Blow up for Systems of Wave Equations in Exterior Domain

Xiuwen LUO

Abstract In this paper the author studies the initial boundary value problem of semilinear wave systems in exterior domain in high dimensions(n≥3).Blow up result is established and what is more,the author gets the upper bound of the lifespan.For this purpose the test function method is used.

Keywords Wave equations,Exterior domain,Blow up,Lifespan

1 Introduction and Main Results

In this paper,we consider the initial boundary value problem of semilinear wave equations in exterior domain:

where u(t,x),v(t,x)are unknown functions of the variable t∈R+and x∈?c.? is a smooth compact obstacle in Rn(n ≥ 3),and ?cis its complement.Without loss of generality,we assume that 0∈?$BR,where BR={x||x|≤R}is a ball of radius R centered at the origin.The smallness of initial data can be measured by the constant ε satisfying 0< ε≤ 1 and u0,u1,v0,v1are compactly supported nonnegative functions,i.e.,

This problem can date back to the famous Strauss’conjecture(see[19]):For the Cauchy problem of the semilinear wave equation with small initial date

there exists a critical power pc(n),which is the positive root of the following quadratic equation:

which means that blow up happens when 1pc(n).Strauss’conjecture was first investigated by John[7]and recently ended after decades by a series of hardworking(see[4–5,9,14,16–17,20–24]).

For the initial boundary value problem of a single wave equation in exterior domain,Zhou and Han[25]gave the lifespan estimation of the solutions when 1

The Cauchy problem for high dimension(n≥4)systems was considered by Georgiev,Takamura and Zhou[3].

For high dimension(n≥3)systems,there exists a curve F(p,q)=0 which separates the domains of existence and blow-up of the solutions in the(p,q)plane(see[15]).And we are concerned for the blow-up and the lifespan T(ε)of the solutions.More precisely,the blow-up happens when

We establish our theorem as follows.

Theorem 1.1 Let ? ? Rn(n ≥ 3)satisfy the exterior ball conditions and the exponents p,q>1 satisfy F(p,q)>0.Suppose that problem(1.1)–(1.2)has a solution(u,ut),(v,vt) ∈C([0,T),H1(?c)×L2(?c))such that

Then T< ∞,and there exists a positive constant C which is independent of ε such that

Here and hereafter,C and Cidenote positive constants and may change from line to line.

2 Preliminaries

To prove our theorem,we first introduce a lemma.

Lemma 2.1 Let 1

Then there exists a positive constant C2independent of C1such that

when F(p,q)>0.

It is easy to prove this lemma by exchanging p,q and set T=τ,F(t)=Y(t),G(t)=X(t),α =n(q?1),β =n(p?1),in Lemma 2.1 of[15].Then the blow-up condition can be rewritten as

which is equivalent to F(p,q)>0 when p≤q.

The following lemmas are from[2,21,25].

Lemma 2.2 There exists a function φ0(x)∈ C2(?c)satisfying the following boundary value problem:

Moreover,φ0(x)satisfies that?x∈?c,0<φ0(x)<1.

Lemma 2.3 There exists a function φ1(x)∈ C2(?c)satisfying the following boundary value problem:

Moreover,there exists a positive constant C3such that φ1(x)satisfies that?x ∈ ?c,0< φ1(x)<

We define a test function ψ1(x,t)as

Then we have the following lemma.

Moreover,we have the following lemmas.

Lemma 2.6 Let p>1.Then ?t≥ 0,

3 Proof of Theorem 1.1

We give the proof of Theorem 1.1 in this section.We first define functions

where u(t,x),v(t,x),φ0(x),ψ1(t,x)are defined as before.The assumptions on u(t,x),v(t,x)imply that F0(t),F1(t),G0(t),G1(t)are well-defined C2functions for all t≥0.We want to derive nonlinear ordinary differential inequalities of those new functions as the form in Lemma 2.1,then we can easily proof Theorem 1.1.

We need one more lemma as below.

Lemma 3.1 Let(u0,u1),(v0,v1)satisfy(1.2).Suppose that problem(1.1)has a solution(u,ut),(v,vt)∈ C([0,T),H1(?c)×L2(?c)),such that

Then we have for all t≥0,

Proof We multiply the first equation in(1.1)by the test function ψ1(t,x)and integrate over[0,t]×?c,then use integration by parts and Lemma 2.3.

First,it is easy to verify that

Noticing that ψ1t= ?ψ1,we have

And

So we get

where

Because of ψ1>0,we have

Multiplying the above inequality with e2t,one has

Then integrating over[0,t],we get

Next we show that F0(t),G0(t)satisfy the ordinary differential inequalities as in(2.1).For this purpose,multiplying the first equation in(1.1)by φ0(x)and integrating over ?c.We note that for a fixed t,u(·,t) ∈(Dt)where Dtis the support of u(·,t).Hence we can use integration by parts and Lemma 2.2.

First

where

So we get

By the H¨older inequality,we get the estimate of the right hand side

i.e.,

So we have

where

Therefore

Similarly,we get

which implies

So we have

Noticing G1(t)≥ εC8>0 by(3.3)and Lemma 2.5,we get

Integrating once more and we reach the final estimate

Hence the following estimation is valid when t is large enough:

Similarly,we have

Therefore,we establish the following inequalities:

Set X(τ)= εaF0(t),Y(τ)= εbG0(t), τ= εdt in(3.11),where a,b,d are constants which should be determined later.(3.11)becomes

To use Lemma 2.1,we need

And by Lemma 2.1,we have

i.e.,

This completes the proof of Theorem 1.1.

Acknowledgement The author would like to show his gratitude to Prof.Yi Zhou and Prof.Ningan Lai for their kindly help and thank the reviewers for advices.