Nonlinear Dyn DOI 10.1007/s11071-013-1116-0

O R I G I N A L PA P E R

Exact conditions of blow-up and global existence for the nonlinear wave equation with damping and source terms Yi Jiang · Yongle Zhang

Received: 4 May 2013 / Accepted: 9 October 2013 © Springer Science+Business Media Dordrecht 2013

Abstract This paper deals with the Cauchy problem of a nonlinear wave equation with damping and source terms. By the exact Gagliardo–Nirenberg inequality connected with the classic nonlinear elliptic equation, we establish new invariant sets of the problem. Then we get the exact conditions of blow-up and global existence. Keywords Nonlinear wave equation · Damping · Blow-up · Global existence

1 Introduction We consider the Cauchy problem for the following nonlinear wave equation with damping and source terms: φtt − φ + φ + φt |φt |m−1 = φ|φ|p−1 , t > 0, x ∈ RN ,

(1.1)

and imposed initial data φ(0, x) = φ0 ,

φt (0, x) = φ1 ,

x ∈ RN .

Y. Jiang (B) · Y. Zhang Department of Mathematics, Sichuan Normal University, Chengdu, Sichuan 610066, China e-mail: [email protected]

(1.2)

Here p > 1, m ≥ 1, φ(t, x) is a complex-valued function of (t, x) ∈ R+ × RN , and φt |φt |m−1 is called damping term. Without the damping term φt |φt |m−1 , (1.1) is reduced to the nonlinear Klein–Gordon equation, which is well studied. Its blow-up properties are investigated by Levine [10], Payee and Scatting [15], Ball [1]. The global existence of solutions with sufficiently small initial data is investigated by Strauss [17]. Some numerical results of the blow-up versus global existence dichotomy are considered by Donninger and Schlag [2]. Global dynamics concerning the ground state energy are studied by Nakanishi, Schlag and Krieger ([8– 13]). With the nonlinear damping term, it creates difficulties in the study of (1.1). Specially, for m = 1, Levine [9] shows that the solution blows up with negative initial energy by the concavity arguments. In general, the abstract version of (1.1) is studied by Levine and Serrin [11] as well as Levine, Pucci and Serrin [12]. Ikehata [6], Ohta [14] and Vitillaro [21] obtain some blow-up results for the case of bounded domain and the absence of the term φ in (1.1). For the Cauchy problem (1.1)–(1.2), Todorova [20] proves that the solution of (1.1) blows up in finite time if the initial energy is negative. Furthermore, Jiang, Gan and He [7] show the instability of the standing wave by the potential well arguments and concavity methods. On the view-point of physics, it is important to find the sharp conditions of blow-up and global existence

Y. Jiang, Y. Zhang

for (1.1). By the potential well arguments, this result is obtained in Todorova [19] as follows. Let m ≥ 1 and 1 < p ≤ N/N − 2 if N > 2 and 1 < p < ∞ if N ≤ 2. Assume that any compactly supported data φ0 ∈ H 1 (R N ), φ1 ∈ L2 (R N ) and 1 1 1 φ1 2L2 + ∇φ0 2L2 + φ0 2L2 2 2 2 1 p+1 − φ0 Lp+1 < d. p+1

Here    p+1 W1 := v ∈ H 1 RN | ∇v2L2 + v2L2 > vLp+1 ,  J (v) < d ∪ {0},    p+1 W2 := v ∈ H 1 RN | ∇v2L2 + v2L2 < vLp+1 ,  J (v) < d , p+1

1 J (v) := 12 ∇u2L2 + 12 u2L2 − p+1 uLp+1 , and d =: 1 N inf{supλ∈R J (λu), u ∈ H (R )\{0}} is called “Potential depth”. Meanwhile, Todorova proves that W1 and W2 are invariant sets. It means that if the initial data are in the set W1 or W2 , then the solution of the Cauchy problem (1.1)–(1.2) also belongs to W1 or W2 , respectively. Since the property (I) and (II), W1 and W2 are also called the stable set and unstable set, respectively. This result is elegant but it is difficult to verify the “Potential depth” d in W1 and W2 . As far as we known, d is not clear and is only given an estimate by Lemma 2.1 in [19]. The goal of this paper is to find a way to give exact characterizations for “Potential depth” d. Then, we would give different invariant sets. Firstly, by the exact Gagliardo–Nirenberg inequality connecting with the classic nonlinear elliptic equation, we give new invariant sets R1 and R2 as follows:    R1 := v ∈ H 1 R N | ∇v2L2

N (p − 1) D2L2 , 2 + N − p(N − 2)

(1.3)

   R2 := v ∈ H 1 R N | ∇v2L2 N (p − 1) D2L2 , 2 + N − p(N − 2)

J (v) <

(I) if φ0 ∈ W1 , the solution φ(t, x) of the Cauchy problem (1.1)–(1.2) globally exists; (II) if φ0 ∈ W2 , the solution φ(t, x) of the Cauchy problem (1.1)–(1.2) blows up in finite time.

 p−1 D2L2 , 2 + N − p(N − 2)

and

>

Then

<

J (v) <

 p−1 D2L2 , 2 + N − p(N − 2)

(1.4) p+1

1 where J (v) := 12 ∇u2L2 + 12 u2L2 − p+1 uLp+1 , and D is the unique solution of the corresponding nonlinear elliptic equation   u − u + u|u|p−1 = 0, u ∈ H 1 R N . (1.5)

Furthermore, we prove that if the initial datum is in the R1 , the solution φ(t, x) of the Cauchy problem (1.1)–(1.2) globally exists; if the initial datum is in the R2 , the solution φ(t, x) of the Cauchy problem (1.1)–(1.2) blows up. This is the result about the sharp conditions of blow-up and global existence for (1.1). It is worth pointing out that the method used in this paper is different from the potential well argument used in [19]. By the exact Gagliardo–Nirenberg inequality connected with the classic nonlinear equation (1.5), we could clearly compute J (v) < p−1 2 2+N −p(N −2) DL2 := dD . But in [19], J (v) < d =: inf{supλ∈R J (λu), u ∈ H 1 (RN ) \ {0}}. Here, d is not clear and is only given an estimate by Lemma 2.1 in [19]. Hence, we could not determine dD > d or dD ≤ d. If dD > d, then R1 and R2 are completely different from W1 and W2 , respectively. So we establish new invariant sets of the problem and generalize the results in [19]. If dD ≤ d, then R1 and R2 are new invariant sets by a clear expression and we improve the results in [19]. In essence, the method used in this paper comes from [22] and [5]. That is the exact Gagliardo– Nirenberg inequality, which is completely different from the constrained variational argument used in the related papers (see [3, 4, 7, 23, 24]) and the potential well argument used in [19]. The rest of this paper is organized as follows. In Sect. 2, we present the Gagliardo–Nirenberg inequality and the well-posedness of the Cauchy problem (1.1)–(1.2) in the energy space H 1 (R N ). In

Exact conditions of blow-up and global existence for the nonlinear wave equation with damping

Sect. 3, we show the new invariant sets. In Sect. 4, we give the main results. For simplicity, throughout   this paper, we denote R N · dx by · dx and H 1 (R N ) by H 1 .

Furthermore, we see that for any t ∈ [0, T ), φ(t, x) satisfies the conservation of energy t



φt (s) m+1 ds = E(0), (2.6) E(t) + Lm+1 0

2 Preliminaries Now, we recall the Gagliardo–Nirenberg inequality [22]. Proposition 2.1 Let N ≥ 3 and 1 < p < u ∈ H 1 , then p+1 uLp+1

p+1− (p−1)N 2 ≤ Cp,N uL2 p−1 DL2

1

N +2 N −2 .

∇u

(p−1)N 2 L2

If

,

(2.1) where Cp,N =

(p + 1)[p + 1 −

(p−1)N (p−1)N ] 4 −1 2

[ (p−1)N ] 2

(p−1)N 4

u ∈ H 1.

(2.3)

if N > 2 and

if N ≤ 2.

Remark 2.1 From Proposition 2.3, it follows that m = p is a critical case, namely for p ≤ m, a weak solution exists globally for any compactly supported initial data. For m < p, blow-up of the solution to the Cauchy problem (1.1)–(1.2) occurs.

3 Invariant sets Now, we define two sets as follows:  R1 := v ∈ H 1 | K(v) < 0,

Proposition 2.2 Let m ≥ 1 and

1
Proposition 2.3 Assume p ≤ m and let the condition (2.2)–(2.5) be fulfilled. Then the Cauchy problem (1.1)–(1.2) has a unique global solution such that   φ(t, x) ∈ C [0, T ); H 1 ,      φt (t, x) ∈ C [0, T ); L2 R N ∩ Lm+1 [0, T ) × R N for any T > 0.

From Todorova [18], we can get the local wellposedness of the solution to the Cauchy problem (1.1)– (1.2) in the energy space H 1 .

1 < p ≤ N/N − 2

Moreover, we give a result in [20].

(2.2)

and D is the unique positive solution of u − u + |u|p−1 u = 0,

where E(t) is defined as follows: 1 1 1 2 2 E(t) = |φt | dx + |∇φ| dx + |φ|2 dx 2 2 2 1 (2.7) − |φ|p+1 dx. p+1

(2.4) J (v) <

Then, for any compactly supported data, φ0 ∈ H 1 ,

  φ1 ∈ L2 R N ,

(3.1)

and (2.5)

the Cauchy problem (1.1)–(1.2) has a unique solution such that   φ(t, x) ∈ C [0, T ); H 1 ,      φt (t, x) ∈ C [0, T ); L2 R N ∩ Lm+1 [0, T × R N ) provided T is small enough.

 p−1 D2L2 , 2 + N − p(N − 2)

 R2 := v ∈ H 1 | K(v) > 0, J (v) <

 p−1 D2L2 . 2 + N − p(N − 2)

Here, K(v) and J (v) are defined as K(v) := |∇v|2 dx

(3.2)

Y. Jiang, Y. Zhang

N(p − 1) 2 + N − p(N − 2)

− and

|D|2 dx,



2

(3.4)

d J (v) = μ dμ

p−1 2 Theorem 3.1 If E(0) < 2+N −p(N −2) DL2 , then R1 and R2 are invariant sets under the flow generated by the Cauchy problem (1.1)–(1.2). That is, if φ0 (x) ∈ R1 or φ0 (x) ∈ R2 , one has φ(t, x) ∈ R1 or φ(t, x) ∈ R2 respectively, for all t ∈ [0, T ). Here φ(t, x) ∈ C([0, T ], H 1 ) is the solution of the Cauchy problem (1.1)–(1.2) and T is the maximal existence time.

Proof Firstly we prove that R1 and R2 are nonempty sets. Let D be the positive solution of (2.3). Then p+1

D2L2 + ∇D2L2 = DLp+1

(3.5)

and the Gagliardo–Nirenberg inequality is optimized as 2− (p−1)N 2

p+1

(p−1)N

∇DL2 2

.

(3.6)

N (p − 1) D2L2 = 0. 2 + N − p(N − 2)

(3.7)

Take v = μD. We get K(v) = μ2 ∇D2L2 N (p − 1) D2L2 . − 2 + N − p(N − 2)

(3.8)

From (3.7), one has K(v) > 0 for μ ∈ (1, ∞); K(v) < 0 for μ ∈ (0, 1). In terms of (3.4), we obtain  μ2  J (v) = |∇D|2 + |D|2 dx 2

|D|p+1 dx.

  |∇D|2 + |D|2 dx p

|D|p+1 dx

  = μ 1 − μp−1

(3.9)



  |∇D|2 + |D|2 dx. (3.10)

This implies the monotonic property of J (·). Then, for all μ > 0, we have J (v) < J (D). Hence, R1 and R2 are nonempty sets. Now we let φ0 ∈ H 1 and φ(t, x) be the solution of (1.1) with the initial datum φ0 . From the Gagliardo– Nirenberg inequality (2.1) and (3.4), one has 1 1 1 p+1 φLp+1 J (φ) = ∇φ2L2 + φ2L2 − 2 2 p+1 1 1 ≥ ∇φ2L2 + φ2L2 2 2 (p−1)N CN,p p+1− (p−1)N 1−p 2 DL2 φL2 − ∇φL2 2 . p+1 Using the Young inequality and taking ε = we have

This implies that K(D) = 0. That is ∇D2L2 −



−μ

D is the positive solution of (2.3) [16]. From Sobolev’s imbedding Theorem, K(v) and J (v) are well defined. Firstly we prove that R1 and R2 are invariant sets of the Cauchy problem (1.1)–(1.2).

DLp+1 = Cp,N DL2



Moreover, (3.5) yields

1 |∇v| dx + |v|2 dx 2 1 − |v|p+1 dx. p+1

1 J (v) := 2

μp+1 − p+1

(3.3)

1 p+1− (p−1)N 2

(p − 1)(N − 2) (p−1)N−2(p+1) ε (p−1)(N−2) 4 4  (p−1)N (p−1)(N−2) CN,p 1−p 2 DL2 ∇φL2 × p+1

J (u) ≥ −

1 1 + ∇φ2L2 + φ2L2 2 2 2(p + 1) − (p − 1)N εφ2L2 − 4  2 (p − 1)N N−2 (p − 1)(N − 2) p+1− ≥− 4 2 −N  −4 2N (p − 1)N N−2 N−2 × DLN−2 2 ∇φL2 2 1 + ∇φ2L2 . 2

,

Exact conditions of blow-up and global existence for the nonlinear wave equation with damping

Let f (x) := [ (p−1)N ] 2

−N N−2

x2 2

D

−2) − (p−1)(N [p +1− (p−1)N ] N−2 × 4 2

−4 N−2 L2

2

x

2N N−2

. Hence



(p − 1)N (p − 1)N p+1− 2 2 −N  −4 N+2 (p − 1)N N−2 N−2 . × DLN−2 2 x 2

f  (x) = x −

(I) If



∇φ0 2L2 < 2 N−2

N (p − 1) D2L2 , 2 + N − p(N − 2)

then the solution φ(t, x) of the Cauchy problem (1.1)–(1.2) globally exists. Moreover, φ(t, x) satisfies (3.11)



∇φ(t, x) 2 2 < L

Then there exist two roots for (3.11) such that

N (p − 1) D2L2 . 2 + N − p(N − 2)

x1 = 0 and

(4.3)



N(p − 1) x2 = 2 + N − p(N − 2)

1 2

(3.12) DL2 .

(II) If ∇φ0 2L2 >

They are two minimizers of f (x). Hence, f (x) is increasing on the interval [x1 , x2 ) and decreasing on the interval [x2 , ∞). p−1 2 From (2.6) and f (x2 ) = 2+N −p(N −2) DL2 , we have   f ∇φL2 ≤ J (φ) ≤ E(t) < E(0) < f (x2 ). (3.13) If φ0 ∈ R1 , then ∇φ0 L2 < x2 and E(0) < p−1 2 2+N −p(N −2) DL2 . From (3.13), we can see that

p−1 2 J (φ) < 2+N −p(N −2) DL2 . on [0, x2 ), we have



∇φ(t, x)

L2

Since f (x) is increasing (3.14)

< x2 .

Hence R1 is invariant. If φ0 ∈ R2 , then ∇φ0 L2 > x2 and E(0) < p−1 2 2+N −p(N −2) DL2 . From (3.13), we can see that p−1 2 2+N −p(N −2) DL2 .

J (φ) < on [x2 , ∞), we have



∇φ(t, x) 2 > x2 . L

Since f (x) is decreasing

Hence R2 is invariant.

(3.15) 

4 Exact conditions of blow-up and global existence Theorem 4.1 Let m ≥ 1 and 1 < p ≤ N/N − 2 for N > 2 and 1 < p < ∞ for N ≤ 2. Assume that we have any compactly supported data φ0 ∈ H 1 , φ1 ∈ L2 (R N ) and 0 ≤ E(0) <

(4.2)

p−1 D2L2 . 2 + N − p(N − 2)

We have the following results.

N (p − 1) D2L2 , 2 + N − p(N − 2)

(4.4)

then the solution φ(t, x) of the Cauchy problem (1.1)–(1.2) blows up in finite time T < ∞, where D is the solution of the following elliptic equation: u − u + |u|p−1 u = 0,

u ∈ H 1.

(4.5)

Proof From [18, 20], we know that for some t1 ≥ 0, E(t1 ) becomes nonnegative and the solution φ(t, x) of the Cauchy problem (1.1)–(1.2) blows up in finite time. So we can suppose that E(t) ≥ 0 for all t > 0, which leas to a constant control of the rate of the energy decrease. That is, from the energy identity, we get t



φt (s) m+1 ds = E(0) − E(t) ≤ d. (4.6) Lm+1 0

(I) By (4.1) and (4.2), we have φ0 ∈ R1 . From Theorem 3.1, the invariant property of R1 implies that the solution φ(t, x) of the Cauchy problem (1.1)– (1.2) globally exists in H˙ 1 . Moreover, φ(t, x) satisfies (p−1) 2 ∇φ(t, x)2L2 < 2+NN−p(N −2) DL2 . (II) By (4.1) and (4.3), we have φ0 ∈ R2 . From Theorem 3.1, the invariant property of R2 implies that



∇φ(t, x) 2 2 > L

N (p − 1) D2L2 . 2 + N − p(N − 2)

(4.7)

Put F (t) = φ(t, ·)2L2 . By simple calculations, we have F  (t) = 2φt 2L2 − 2∇φ2L2 − 2φ2L2 + 2φLp+1 − 2 Re φφt |φt |m−1 dx. (4.8) p+1

(4.1)

Y. Jiang, Y. Zhang

=

(1) m = p; (2) m = 1; (3) 1 < m < p. Case (1) When m = p, using the Hölder inequality and the Young inequality, we get





φφt |φt |m−1 dx

≤ φLm+1 φt m Lm+1 ≤ εφm+1 + c(ε)φt m+1 Lm+1 Lm+1 p+1

= εφLp+1 + c(ε)φt m+1 . Lm+1

(4.9)

Case (3) When 1 < m < p, applying the Hölder inequality, the Young inequality, and the so-called interpolation inequality, we obtain by φ ∈ Lp (R N ) that





φφt |φt |m−1 dx

From (2.6) and (4.7), we have F  (t) + c(ε)φt m+1 Lm+1 ≥ 2φt 2L2 − 2∇φ2L2 − 2φ2L2   p+1 + (2 − 2ε)φLp+1 + δ E(t) − E(0)     δ δ = − 2 ∇φ2L2 + − 2 φ2L2 2 2   δ p+1 − 2ε φLp+1 − δE(0) + 2− p+1   δ φt 2L2 + 2+ 2     δ δ φt 2L2 + − 2 φ2L2 ≥ 2+ 2 2   δ p+1 − 2ε φLp+1 − δE(0) + 2− p+1   N (p − 1) δ −2 D2L2 . + 2 2 + N − p(N − 2)

≤ φLm+1 φt m Lm+1 1−θ m ≤ φθL2 φL p+1 φt Lm+1 m+1  m+1 ≤ εφθL2 φ1−θ + c(ε)φt L m+1 Lp+1 p+1

m+1 ≤ ε1 φ2L2 + ε2 φLp+1 + c(ε)φt L m+1 , 1 m+1 1 p+1 ).

where

F  (t) + c(ε)φt m+1 Lm+1 ≥ 2φt 2L2 − 2∇φ2L2 − (2 + 2ε)φ2   p+1 + 2φLp+1 + δ E(t) − E(0)

=

θ 2

+

1−θ p+1

1 and also θ = ( m+1 −

( 12 − From (2.6) and (4.7), we obtain F  (t) + c(ε)φt m+1 Lm+1 (4.10)

Case (2) When m = 1, using the Young inequality, we obtain





φφt dx ≤ εφ2 2 + c(ε)φt 2 2 . (4.11)

L L From (2.6) and (4.7), we get

   δ δ − 2 ∇φ2L2 + − 2 − 2ε φ2 2 2   δ p+1 φLp+1 − δE(0) + 2− p+1   δ φt 2L2 + 2+ 2     δ δ 2 φt L2 + − 2 − 2ε φ2 ≥ 2+ 2 2   δ p+1 φLp+1 − δE(0) + 2− p+1   N (p − 1) δ −2 D2L2 . (4.12) + 2 2 + N − p(N − 2) 

In the following, we discuss tree cases:

≥ 2φt 2L2 − 2∇φ2L2 − (2 + 2ε1 )φ2   p+1 + (2 − 2ε2 )φLp+1 + δ E(t) − E(0)     δ δ − 2 ∇φ2L2 + − 2 − 2ε1 φ2 = 2 2   δ p+1 + 2− − 2ε2 φLp+1 − δE(0) p+1   δ φt 2L2 + 2+ 2     δ δ 2 φt L2 + − 2 − 2ε1 φ2 ≥ 2+ 2 2   δ p+1 − 2ε2 φLp+1 − δE(0) + 2− p+1

1 p+1 )/

Exact conditions of blow-up and global existence for the nonlinear wave equation with damping

 +

 δ N (p − 1) −2 D2L2 . 2 2 + N − p(N − 2)

(4.13)

We choose δ satisfies 4(p − 1)N [2 + N − p(N − 2)]−1 D2L2

Since  − δ−4  F 8 (t) δ − 4 − δ+12 F 8 (t) 8  4+δ  2 × F (t)F  (t) − F (t) , 8

=−

(p − 1)N [2 + N − p(N − 2)]−1 D2L2 − 2E(0) < δ < 2(p + 1),

(4.14)

p−1 2 which is possible as E(0) < 2+N −p(N −2) DL2 and guarantees that δ > 4. Thus   δ − 2 ∇φ2L2 − δE(0) 2   N (p − 1) δ −2 D2L2 ≥ 2 2 + N − p(N − 2)

− δE(0) ≥ 0.

(4.15)

2−

δ − 2ε > 0, p+1

2−

δ − 2ε2 > 0. p+1

δ − 2 − 2ε1 > 0, 2

From (4.10)–(4.13), we get   δ m+1  F (t) + c(ε)φt Lm+1 ≥ 2 + φt 2L2 . 2

lim F −

t→T ∗

δ−4 8

(t) = 0.

δ−4

(4.21)

Thus one has T < ∞ and lim φH 1 = ∞.

(4.22) 

(4.16)

Remark 4.1 In term of the definitions in [19], Theorem 4.1 implies that R1 and R2 are stable set and unstable set of the Cauchy problem (1.1)–(1.2), respectively.

(4.17)

Acknowledgements This work was partially supported by National Natural Science Foundation of China (Nos. 11126336 and 11201324), New Teachers’ Fund for Doctor Stations, Ministry of Education (No. 20115134120001).

It concludes that there exists a t1 such that F  (t)|t=t1 > 0. Then F (t) is increasing for t > t1 . is decreasing. OthMoreover, the quantity of φt m+1 Lm+1 ∗ > erwise, assume there is t such that φ(t, ·)t m+1 Lm+1 m+1 ∗ ∗ φ(t , ·)t Lm+1 for all t > t . By integrating the inequality, we get contradiction from (4.6). Thus in these cases, the quantity of φ2 − will eventually become positive. Therec(ε)φt m+1 Lm+1 fore for large enough t, from (4.10)–(4.13) we get   δ (4.18) F  (t) ≥ 2 + φt 2L2 . 2 Using the Hölder inequality, one has 4 + δ   2 F (t) . 8

δ−4

and

Integrating (4.17) over [0, t] and taking into account (4.6), we arrive at  t

δ



φt (s, x) 2 2 ds − c(ε)d + F  (0). 2+ F (t) ≥ L 2 0

F (t)F  (t) ≥

we obtain [F − 8 (t)] ≤ 0. Therefore F − 8 (t) is concave for sufficiently large t, and there exists a finite time T ∗ such that

t→T −

Then, let ε, ε1 and ε2 be so small that

(4.20)

(4.19)

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