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

Authors: Zhao Junsheng; Li Shufeng
Journal: Mathematical Problems in Engineering
 (2009)

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## 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.


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## Body

## 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:(1.1)φtt-Δφ+φ=|φ|p-1φ,
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+up=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(2.1)φtt-∆φ+φ=|φ|p-1φ,x∈Ω,t>0,φ(x,0)=φ0(x),φt(x,0)=φ1(x),x∈Ω,φ(x,t)=0,x∈∂Ω,t⩾0,where1<p<∞ for n⩽2; 1<p<(n+2)/(n-2) for n⩾3.For problem (2.1), we define the energy function and some functionals as follows:
(2.2)E(t)=12∥φt∥2+12∥φ∥2+12∥∇φ∥2-1p+1∥φ∥p+1p+1,J(φ)=12∥φ∥2+12∥∇φ∥2-1p+1∥φ∥p+1p+1,I(φ)=∥φ∥2+∥∇φ∥2-∥φ∥p+1p+1,Iδ(φ)=δ∥∇φ∥2+∥φ∥2-∥φ∥p+1p+1.
In aid of  the above functionals, we define the potential well as follows:
(2.3)W={φ∈H01(Ω)∣I(φ)>0,J(φ)<d}∪{0},
where
(2.4)d=infφ∈𝒩J(φ),𝒩:={φ∈H01(Ω)∣I(φ),∥∇φ∥≠0}.
Then we further give the following definitions
(2.5)d(δ)=infφ∈𝒩δJ(φ),𝒩δ:={φ∈H01(Ω)∣Iδ(φ)=0,∥∇φ∥≠0}.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:(2.6)Wδ={φ∈H01(Ω)∣Iδ(φ)>0,J(φ)<d(δ)}∪{0},0<δ<1,V={φ∈H01(Ω)∣I(φ)<0,J(φ)<d},Vδ={φ∈H01(Ω)∣Iδ(φ)<0,J(φ)<d(δ)}.The following lemmas are given to show the relations between the functionalIδ(φ) and ∥∇φ∥.Lemma 2.1.
If0<∥∇φ∥<r(δ) then Iδ(φ)>0, where r(δ)=(δ/C*p+1)1/(p-1) and C*=sup(∥φ∥p+1/∥∇φ∥).Proof.
If0<∥∇φ∥<r(δ), then
(2.7)Iδ(φ)=δ∥∇φ∥2-(∥φ∥p+1p+1-∥φ∥2),∥φ∥p+1p+1-∥φ∥2⩽C*p+1∥∇φ∥p+1⩽δ∥∇φ∥2.
Hence Iδ(φ)>0.Lemma 2.2.
IfIδ(φ)<0 then ∥∇φ∥>r(δ).Proof.
Note thatIδ(φ)<0 gives
(2.8)δ∥∇φ∥2<∥φ∥p+1p+1-∥φ∥2<∥φ∥p+1p+1⩽C*p+1∥∇φ∥p-1∥∇φ∥2,
which implies
(2.9)∥∇φ∥p-1>δC*p+1=rp-1(δ),
that can be deduced from (2.8).Lemma 2.3.
As the function ofδ,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 thatd(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∈𝒩δ2, there exist a v∈𝒩δ1 and a constant ε(δ1,δ2)≥𝒩δ1 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+1p+1, then Iδ(λ(δ)φ)=0, λ(δ2)=1. Let g(λ)=J(λφ), then
(2.10)ddλg(λ)=1λ(∥λφ∥2+∥λ∇φ∥2-∥λφ∥p+1p+1)=1λ((1-δ)∥λ∇φ∥2+Iδ(λφ))=(1-δ)∥λ∇φ∥2.
Takev=λ(δ1)φ, then v∈𝒩δ1.
For0<δ1<δ2<1, we have
(2.11)J(φ)-J(v)=g(1)-g(λ(δ1))>(1-δ2)r2(δ2)λ(δ1)(1-λ(δ1))≡ε(δ1,δ2).
For 1<δ2<δ1<(p+1)/2, we have
(2.12)J(φ)-J(v)=g(1)-g(λ(δ1))>(δ2-1)r2(δ2)λ(δ2)(λ(δ1)-1)≡ε(δ1,δ2).
These give the conclusion.


## 3. Sharp Condition for Global Existence and Blow-Up

Definition 3.1 (weak solution).
The functionφ(x,t)∈L∞(0,T;H01(Ω)) with φt(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)
(φt,v)+∫0T(∇φ,∇v)dτ+∫0T(φ,v)dτ=∫0T(φ|φ|p-1,v)dτ+(φ1(x),v),(2)
φ(x,0)=φ0(x)inH01(Ω),φt(x,0)=φ1(x)inL2(Ω),for all v∈H01(Ω).Theorem 3.2 (global existence).
Letp satisfy
(H)1<p<∞ for n≤2; 1<p<(n+2)/(n-2) for n≥3.
Letφ0(x)∈H01(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,∞;H01(Ω)), φt(x,t)∈L∞(0,∞;L2(Ω)) with φ(t)∈W.Proof.
Let{wj(x)} be a system of base functions in H01(Ω). Construct the approximate solutions φm(x,t) of problem (2.1) as done in [7]
(3.1)φm(x,t)=∑j=1mgjm(t)wj(x),m=1,2,…,
satisfying
(3.2)(φmtt,ws)+(∇φm,∇ws)+(φm,ws)=(φm|φm|p-1,ws),s=1,2,…,φm(x,0)=∑j=1majmwj(x)→φ0(x)inH01(Ω),φmt(x,0)=∑j=1mbjmwj(x)→φ1(x)inL2(Ω).
Multiplying (3.2) by gsm′(t) and summing for s we can obtain
(3.3)12ddt∥φmt∥2+12ddt∥∇φm∥2+12ddt∥φm∥2-1p+1ddt∥φm∥p+1p+1=0.
Integrating with respect tot we obtain
(3.4)Emt(t)=12∥φmt∥2+12∥∇φm∥2+12∥φm∥2-1p+1∥φm∥p+1p+1=12∥φmt(0)∥2+12∥∇φm(0)∥2+12∥φm(0)∥2-1p+1∥φm(0)∥p+1p+1=Em(0).
For the casesE(0)<d and I(φ0)>0 or ∥∇φ0∥=0, we have
(3.5)12∥φmt∥2+J(φm)=Em(0)<d,0⩽t<∞,J(φm)=12∥∇φm∥2+12∥φm∥2-1p+1∥φm∥p+1p+1=(12-1p+1)(∥∇φm∥2+∥φm∥2)+1p+1I(φm)⩾p-12(p+1)(∥∇φm∥2+∥φm∥2).
Hence we arrive at(3.6)12∥φmt∥2+p-12(p+1)(∥∇φm∥2+∥φm∥2)<d,
then
(3.7)∥φmt∥2⩽2d,∥∇φm∥2⩽2(p+1)p-1d,∥φm∥2⩽2(p+1)p-1d.
Hence, there exist aφ and a subsequence φv such that φv→φ in L∞(0,∞;H1(Ω)) weak star and a.e. in Q=Ω×[0,∞), |φv|p-1φv→|φ|p-1φ in L∞(0,∞;Lq(Ω)) weak star, φvt→φt in L2(0,∞;L2(Ω)) weakly.
In (3.2) for fixed s, letting m=v→∞, we get
(3.8)(φt,ws)+(∇φ,∇ws)+(φ,ws)=(|φ|p-1φ,ws),∀s.
Integratingt from 0 to t, we obtain that φ(t,x)∈L∞(0,∞;H02(Ω)), φt(x,t)∈L∞(0,∞;L2(Ω)) is a global weak solution of problem (2.1).
Next we prove the fact thatφ(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
(3.9)E(0)=12∥φ1∥2+J(φ0)<d,
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 φm0(x)∈W for sufficiently large m.
It is enough for us to proveφm(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 φm(t0)∈∂W, that is,
(3.10)I(φm(t0))=0,∥∇φm(t0)∥≠0,orJ(φm(t0))=d.
From the energy inequality E(0)<d, we get Em(0)<d for sufficiently large m, that is,
(3.11)12∥φmt∥2+J(φm)=Em(0)<d.
Then we can see that J(φm(t0))=d is impossible. On the other hand, if I(φm(t0))=0, ∥∇φm(t0)∥≠0 we obtain φm(t0)∈𝒩. By the definition of 𝒩, we get J(φm(t0))⩾d, which contradicts (3.11). Hence φm(t)∈W is true.Theorem 3.3 (blow-up).
Assume thatφ0(x)∈H01(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 limt→T∥φ(t)∥=+∞.Proof.
Letφ(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(φt,φ),(3.12)(F(t))′′=2∥φt∥2+2(φtt,φ)=2∥φt∥2-2I(φ),(3.13)(F(t))′′⩾(p+3)∥φt∥2+(p-1)(λ1+1)F(t)-2(p+1)E(0),
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)
ifE(0)⩽0, then (F(t))′′⩾(p+3)∥φt∥2;(ii)
if0<E(0)<d, we should discuss this case in aid of set Vδ.
Letδ1<δ2 be two roots of equation d(δ)=E(0). For any δ∈(δ1,δ2) we will prove φ(t)∈Vδ.
First let us proveφ0∈Vδ. From the energy equality
(3.14)12∥φ1∥2+J(φ0)=E(0)<d(δ),
we get
(3.15)J(φ0)<d(δ)forδ1<δ<δ2,
and I(φ0)<0 gives Iδ(φ0)<0 for δ1<δ<δ2. Thereby we obtain φ0∈Vδ.
Next let us show thatφ(t)∈Vδ 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)∈∂Vδ, that is,  J(φ(t0))=d(δ) or Iδ(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 Iδ(φ(t0))=0 then Iδ(φ(t))<0 for 0⩽t<t0. At the same time, Lemma 2.2 yields that ∥∇φ(t)∥⩾r(δ)>0 and ∥∇φ(t0)∥⩾r(δ). Hence, by the definition of d(δ) we get J(φ(t))⩾d(δ), which contradicts J(φ)<d(δ). So we obtain φ(t)∈Vδ for δ1<δ<δ2 and t⩾0. Hence, Iδ(φ)<0 and ∥∇φ∥⩾r(δ). Let δ→δ2, then Iδ2(φ)⩽0 and ∥∇φ∥⩾r(δ2). By (3.12) we obtain
(3.16)(F(t))′′=2∥φt∥2-2I(φ)⩾-2I(φ)=2(δ2-1)∥∇φ∥2-2Iδ2(φ)⩾2(δ2-1)r2(δ2)≡a0,t⩾0.
Fora=min{(p+3)∥φt∥2,a0} we have
(3.17)(F(t))′⩾at+(F(0))′,∀t⩾0.
Hence there exists a t0⩾0 such that
(3.18)(F(t))′⩾(F(t0))′>0,t⩾t0,
which gives
(3.19)F(t)⩾(F(t0))′(t-t0)+F(t0),t⩾t0.
By (3.13) for sufficiently large t, we obtain
(3.20)(F(t))′′F(t)-p+34F(t)′⩾(p+3)(∥φt∥2∥φ∥2-(φt,φ)2)⩾0.
By a direct computation we can see that
(3.21)(F(t)-α)′′=-αF-α-2(FF′′+(-α-1)(F′)2).
Let α=(p-1)/4, then we get
(3.22)FF′′+(-α-1)(F′)2⩾0,
that is,
(3.23)(F(t)-α)′′⩽0.
Applying properties of concave function we can get that there exists a bounded T>0 such that
(3.24)limt→TF(t)=+∞.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.Letp satisfy (H). Assume that φ0(x)∈H01(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).



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*Source: 164303-2009-07-02.xml*