Global existence and nonexistence for a nonlinear wave equation with damping and source terms, Math. Nachr. 278 (11) (2005) 1341–1358.