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:
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
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,
where 1<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:
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:
W={φ∈H01(Ω)∣I(φ)>0,J(φ)<d}∪{0},
where
d=infφ∈𝒩J(φ),𝒩:={φ∈H01(Ω)∣I(φ),∥∇φ∥≠0}.
Then we further give the following definitions
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:
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 functional Iδ(φ) and ∥∇φ∥.
Lemma 2.1.
If 0<∥∇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.
If 0<∥∇φ∥<r(δ), then
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.
If Iδ(φ)<0 then ∥∇φ∥>r(δ).
Proof.
Note that Iδ(φ)<0 gives
δ∥∇φ∥2<∥φ∥p+1p+1-∥φ∥2<∥φ∥p+1p+1⩽C*p+1∥∇φ∥p-1∥∇φ∥2,
which implies
∥∇φ∥p-1>δC*p+1=rp-1(δ),
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∈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
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.
Take v=λ(δ1)φ, then v∈𝒩δ1.
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).
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).
These give the conclusion.
3. Sharp Condition for Global Existence and Blow-UpDefinition 3.1 (weak solution).
The function φ(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:
Let φ0(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]
φm(x,t)=∑j=1mgjm(t)wj(x),m=1,2,…,
satisfying
(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
12ddt∥φmt∥2+12ddt∥∇φm∥2+12ddt∥φm∥2-1p+1ddt∥φm∥p+1p+1=0.
Integrating with respect to t we obtain
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 cases E(0)<d and I(φ0)>0 or ∥∇φ0∥=0, we have
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
12∥φmt∥2+p-12(p+1)(∥∇φm∥2+∥φm∥2)<d,
then
∥φmt∥2⩽2d,∥∇φm∥2⩽2(p+1)p-1d,∥φm∥2⩽2(p+1)p-1d.
Hence, there exist a φ 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
(φt,ws)+(∇φ,∇ws)+(φ,ws)=(|φ|p-1φ,ws),∀s.
Integrating t 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 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
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 prove φm(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,
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,
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 that φ0(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.
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(φt,φ),(F(t))′′=2∥φt∥2+2(φtt,φ)=2∥φt∥2-2I(φ),(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:
if E(0)⩽0, then (F(t))′′⩾(p+3)∥φt∥2;
if 0<E(0)<d, we should discuss this case in aid of set Vδ.
Let δ1<δ2 be two roots of equation d(δ)=E(0). For any δ∈(δ1,δ2) we will prove φ(t)∈Vδ.
First let us prove φ0∈Vδ. From the energy equality
12∥φ1∥2+J(φ0)=E(0)<d(δ),
we get
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 that φ(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
(F(t))′′=2∥φt∥2-2I(φ)⩾-2I(φ)=2(δ2-1)∥∇φ∥2-2Iδ2(φ)⩾2(δ2-1)r2(δ2)≡a0,t⩾0.
For a=min{(p+3)∥φt∥2,a0} we have
(F(t))′⩾at+(F(0))′,∀t⩾0.
Hence there exists a t0⩾0 such that
(F(t))′⩾(F(t0))′>0,t⩾t0,
which gives
F(t)⩾(F(t0))′(t-t0)+F(t0),t⩾t0.
By (3.13) for sufficiently large t, we obtain
(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
(F(t)-α)′′=-αF-α-2(FF′′+(-α-1)(F′)2).
Let α=(p-1)/4, then we get
FF′′+(-α-1)(F′)2⩾0,
that is,
(F(t)-α)′′⩽0.
Applying properties of concave function we can get that there exists a bounded T>0 such that
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.
Let p 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).
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).
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