Sharp Condition for Global Existence and Blow-Up on Klein-Gordon Equation

Abstract

We study the initial boundary value problem of the nonlinear Klein-Gordon equation. First we introduce a family of potential wells. By using them, we obtain a new existence theorem of global solutions and show the blow-up in finite time of solutions. Especially the relation between the above two phenomena is derived as a sharp condition.

1. Introduction

Klein-Gordon equation is one of the famous evolution equations arising in relativistic quantum mechanics. There are a lot of literature giving the outline of its study trace. For the following type nonlinear Klein-Gordon (NLKG) equation: 𝜑_𝑡𝑡−Δ𝜑+𝜑=|𝜑|^𝑝−1𝜑, (1.1) a lot of papers show the global and local well-posedness and blow-up properties for the Cauchy problem of the above NLKG equation, which can be found in [1–5]. Especially Zhang derived a sharp condition for the global existence of the Cauchy problem of the above NLKG equation in [6]. By introducing a so-called ground state solution, which is the positive solution of the nonlinear Euclidean scalar field equation Δ𝑢−𝑢+𝑢^𝑝=0, he applied a host of very useful properties of the ground state solution to show the sharp condition for this Cauchy problem. In the present paper, we try to make use of the classical potential wells argument [7], which is different from that in [6], to clarify the sharp condition for initial boundary value problem (IBVP) of the same NLKG equation.

2. Potential Wells and Their Properties

In this paper, we study the initial boundary value problem of nonlinear Klein-Gordon equation 𝜑_𝑡𝑡−▵𝜑+𝜑=|𝜑|^𝑝−1𝜑,𝑥∈Ω,𝑡>0,𝜑(𝑥,0)=𝜑₀(𝑥),𝜑_𝑡(𝑥,0)=𝜑₁(𝑥),𝑥∈Ω,𝜑(𝑥,𝑡)=0,𝑥∈𝜕Ω,𝑡⩾0, (2.1)

where 1<𝑝<∞ for 𝑛⩽0𝑥000842; 1<𝑝<(𝑛+2)/(𝑛−2) for 𝑛⩾3.

For problem (2.1), we define the energy function and some functionals as follows: 1𝐸(𝑡)=2‖𝜑_𝑡‖²+12‖𝜑‖²+12‖∇𝜑‖²−1𝑝+1‖𝜑‖^𝑝+1_𝑝+1,1𝐽(𝜑)=2‖𝜑‖²+12‖∇𝜑‖²−1𝑝+1‖𝜑‖^𝑝+1_𝑝+1,𝐼(𝜑)=‖𝜑‖²+‖∇𝜑‖²−‖𝜑‖^𝑝+1_𝑝+1,𝐼_𝛿(𝜑)=𝛿‖∇𝜑‖²+‖𝜑‖²−‖𝜑‖^𝑝+1_𝑝+1. (2.2) In aid of the above functionals, we define the potential well as follows: 𝑊={𝜑∈𝐻¹₀(Ω)∣𝐼(𝜑)>0,𝐽(𝜑)<𝑑}∪{0}, (2.3) where 𝑑=inf_𝜑∈𝒩𝐽(𝜑),𝒩∶={𝜑∈𝐻¹₀(Ω)∣𝐼(𝜑),‖∇𝜑‖≠0}. (2.4) Then we further give the following definitions 𝑑(𝛿)=inf_{𝜑∈𝒩𝛿}𝒩𝐽(𝜑),_𝛿∶={𝜑∈𝐻¹₀(Ω)∣𝐼_𝛿(𝜑)=0,‖∇𝜑‖≠0}. (2.5)

Now, it is ready for us to define a family of potential wells and the outside sets of the corresponding potential wells sets as follows: 𝑊_𝛿={𝜑∈𝐻¹₀(Ω)∣𝐼_𝛿(𝜑)>0,𝐽(𝜑)<𝑑(𝛿)}∪{0},0<𝛿<1,𝑉={𝜑∈𝐻¹₀𝑉(Ω)∣𝐼(𝜑)<0,𝐽(𝜑)<𝑑},_𝛿={𝜑∈𝐻¹₀(Ω)∣𝐼_𝛿(𝜑)<0,𝐽(𝜑)<𝑑(𝛿)}. (2.6)

The following lemmas are given to show the relations between the functional 𝐼_𝛿(𝜑) and ‖∇𝜑‖.

Lemma 2.1. If 0<‖∇𝜑‖<𝑟(𝛿) then 𝐼_𝛿(𝜑)>0, where 𝑟(𝛿)=(𝛿/𝐶_∗^𝑝+1)^1/(𝑝−1) and 𝐶_∗=sup(‖𝜑‖_𝑝+1/‖∇𝜑‖).

Proof. If 0<‖∇𝜑‖<𝑟(𝛿), then 𝐼_𝛿(𝜑)=𝛿‖∇𝜑‖²−(‖𝜑‖^𝑝+1_𝑝+1−‖𝜑‖²),‖𝜑‖^𝑝+1_𝑝+1−‖𝜑‖²⩽0𝑥00084𝐶_∗^𝑝+1‖∇𝜑‖^𝑝+1⩽𝛿‖∇𝜑‖². (2.7) Hence 𝐼_𝛿(𝜑)>0.

Lemma 2.2. If 𝐼_𝛿(𝜑)<0 then ‖∇𝜑‖>𝑟(𝛿).

Proof. Note that 𝐼_𝛿(𝜑)<0 gives 𝛿‖∇𝜑‖²<‖𝜑‖^𝑝+1_𝑝+1−‖𝜑‖²<‖𝜑‖^𝑝+1_𝑝+1⩽0𝑥00084𝐶_∗^𝑝+1‖∇𝜑‖^𝑝−1‖∇𝜑‖², (2.8) which implies ‖∇𝜑‖^𝑝−1>𝛿𝐶_∗^𝑝+1=𝑟^𝑝−1(𝛿), (2.9) that can be deduced from (2.8).

Lemma 2.3. As the function of 𝛿,𝑑(𝛿) is increasing on 0<𝛿≤1, decreasing on 1≤𝛿≤(𝑝+1)/2, and takes the maximum 𝑑=𝑑(1) at 𝛿=1.

Proof. Now we prove that 𝑑(𝛿¹)<𝑑(𝛿²) for any 0<𝛿¹<𝛿²<1 or 1<𝛿²<𝛿¹<(𝑝+1)/2. Clearly, it is enough to prove that for any 0<𝛿¹<𝛿²<1 or 1<𝛿²<𝛿¹<(𝑝+1)/2 and for any 𝜑∈𝒩_𝛿², there exist a 𝑣∈𝒩_𝛿¹ and a constant 𝜀(𝛿¹,𝛿²)≥𝒩_𝛿¹ such that 𝐽(𝑣)<𝐽(𝜑)−𝜀(𝛿¹,𝛿²). In fact, for the previous 𝜑 we also define 𝜆(𝛿) by 𝛿(‖𝜆𝜑‖²+‖𝜆∇𝜑‖²)=‖𝜆𝜑‖^𝑝+1_𝑝+1, then 𝐼_𝛿(𝜆(𝛿)𝜑)=0, 𝜆(𝛿²)=1. Let 𝑔(𝜆)=𝐽(𝜆𝜑), then 𝑑1𝑑𝜆𝑔(𝜆)=𝜆(‖𝜆𝜑‖²+‖𝜆∇𝜑‖²−‖𝜆𝜑‖^𝑝+1_𝑝+1)=1𝜆((1−𝛿)‖𝜆∇𝜑‖²+𝐼_𝛿(𝜆𝜑))=(1−𝛿)‖𝜆∇𝜑‖². (2.10)
Take 𝑣=𝜆(𝛿¹)𝜑, then 𝑣∈𝒩_𝛿¹.
For 0<𝛿¹<𝛿²<1, we have 𝐽(𝜑)−𝐽(𝑣)=𝑔(1)−𝑔(𝜆(𝛿¹))>(1−𝛿²)𝑟²(𝛿²)𝜆(𝛿¹)(1−𝜆(𝛿¹))≡𝜀(𝛿¹,𝛿²). (2.11) For 1<𝛿²<𝛿¹<(𝑝+1)/2, we have 𝐽(𝜑)−𝐽(𝑣)=𝑔(1)−𝑔(𝜆(𝛿¹))>(𝛿²−1)𝑟²(𝛿²)𝜆(𝛿²)(𝜆(𝛿¹)−1)≡𝜀(𝛿¹,𝛿²). (2.12) These give the conclusion.

3. Sharp Condition for Global Existence and Blow-Up

Definition 3.1 (weak solution). The function 𝜑(𝑥,𝑡)∈𝐿^∞(0,𝑇;𝐻¹₀(Ω)) with 𝜑_𝑡(𝑡,𝑥)∈𝐿^∞(0,𝑇;𝐿²(Ω)) is called a weak solution of problem (2.1) for 𝑡∈[0,𝑇) if the following conditions are satisfied:(1)(𝜑_𝑡,𝑣)+∫^𝑇₀(∇𝜑,∇𝑣)𝑑𝜏+∫^𝑇₀(𝜑,𝑣)𝑑𝜏=∫^𝑇₀(𝜑|𝜑|^𝑝−1,𝑣)𝑑𝜏+(𝜑₁(𝑥),𝑣),(2)𝜑(𝑥,0)=𝜑₀(𝑥)in𝐻¹₀(Ω),𝜑_𝑡(𝑥,0)=𝜑₁(𝑥)in𝐿²(Ω),for all 𝑣∈𝐻¹₀(Ω).

Theorem 3.2 (global existence). Let 𝑝 satisfy
(H)1<𝑝<∞ for 𝑛≤2; 1<𝑝<(𝑛+2)/(𝑛−2) for 𝑛≥3.
Let 𝜑₀(𝑥)∈𝐻¹₀(Ω), 𝜑₁(𝑥)∈𝐿²(Ω). Suppose that 0<𝐸(0)<𝑑, 𝐼(𝜑₀)>0, or ‖∇𝜑₀‖=0. Then problem (2.1) admits a global weak solution 𝜑(𝑥,𝑡)∈𝐿^∞(0,∞;𝐻¹₀(Ω)), 𝜑_𝑡(𝑥,𝑡)∈𝐿^∞(0,∞;𝐿²(Ω)) with 𝜑(𝑡)∈𝑊.

Proof. Let {𝑤_𝑗(𝑥)} be a system of base functions in 𝐻¹₀(Ω). Construct the approximate solutions 𝜑_𝑚(𝑥,𝑡) of problem (2.1) as done in [7] 𝜑_𝑚(𝑥,𝑡)=^𝑚_𝑗=1𝑔_𝑗𝑚(𝑡)𝑤_𝑗(𝑥),𝑚=1,2,…, (3.1) satisfying (𝜑_𝑚𝑡𝑡,𝑤_𝑠)+(∇𝜑_𝑚,∇𝑤_𝑠)+(𝜑_𝑚,𝑤_𝑠)=(𝜑_𝑚|𝜑_𝑚|^𝑝−1,𝑤_𝑠𝜑),𝑠=1,2,…,_𝑚(𝑥,0)=^𝑚_𝑗=1𝑎_𝑗𝑚𝑤_𝑗(𝑥)→𝜑₀(𝑥)in𝐻¹₀𝜑(Ω),_𝑚𝑡(𝑥,0)=^𝑚_𝑗=1𝑏_𝑗𝑚𝑤_𝑗(𝑥)→𝜑₁(𝑥)in𝐿²(Ω). (3.2)
Multiplying (3.2) by 𝑔^′_𝑠𝑚(𝑡) and summing for 𝑠 we can obtain 12𝑑𝑑𝑡‖𝜑_𝑚𝑡‖²+12𝑑𝑑𝑡‖∇𝜑_𝑚‖²+12𝑑𝑑𝑡‖𝜑_𝑚‖²−1𝑑𝑝+1𝑑𝑡‖𝜑_𝑚‖^𝑝+1_𝑝+1=0. (3.3)
Integrating with respect to 𝑡 we obtain 𝐸_𝑚𝑡1(𝑡)=2‖𝜑_𝑚𝑡‖²+12‖∇𝜑_𝑚‖²+12‖𝜑_𝑚‖²−1𝑝+1‖𝜑_𝑚‖^𝑝+1_𝑝+1=12‖𝜑_𝑚𝑡(0)‖²+12‖∇𝜑_𝑚(0)‖²+12‖𝜑_𝑚(0)‖²−1𝑝+1‖𝜑_𝑚(0)‖^𝑝+1_𝑝+1=𝐸_𝑚(0). (3.4)
For the cases 𝐸(0)<𝑑 and 𝐼(𝜑₀)>0 or ‖∇𝜑₀‖=0, we have 12‖𝜑_𝑚𝑡‖²+𝐽(𝜑_𝑚)=𝐸_𝑚(0)<𝑑,0⩽0𝑥00084𝑡<∞,𝐽(𝜑_𝑚1)=2‖∇𝜑_𝑚‖²+12‖𝜑_𝑚‖²−1𝑝+1‖𝜑_𝑚‖^𝑝+1_𝑝+11=(2−1𝑝+1)(‖∇𝜑_𝑚‖²+‖𝜑_𝑚‖²1)+𝑝+1𝐼(𝜑_𝑚)⩾𝑝−12(𝑝+1)(‖∇𝜑_𝑚‖²+‖𝜑_𝑚‖²). (3.5)
Hence we arrive at 12‖𝜑_𝑚𝑡‖²+𝑝−12(𝑝+1)(‖∇𝜑_𝑚‖²+‖𝜑_𝑚‖²)<𝑑, (3.6) then ‖𝜑_𝑚𝑡‖²⩽2𝑑,‖∇𝜑_𝑚‖²⩽2(𝑝+1)𝑝−1𝑑,‖𝜑_𝑚‖²⩽2(𝑝+1)𝑝−1𝑑. (3.7)
Hence, there exist a 𝜑 and a subsequence 𝜑_𝑣 such that 𝜑_𝑣→𝜑 in 𝐿^∞(0,∞;𝐻¹(Ω)) weak star and a.e. in 𝑄=Ω×[0,∞), |𝜑_𝑣|^𝑝−1𝜑_𝑣→|𝜑|^𝑝−1𝜑 in 𝐿^∞(0,∞;𝐿^𝑞(Ω)) weak star, 𝜑_𝑣𝑡→𝜑_𝑡 in 𝐿²(0,∞;𝐿²(Ω)) weakly.
In (3.2) for fixed 𝑠, letting 𝑚=𝑣→∞, we get (𝜑_𝑡,𝑤_𝑠)+(∇𝜑,∇𝑤_𝑠)+(𝜑,𝑤_𝑠)=(|𝜑|^𝑝−1𝜑,𝑤_𝑠),∀𝑠. (3.8)
Integrating 𝑡 from 0 to 𝑡, we obtain that 𝜑(𝑡,𝑥)∈𝐿^∞(0,∞;𝐻²₀(Ω)), 𝜑_𝑡(𝑥,𝑡)∈𝐿^∞(0,∞;𝐿²(Ω)) is a global weak solution of problem (2.1).
Next we prove the fact that 𝜑(𝑡)∈𝑊 for 0≤𝑡<∞. First of all, we will show that 𝜑₀(𝑥)∈𝑊. Let 𝜑(𝑡) be any solution of problem (2.1) with 1𝐸(0)=2‖𝜑₁‖²+𝐽(𝜑₀)<𝑑, (3.9) which gives that 𝐽(𝜑₀)<𝑑. If 𝐼(𝜑₀)>0 then from the definition of 𝑊 we obtain 𝜑₀(𝑥)∈𝑊. If ‖∇𝜑₀‖=0 then 𝜑₀(𝑥)∈𝑊 also. It is easy to see 𝜑_𝑚0(𝑥)∈𝑊 for sufficiently large 𝑚.
It is enough for us to prove 𝜑_𝑚(𝑡)∈𝑊 for sufficiently large 𝑚 and 𝑡>0. If it is false, then there must exist a 𝑡₀>0 for sufficiently large 𝑚 such that 𝜑_𝑚(𝑡₀)∈𝜕𝑊, that is, 𝐼(𝜑_𝑚(𝑡₀))=0,‖∇𝜑_𝑚(𝑡₀)‖≠0,or𝐽(𝜑_𝑚(𝑡₀))=𝑑. (3.10) From the energy inequality 𝐸(0)<𝑑, we get 𝐸_𝑚(0)<𝑑 for sufficiently large 𝑚, that is, 12‖𝜑_𝑚𝑡‖²+𝐽(𝜑_𝑚)=𝐸_𝑚(0)<𝑑. (3.11) Then we can see that 𝐽(𝜑_𝑚(𝑡₀))=𝑑 is impossible. On the other hand, if 𝐼(𝜑_𝑚(𝑡₀))=0, ‖∇𝜑_𝑚(𝑡₀)‖≠0 we obtain 𝜑_𝑚(𝑡₀)∈𝒩. By the definition of 𝒩, we get 𝐽(𝜑_𝑚(𝑡₀))0𝑥00085⩾𝑑, which contradicts (3.11). Hence 𝜑_𝑚(𝑡)∈𝑊 is true.

Theorem 3.3 (blow-up). Assume that 𝜑₀(𝑥)∈𝐻¹₀(Ω), 𝜑₁(𝑥)∈𝐿²(Ω), 𝐸(0)<𝑑, and 𝐼(𝜑₀)<0, then the solution of problem (2.1) must blow up in finite time, that is, there exists a 𝑇>0 such that lim_𝑡→𝑇‖𝜑(𝑡)‖=+∞.

Proof. Let 𝜑(𝑡) be any solution of problem (2.1) with 𝐸(0)<𝑑 and 𝐼(𝜑₀)<0. Set 𝐹(𝑡)=‖𝜑‖², then (𝐹(𝑡))^′=2(𝜑_𝑡,𝜑),(𝐹(𝑡))^′′=2‖𝜑_𝑡‖²+2(𝜑_𝑡𝑡,𝜑)=2‖𝜑_𝑡‖²−2𝐼(𝜑),(𝐹(𝑡))^′′⩾(𝑝+3)‖𝜑_𝑡‖²+(𝑝−1)(𝜆₁+1)𝐹(𝑡)−2(𝑝+1)𝐸(0), (3.12)(3.13) where 𝜆₁>0 is the first eigenvalue of problem Δ𝜑+𝜆𝜑=0,𝜑(𝑥,𝑡)|_𝜕Ω=0.
Now we will consider the following two cases to finish the proof:(i)if 𝐸(0)⩽0, then (𝐹(𝑡))^′′⩾0𝑥00085(𝑝+3)‖𝜑_𝑡‖²;(ii)if 0<𝐸(0)<𝑑, we should discuss this case in aid of set 𝑉_𝛿.
Let 𝛿₁<𝛿₂ be two roots of equation 𝑑(𝛿)=𝐸(0). For any 𝛿∈(𝛿₁,𝛿₂) we will prove 𝜑(𝑡)∈𝑉_𝛿.
First let us prove 𝜑₀∈𝑉_𝛿. From the energy equality 12‖𝜑₁‖²+𝐽(𝜑₀)=𝐸(0)<𝑑(𝛿), (3.14) we get 𝐽(𝜑₀)<𝑑(𝛿)for𝛿₁<𝛿<𝛿₂, (3.15) and 𝐼(𝜑₀)<0 gives 𝐼_𝛿(𝜑₀)<0 for 𝛿₁<𝛿<𝛿₂. Thereby we obtain 𝜑₀∈𝑉_𝛿.
Next let us show that 𝜑(𝑡)∈𝑉_𝛿 for 𝛿₁<𝛿<𝛿₂ and 𝑡⩾0. If it is false, we can find a 𝑡₀∈(0,+∞) as the first time such that 𝜑(𝑡₀)∈𝜕𝑉_𝛿, that is, 𝐽(𝜑(𝑡₀))=𝑑(𝛿) or 𝐼_𝛿(𝜑(𝑡₀))=0 for some 𝛿₁<𝛿<𝛿₂. However from the conservation law we can see that 𝐽(𝜑(𝑡))=𝑑(𝛿) is impossible. If 𝐼_𝛿(𝜑(𝑡₀))=0 then 𝐼_𝛿(𝜑(𝑡))<0 for 0⩽𝑡<𝑡₀. At the same time, Lemma 2.2 yields that ‖∇𝜑(𝑡)‖0𝑥00085⩾𝑟(𝛿)>0 and ‖∇𝜑(𝑡₀)‖⩾𝑟(𝛿). Hence, by the definition of 𝑑(𝛿) we get 𝐽(𝜑(𝑡))0𝑥00085⩾𝑑(𝛿), which contradicts 𝐽(𝜑)<𝑑(𝛿). So we obtain 𝜑(𝑡)∈𝑉_𝛿 for 𝛿₁<𝛿<𝛿₂ and 𝑡⩾0𝑥000850. Hence, 𝐼_𝛿(𝜑)<0 and ‖∇𝜑‖0𝑥00085⩾𝑟(𝛿). Let 𝛿→𝛿₂, then 𝐼_𝛿2(𝜑)0𝑥00084⩽0 and ‖∇𝜑‖⩾0𝑥00085𝑟(𝛿₂). By (3.12) we obtain (𝐹(𝑡))^′′=2‖𝜑_𝑡‖²−2𝐼(𝜑)0𝑥00085⩾−2𝐼(𝜑)=2(𝛿₂−1)‖∇𝜑‖²−2𝐼_𝛿2(𝜑)⩾2(𝛿₂−1)𝑟²(𝛿₂)≡𝑎⁰,𝑡⩾0.0𝑥00085 (3.16)
For 𝑎=min{(𝑝+3)‖𝜑_𝑡‖²,𝑎⁰} we have (𝐹(𝑡))^′⩾𝑎𝑡+(𝐹(0))^′,∀𝑡⩾0. (3.17) Hence there exists a 𝑡₀⩾0𝑥000850 such that (𝐹(𝑡))^′⩾0𝑥00085(𝐹(𝑡₀))^′>0,𝑡0𝑥00085⩾𝑡₀, (3.18) which gives 𝐹0𝑥00085(𝑡)⩾(𝐹(𝑡₀))^′(𝑡−𝑡₀)+𝐹(𝑡₀),𝑡⩾𝑡₀. (3.19) By (3.13) for sufficiently large 𝑡, we obtain (𝐹(𝑡))^′′𝐹(𝑡)−𝑝+34𝐹(𝑡)^′0𝑥00085⩾(𝑝+3)(‖𝜑_𝑡‖²‖𝜑‖²−(𝜑_𝑡,𝜑)²)⩾0𝑥000850. (3.20) By a direct computation we can see that (𝐹(𝑡)^−𝛼)^′′=−𝛼𝐹^−𝛼−2(𝐹𝐹^′′+(−𝛼−1)(𝐹^′)²). (3.21) Let 𝛼=(𝑝−1)/4, then we get 𝐹𝐹^′′+(−𝛼−1)(𝐹^′)²0𝑥00085⩾0, (3.22) that is, (𝐹(𝑡)^−𝛼)^′′⩽0. (3.23) Applying properties of concave function we can get that there exists a bounded 𝑇>0 such that lim_𝑡→𝑇𝐹(𝑡)=+∞. (3.24)

From the above two theorems we can easily get a sharp condition for global existence and blow-up of solutions to problem (2.1) like the following.

Let 𝑝 satisfy (H). Assume that 𝜑₀(𝑥)∈𝐻¹₀(Ω), 𝜑₁(𝑥)∈𝐿²(Ω), 0<𝐸(0)<𝑑. Then 𝐼(𝜑₀)>0 supports problem (2.1) to admit a global weak solution, and 𝐼(𝜑₀)<0 leads blow-up of solutions for problem (2.1).

Acknowledgments

This work is supported by Natural Science Foundation of Heilongjiang Province (A200702; A200810); Science and Technology Foundation of Education Office of Heilongjiang Province (11541276).