A note on finite time blowup for dissipative Klein–Gordon equation

Nonlinear Analysis 195 (2020) 111729

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A note on finite time blowup for dissipative Klein–Gordon equation Yue Pang, Yanbing Yang ∗ College of Mathematical Sciences, Harbin Engineering University, 150001, People’s Republic of China

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Article history: Received 2 October 2019 Accepted 7 December 2019 Communicated by Vicentiu D. Radulescu MSC: 35Q53 35B65 35A01

This is a continuation of the study on an arbitrarily positive initial energy finite time blowup result of the solution to the initial value problem of strongly damped Klein-Gordon equation. Based on an adapted concavity method, both the conclusions and arguments of the present paper are not bounded by the coefficients of dissipative damping terms. © 2019 Elsevier Ltd. All rights reserved.

Keywords: Klein–Gordon equation Finite time blowup Dissipative term Arbitrarily positive initial energy

1. Introduction The dissipative Klein–Gordon equation over Rn can be viewed as utt − ∆u + u − µ∆ut + γut = |u|

p−1

u.

(1.1)

We should consider Eq. (1.1) supplemented with the following conditions (H1) Initial data: u(x, 0) = u0 (x) ∈ H 1 (Rn ),

ut (x, 0) = u1 (x) ∈ L2 (Rn );

(H2) Power index: p ∈ (1, +∞)

if

n = 2,

( n + 2) p ∈ 1, n−2

(H3) Coefficients: µ ≥ 0, ∗ Corresponding author. E-mail address: [email protected] (Y. Yang).

https://doi.org/10.1016/j.na.2019.111729 0362-546X/© 2019 Elsevier Ltd. All rights reserved.

γ ≥ 0.

if

n ≥ 3;

(1.2)

2

Y. Pang and Y. Yang / Nonlinear Analysis 195 (2020) 111729

In the absence of both µ and γ, Eq. (1.1) with some specific nonlinearities becomes the classical nonlinear Klein–Gordon equations that appear in various branches of science including quantum mechanics, nonlinear optics and solid physics, see, for instance, [18] and the references therein. Also, there have been many works on qualitative properties for the classical Klein–Gordon equations associated with the initial profile, see, for example, [17] for the local existence, [8,9,19,25] for finite time blowup and [5,6,14] for global existence and behavior. For certain remarkable works on Klein–Gordon equation with the weak damping term (γ > 0), we refer to [7,13,15,20] and the references thereof. In the presence of both µ and γ, the strongly damped Klein–Gordon Eq. (1.1) was first posed by P. Aviles and J. Sandefur [2] over R3 . Later, Avrin [3] pointed out that a global weak solution for the considered Eq. (1.1) without dissipative term supplemented with the initial profile possessing radial symmetry can be arbitrarily closely approximated by a global strong solution for Eq. (1.1), which extended the previous result for this equation constructed in [4]. Since the aim of this note is to deal with an unsolved problem left in [22], we quickly turn to the motivation of our work. It is recently demonstrated that the exponential decay and finite time blowup for Eq. (1.1) supplemented with the conditions (H1)–(H3) were established by the authors in [22] when the total initial energy is less than or equal to the mountain pass level in view of mountain pass theorem. An entire list about the variational method (mountain pass theorem or critical point theory) herein is not possible and we may only refer to [1,16] for a detailed description on the theory of variational method. And about its applications on the qualitative properties for various partial differential equations, the interested readers can see [21] for nonlinear wave equation, [23] for pseudo-parabolic equation, [10,12] for logarithmic nonlinearity, [24] for viscous term, [11] for strain term and the references therein, for instance. Additionally, the authors in [22] constructed a finite time blowup result when E(0) > 0 under the situation that the coefficient of the so-called strongly damping term ∆ut vanishes (µ = 0), but left an interesting issue, that is, what the corresponding situation is if µ > 0. This motivates us to write this short paper. The goal of this short paper is to extend an arbitrarily positive initial energy finite time blowup result derived in [22] from µ = 0 to µ > 0. Inspired by [26], we figure out what kind of initial data can cause the finite time blowup for this considered problem (1.1)–(1.2) in the presence of µ > 0 when E(0) > 0. For the sake of the precise statement of the main result described in Theorem 1.2, both the definition and local existence of weak solutions that constructed in [22] will be ahead provided. Definition 1.1 (Weak Solution, [22]). A function u ∈ C([0, T˜); H 1 (Rn )) ∩ C 1 ([0, T˜); L2 (Rn )), ut ∈ L2 ([0, T˜); H 1 (Rn )), utt ∈ L2 ([0, T˜); H −1 (Rn )) satisfying p−1

⟨utt , ϕ⟩ + (∇u, ∇ϕ) + (u, ϕ) + µ(∇ut , ∇ϕ) + γ(ut , ϕ) = (|u|

, ϕ)

(1.3)

for all ϕ ∈ H 1 (Rn ) and a.e. t ∈ [0, T˜) with u(0) = u0 (x) and ut (0) = u1 (x) represents a weak solution to Eq. (1.1) involving condition (1.2) over [0, T˜). Hereafter, ∫ (u, v) = uvdx Rn

denotes the inner product over Rn , the symbol T˜ represents the lifespan of u(x, t) and ⟨·, ·⟩ means the duality pairing between H −1 (Rn ) and H 1 (Rn ). Proposition 1.1 (Local Solution, [22]). Let (H1)–(H3) hold. Then, problem (1.1)–(1.2) allows a uniquely local solution ( ) ( ) u(x, t) ∈ C [0, T˜), H 1 (Rn ) ∩ C 1 [0, T˜), L2 (Rn ) .

Y. Pang and Y. Yang / Nonlinear Analysis 195 (2020) 111729

3

Also, there appears either T˜ = +∞ or T˜ < +∞

lim ∥u(t)∥H 1 (Rn ) = +∞.

and

t→T˜

Additionally, the solution u(x, t) possesses ∫ E(0) = E(t) +

t

∥us ∥2∗ ds,

t ∈ [0, T˜)

(1.4)

0

with E(t) =

1 1 1 ∥ut ∥2 + ∥u∥2H 1 (Rn ) − ∥u∥p+1 p+1 . 2 2 p+1

(1.5)

Henceforth, let ∥ · ∥p = ∥ · ∥Lp (Rn ) ,

∥ · ∥ = ∥ · ∥L2 (Rn )

and for ψ, ν ∈ H 1 (Rn ), ∫

∫ ∇ψ∇νdx + γ

(ψ, ν)∗ = µ Rn

ψνdx Rn

with ∥ψ∥2∗ = (ψ, ψ)∗ . Now we come to the result of this note, whose detailed statement is presented as follows. Theorem 1.2.

Let (H1)–(H3) hold. Assume I(u0 ) < 0, where I(u) = ∥u∥2H 1 (Rn ) − ∥u∥p+1 p+1

and (u0 , u1 ) >

(1.6)

α(p + 1) E(0) > 0 p−1

(1.7)

with α = max{µ, γ + 1, p},

(1.8)

then the solution to Eq. (1.1) involving condition (1.2) blows up in finite time. The method of proving Theorem 1.2 is based on an adapted concavity method. The key is to introduce a new auxiliary function and a flexible parameter. Theorem 1.2 also reveals an arbitrarily positive initial energy finite time blowup result for Eq. (1.1) supplemented with the initial profile (1.2) whenever γ ≥ 0 and µ ≥ 0, which covered the previous results derived in [19,20,25]. The detailed proof of Theorem 1.2 will be illustrated in the coming section. 2. Proof of Theorem 1.2 The proof will be performed in the subsequent two steps. Step I. It is first claimed that I(u(t)) < 0, t ∈ [0, T˜) and µ∥∇u(t)∥2 + γ∥u(t)∥2 + 2(u, ut ) >

α(p + 1) E(0), p−1

t ∈ [0, T˜).

(2.1)

4

Y. Pang and Y. Yang / Nonlinear Analysis 195 (2020) 111729

For this purpose, set A(t) := 2(u, ut ) + ∥u∥2∗ .

(2.2)

An injection of ϕ = u into (1.3) along with (1.6) then tells A′ (t) =2µ(∇u, ∇ut ) + 2γ(u, ut ) + 2∥ut ∥2 + 2⟨utt , u⟩ =2∥ut ∥2 − 2I(u), t ∈ [0, T˜).

(2.3)

Suppose by contradiction that I(u(t)) < 0

for all

0 ≤ t < t0

(2.4)

and I(u(t0 )) = 0.

(2.5)

Hereafter 0 < t0 < T˜. A substitution of (2.4) into (2.3) straightly shows A′ (t) > 0 over [0, t0 ) and, consequently by (1.7) and (1.8), A(t) > A(0) > 2(u0 , u1 ) >

2α(p + 1) E(0), p−1

0 < t < t0 .

Recalling Definition 1.1, it is found that u(t) and ut (t) are both continuous about t, which allows A(t0 ) >

2α(p + 1) E(0). p−1

(2.6)

The later aim is to construct a contradiction with (2.6). In fact, (1.4) and (1.5) both enjoy ∫

t

E(0) =E(t) +

∥us ∥2∗ ds

0

1 = ∥ut ∥2 + 2 1 = ∥ut ∥2 + 2

∫ t 1 1 ∥u∥2H 1 (Rn ) − ∥u∥p+1 + ∥us ∥2∗ ds p+1 2 p+1 0 ∫ t 1 p−1 2 I(u) + ∥u∥H 1 (Rn ) + ∥us ∥2∗ ds, p+1 2(p + 1) 0

(2.7)

which together with (2.5) and µ ≥ 0, γ ≥ 0 indicates that ) p−1 ( 1 ∥ut (t0 )∥2 + ∥u(t0 )∥2 + ∥∇u(t0 )∥2 2 2(p + 1) ) p−1 ( ≥ ∥ut (t0 )∥2 + µ∥∇u(t0 )∥2 + (γ + 1)∥u(t0 )∥2 2α(p + 1) ) p−1 ( ≥ 2(u(t0 ), ut (t0 )) + µ∥∇u(t0 )∥2 + γ∥u(t0 )∥2 . 2α(p + 1)

E(0) ≥

(2.8)

Clearly (2.8) contradicts (2.6). Again by the argument mentioned above, obviously A′ (t) > 0 if I(u(t)) < 0 for 0 ≤ t < T˜, which transpires that the desired claim as indicated above is complete. Step II. It is now in a position to establish the finite time blowup. The argument of this proof is based on Proposition 1.1. Proposition 1.1 allows us to perform an adapted concavity inequality by using a contradiction argument. Suppose, to the contrary, that u(t) is a global solution, i.e., T˜ = +∞. Then for any T0 ∈ (0, T˜), an auxiliary function is defined by 2

∫

H(t) = ∥u∥ + 0

t

∥u(s)∥2∗ ds + (T0 − t)∥u0 ∥2∗ ,

0 ≤ t ≤ T0 ,

(2.9)

Y. Pang and Y. Yang / Nonlinear Analysis 195 (2020) 111729

5

where the last term in (2.9), (T0 − t)∥u0 ∥2∗ , will be used to make sure H′ (t) = 2(u, ut ) + 2

∫

t

(u(s), us (s))∗ ds

0

(2.10)

and thus the second-order derivative of H(t) can play the role of the anti-derivative of A(t). So, the fact that H(t) is continuous over [0, T0 ] directly reveals that H(t) ≥ η > 0,

0 ≤ t ≤ T0 ,

(2.11)

where the constant η is independent of the choice of T0 . Further, applying (u, ut )2 ≤ ∥ut ∥2 ∥u∥2 , t

(∫ 0

)2 ∫ t ∫ t 2 (u(s), us (s))∗ ds ≤ ∥u(s)∥∗ ds ∥us (s)∥2∗ ds 0

0

and ∫ 2(u, ut ) 0

t

(u(s), us (s))∗ ds

(∫ t )1/2 (∫ t )1/2 2 2 ≤2∥u∥∥ut ∥ ∥u(s)∥∗ ds ∥us (s)∥∗ ds 0 0 ∫ t ∫ t ≤∥u∥2 ∥us (s)∥2∗ ds + ∥ut ∥2 ∥u(s)∥2∗ ds 0

0

on (2.10) yields (

( ) ∫ t (∫ t ( )2 ( ) ) )2 2 H (t) =4 (u, ut ) + 2(u, ut ) u(s), us (s) ∗ ds + u(s), us (s) ∗ ds 0 0 ∫ t ∫ t ( )( ) ≤4 ∥u∥2 + ∥u(s)∥2∗ ds ∥ut ∥2 + ∥us (s)∥2∗ ds , t ∈ [0, T0 ]. ′

0

0

In conjunction with H′′ (t) = 2⟨utt , u⟩ + 2∥ut ∥2 + 2(u, ut )∗ = 2∥ut ∥2 − 2I(u),

0 ≤ t ≤ T0 ,

it is inferred that for t ∈ [0, T0 ] ϖ+3 2 (H′ (t)) H′′ (t)H(t) − 4 ( ( )) ∫ t ′′ 2 2 ≥H(t) H (t) − (ϖ + 3) ∥ut ∥ + ∥us (s)∥∗ ds 0 ( ) ∫ t 2 2 ≥H(t) −(ϖ + 1)∥ut ∥ − (ϖ + 3) ∥us (s)∥∗ ds − 2I(u)

(2.12)

0

with a parameter ϖ > 1. The subsequent aim is to control the term ζ(t) := −(ϖ + 1)∥ut ∥2 − (ϖ + 3)

∫ 0

t

∥us (s)∥2∗ ds − 2I(u).

(2.13)

Y. Pang and Y. Yang / Nonlinear Analysis 195 (2020) 111729

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(2.7) and p > 1 along with Cauchy–Schwarz inequality deduce (2.13) to ζ(t) =(p − ϖ)∥ut ∥2 + (p − 1)∥u∥2H 1 (Rn ) ∫ t ∥us (s)∥2∗ ds − 2(p + 1)E(0) + (2p − 1 − ϖ) 0

=(p − ϖ)µ∥∇u∥2 + (p − ϖ)∥ut ∥2 + (p − ϖ)(γ + 1)∥u∥2 ∫ t − 2(p + 1)E(0) + (2p − 1 − ϖ) ∥us (s)∥2∗ ds 0 ( ) ( ) 2 + p − 1 − µ(p − ϖ) ∥∇u∥ + p − 1 − (p − ϖ)(γ + 1) ∥u∥2 ) ( ≥(p − ϖ) µ∥∇u∥2 + 2(u, ut ) + γ∥u∥2 ∫ t − 2(p + 1)E(0) + (2p − 1 − ϖ) ∥us (s)∥2∗ ds 0 ( ) ( ) 2 + p − 1 − (p − ϖ)(γ + 1) ∥u∥ + p − 1 − µ(p − ϖ) ∥∇u∥2 .

(2.14)

The condition (1.8) with p > 1 shall allow us to select ϖ = p − p−1 α > 1, then a straightforward computation with the advertised result claimed in Step I on (2.14) reveals ) p − 1( 2(u, ut ) + µ∥∇u∥2 + γ∥u∥2 − 2(p + 1)E(0) α ) p − 1( > 2(u0 , u1 ) + µ∥∇u0 ∥2 + γ∥u0 ∥2 − 2(p + 1)E(0) α 2(p − 1) > (u0 , u1 ) − 2(p + 1)E(0) = ϱ > 0. α

ζ(t) >

(2.15)

From the above arguments provided in (2.9)–(2.15), it can be concluded that H′′ (t)H(t) − and hence by z(t) = H(t)−

ϖ−1 4

ϖ+3 ′ 2 H (t) > ηϱ > 0, 4

a.e. 0 ≤ t ≤ T0

(2.16)

,

z ′′ (t) < −

ϖ+7 ϖ−1 ηϱz(t) ϖ−1 , 4

a.e. 0 ≤ t ≤ T0

(2.17)

∗ with ϖ = p − p−1 α . This implies that z(t) arrives at 0 in finite time, i.e., there exists a T < T0 such that lim H(t) = +∞, which contradicts T˜ = +∞. This hence finishes the proof of Theorem 1.2. t→T ∗ Here it is necessary to show clearly the relations between T˜, T0 and T ∗ , as T ∗ ∈ (0, T0 ) ⊂ (0, T˜) is the so-called blowup time, and its existence ruins the global time assumption T˜ = +∞, which means that the

local solution in this case cannot be extended to the global one in time, and it blows up in finite time. As no global estimate can be based on at the beginning of the arguments, we just started from the local solution in Step II, that is also why we introduce the local interval (0, T0 ) ⊂ (0, T˜), here T0 < +∞. But when we ϖ−1 arrive at the differential inequalities (2.16) and (2.17), which indicate that H(t)− 4 can reach zero as t goes to a certain point T ∗ ∈ (0, T0 ) ⊂ (0, T˜), but all of these estimates never ensure or concern the uniform boundness of such T ∗ , that is why we have to assume the infinite existence time first in order to ensure that such T ∗ can be found in the time interval we supposed, and then by the discontinuous arguments we prove such assumption is impossible.

Y. Pang and Y. Yang / Nonlinear Analysis 195 (2020) 111729

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Acknowledgments The work of Yang is partially supported by the National Natural Science Foundation of China (No. 11801114), the Heilongjiang Postdoctoral Foundation (No. LBH-Z15036), the Fundamental Research Funds for the Central Universities and the China Scholarship Council (No. 201706685064). References [1] A. Ambrosetti, P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973) 349–381. [2] P. Aviles, J. Sandefur, Nonlinear second-order equations with applications to partial differential equations, J. Differential Equations 58 (1985) 404–427. [3] J.D. Avrin, Convergence of the strongly damped nonlinear Klein–Gordon equation in Rn with radial symmetry, Proc. Roy. Soc. Edinburgh Sect. A 107 (1/2) (1987) 169–174. [4] J.D. Avrin, Convergence properties of the strongly damped nonlinear Klein–Gordon equation, J. Differential Equations 67 (1987) 243–255. [5] T. Cazenave, Uniform estimates for solutions of nonlinear Klein–Gordon equations, J. Funct. Anal. 60 (1985) 36–55. [6] J. Ginibre, G. Velo, The global Cauchy problem for the nonlinear Klein–Gordon equation. II, Ann. Inst. Henri Poincar´ e Anal. Non Lin´ eaire 6 (1989) 15–35. [7] T.G. Ha, J.Y. Park, Global existence and uniform decay of a damped Klein–Gordon equation in a noncylin-drical domain, Nonlinear Anal. 74 (2011) 577–584. [8] M. Keel, T. Tao, Small data blow-up for semilinear Klein–Gordon equations, Amer. J. Math. 121 (1999) 629–669. [9] H.A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form P utt = −Au+F (u), Trans. Amer. Math. Soc. 192 (1974) 1–21. [10] W. Lian, M.S. Ahmed, R.Z. Xu, Global existence and blow up of solution for semilinear hyperbolic equation with logarithmic nonlinearity, Nonlinear Anal. 184 (2019) 239–257. [11] W. Lian, V.D. R˘ adulescu, R.Z. Xu, Y.B. Yang, N. Zhao, Global well-posedness for a class of fourth-order nonlinear strongly damped wave equations, Adv. Calc. Var. (2019) http://dx.doi.org/10.1515/acv-2019-0039. [12] W. Lian, R.Z. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal. 9 (2020) 613–632. [13] Y.Z. Lin, M.C. Cui, A new method to solve the damped nonlinear Klein–Gordon equation, Sci. China Ser. A 51 (2008) 304–313. [14] C.S. Morawetz, W.A. Strauss, Decay and scattering of solutions of a nonlinear relativistic wave equation, Comm. Pure Appl. Math. 25 (1972) 1–31. [15] N.S. Papageorgiou, V.D. R˘ adulescu, D.D. Repov˘s, Nonlinear second order evolution inclusions with noncoercive viscosity term, J. Differential Equations 264 (2018) 4749–4763. [16] N.S. Papageorgiou, V.D. R˘ adulescu, D.D. Repov˘s, Nonlinear Analysis-Theory and Methods, in: Springer Monographs in Mathematics, Springer, Cham, 2019. [17] H. Pecher, Lp -Absch¨ azungen und klassiche L¨ osungen f¨ ur nichtlineare Wellengeichungen I, Math. Z. 150 (1976) 159–183. [18] J.K. Perring, T.H. Skyrme, A model unified field equation, Nuclear Phys. 31 (1962) 550–555. [19] Y.J. Wang, A sufficient condition for finite time blow up of the nonlinear Klein–Gordon equations with arbitrarily positive initial energy, Proc. Amer. Math. Soc. 136 (2008) 3477–3482. [20] R.Z. Xu, Global existence, blow up and asymptotic behaviour of solutions for nonlinear Klein–Gordon equation with dissipative term, Math. Methods Appl. Sci. 33 (2010) 831–844. [21] R.Z. Xu, Y.X. Chen, Y.B. Yang, S.H. Chen, J.H. Shen, T. Yu, Z.S. Xu, Global well-posedness of semilinear hyperbolic equations, parabolic equations and Schr¨ odinger equations, Electron. J. Differential Equations 2018 (2018) 55. [22] R.Z. Xu, Y.H. Ding, Global solutions and finite time blow up for damped Klein–Gordon equation, Acta Math. Sci. Ser. B Engl. Ed. 33 (2013) 643–652. [23] R.Z. Xu, J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal. 264 (2013) 2732–2763. [24] R.Z. Xu, X.C. Wang, Y.B. Yang, S.H. Chen, Global solutions and finite time blow-up for fourth order nonlinear damped wave equation, J. Math. Phys. 59 (2018) 061503. [25] Y.B. Yang, R.Z. Xu, Finite time blow up for nonlinear Klein–Gordon equations with arbitrarily positive initial energy, Appl. Math. Lett. 77 (2018) 21–26. [26] Y.B. Yang, R.Z. Xu, Supcritical initial energy blowup for nonlinear wave equation with both strongly and weakly damped terms, Commun. Pure Appl. Anal. 18 (2019) 1351–1358.