Qian LEI Han YANG

Abstract The authors consider the critical exponent problem for the variable coefficients wave equation with a space dependent potential and source term.For sufficiently small data with compact support,if the power of nonlinearity is larger than the expected exponent,it is proved that there exists a global solution.Furthermore,the precise decay estimates for the energy,L2and Lp+1norms of solutions are also established.In addition,the blow-up of the solutions is proved for arbitrary initial data with compact support when the power of nonlinearity is less than some constant.

Keywords Semilinear wave equations,Global existence,Energy decay,L2and Lp+1 estimates,Blow up

1 Introduction

We consider the following Cauchy problem for the semilinear wave equation with variable coefficients:

where ε>0,the coefficients a(x)∈ C0(Rn),b(x)∈ C1(Rn)are positive functions which will be specified later and the initial data u0∈H1(Rn),u1∈L2(Rn)have compact support

where the exponent p of nonlinearity satisfies

Such a system is generally accepted as models for travelling waves in a nonhomogeneous gas with damping changing with the position.The unknown u denotes the displacement,the coefficient b called the bulk modulus,accounts for changes of the temperature depending on the location,while a is referred as the friction coefficient or potential(see[5]).This problem has been studied intensively for the homogenous medium,but the result are scarce for the variable coefficient case.In[11]there is the authors find decay estimates for wave equations with variable coefficient,however,no nonlinearity is present.In addition,[1],looked at an equation with nonlinear internal damping but for bounded domains.To our knowledge,the results of this paper are the first to be obtained for semilinear wave equations which exhibit space dependent hyperbolic operators and space dependent potential on the entire space Rn.

Our aim is to determine the critical exponent pc,which is a number defined by the following property:

If pc

It is of interest to compare the semilinear wave equations(1.1)–(1.2)for different coefficients.When a(x)=0 namely the damping term is missing and b(x)=1,that is

for small data with compact support,there exists a critical exponent pw(n)such that the solutions of(1.3)–(1.4)are global if p>pw(n),and the solutions of(1.3)–(1.4)blow up if 1

There are many results for the semilinear damped wave equation.Todorova and Yordanov[17–18]studied the constant coefficients case of(1.1)–(1.2),that is

They developed a weighted energy method and determined that the critical exponent is pc(n)=1+more precisely,they proved small data global existence in the case p>pc(n)and blow-up for all solutions of(1.5)–(1.6)with positive on average data in the case 1

On the other hand,Ikehata,Todorova and Yordanov[7]solved the critical exponent problem for the wave equation(1.1)–(1.2)when b(x)=1,

They determined that the critical exponent isby using a refined multiplier method,where a(x)∈C1(Rn)is a radially symmetric function satisfying

with a0>0 and α∈[0,1).They derived the global existence of the sufficiently small data for p>pc(n,α),also obtained precise decay estimates for the energy,L2and Lp+1norms of solutions.Moreover,they proved that the solutions blow up for 1

where a(t)=a0(1+t)−β, β ∈ (−1,1).He proved that the critical exponent is pc(n)=1+2n.Wakasugi[20]considered the Cauchy problem for the semilinear wave equation with space-time dependent damping

where a(x)=a0(1+|x|2)−α2,b(t)=(1+t)−β,with a0>0, α,β ≥ 0, α + β <1,and proved that the expected exponent is given by

which is the critical exponent for the semilinear wave equations with space dependent potential.This shows that,roughly speaking,time-dependent coefficient of damping term does not influence the critical exponent.This is also why we consider wave equation with only space dependent potential.

The main innovation in this paper is that we find the exponent pcr(n,α,β)such that for sufficiently small data and pcr(n,α,β)

For the potential a(x)and the bulk modulus b(x),we assume that

and

where a0,a1,b0,b1>0 are constants,and α and β belong to the following range of exponents

the exponent of focusing nonlinearity is given explicitly by

In the case of variable coefficient wave equation we observe some new phenomena.The decay rates pinpoint the interaction between the coefficients a and b.It is worthwhile to mention that the energy decay rate goes to infinity(see Corollary 1.3)when β → 2−and α → 0+.This shows that the range of the exponent β in(1.9)is natural.On the other hand,in the case of β → 0+,α → 1−,the energy of global solutions decays polynomially like t−nas t→ ∞ (see Corollary 1.3).

Before stating the main results,we need some preparations concerning the space dependent on factors a(x)and b(x).

Hypothesis A(see[11])Under the above assumptions(1.7)and(1.8),there exists a subsolution A(x)which satisfies

and has the following properties:

Here we announce our main results for the existence of the global solutions for sufficiently small data.

Let us denote X1(0,T):=C([0,T);H1(Rn))∩C1([0,T);L2(Rn)).

Theorem 1.1 Let pcr(n,α,β)be defined in(1.10),a(x),b(x)satisfy(1.7),(1.8)respectively and the exponents α,β belong to(1.9).If pcrfor n ≥ 3 and pcr(n,α,β)0 such that for any 0< ε< ε0,the problems(1.1)–(1.2)has a solution u ∈ X1(0,∞)satisfying

for large t≫1,where and δ>0 is an arbitrarily small number.

Remark 1.1 In the case of constant coefficients a(x)=1,b(x)=1,we have α =0,β =0.Thus pcr(n,α,β)becomes the Fujita’s critical exponent 1+Furthermore,in the case of constant bulk modulus b(x)=1,namely,β=0,then the exponent agrees with that of only the space dependent coefficient case in the literature[7].

Proposition 1.1(see[11])Let a(x),b(x)satisfy(1.7),(1.8)respectively.

(i)(1.11)admits a solution A(x)such that

where A0and A1are positive constants.

(ii)In the special case,

with a2>0,b2>0,(1.11)has a solution with the following properties:

Combining the above proposition with Theorem 1.1,we can give more explicit weighted estimates.

Corollary 1.1 Under the assumptions in Theorem 1.1,the following estimates hold:

for large t≫ 1,where ρ is defined as Theorem 1.1.

Corollary 1.2 Assume that a(x),b(x)satisfy the condition(1.15).Then for every δ>0,the solution of(1.1)–(1.2)satisfies

for large t≫ 1,where ρ is defined as Theorem 1.1.

Corollary 1.3 Assume that a(x),b(x)satisfy the condition(1.15).Then for every δ>0,the solution of(1.1)–(1.2)satisfies

for large t≫ 1,where ρ is defined as Theorem 1.1.

Another important consequence of main conclusions is that the energy estimate under consideration,restricted to{x:A(x)≥ t1+κ}with κ >0,decays exponentially.

Corollary 1.4 Under the assumptions in Theorem 1.1,for arbitrary fixed κ>0 and every δ>0,the solution of(1.1)–(1.2)satisfies

for large t≫1,where A(x)andµare given by Hypothesis A.

Thus,the local energy in{x:A(x)≥ t1+κ}decays exponentially fast as t→ ∞.This observation confirms that for small data the global solutions of(1.1)–(1.2)have parabolic asymptotic profiles.

The blow up result in the case when 1

Theorem 1.2 Let a(x)and b(x)satisfy(1.7)and(1.8)respectively,and let the exponents α,β belong to

When 1

then the solution of problem(1.1)–(1.2)does not exist globally for any ε >0.

Remark 1.2 In the case of constant a(x)=1,b(x)=1,we obtain α =0,β =0.Then the exponent p2(n,0,0)=1+becomes the Fujita’s critical exponent.In addition,when b(x)=1,namely,β =0,the exponent p2(n,α,0)=1+coincides with the blow up result in the literature[7].

Remark 1.3 When p2(n,α,β)

2 Small Data Global Solutions

We first state a proposition about the support of the solutions for the wave equation with variable coefficients.Fortunately,the argument has presented in[11].

Proposition 2.1(Finite Speed of Propagation)Assume that b(x)satisfies(1.8).If u0,u1are supported inside the ball|x|

Moreover,one has that the radius Rtfor a general b(x)satisfies the following estimates:

Proposition 2.2(see[19])Define γ :=,then γ ∈ [0,1]and

Then

where g0and G0are positive constants.

To show the global existence of solutions of problem(1.1)–(1.2)for sufficiently small data,we rely on a modification of technique developed by Todorova and Yordanov[17–18].Indeed,for the solution u(x,t)of problem(1.1)–(1.2)we set v=uw−1,where w is an approximate solution of linear part of(1.1)–(1.2)and can be defined by

where the parameters m:=µ−2δ,m1:=µ−δ and A(x)is determined in Hypothesis A,where δ∈ (0,µ)is a small number.

We also set

where

We consider the semilinear wave equation of the form

where the coefficients a(x)∈C0(Rn),b(x)∈C1(Rn).

Our goal is to derive a weighted energy identity for u.Let v=w−1u and substitute u=wv into(2.5),we have

where

Multiplying both sides of(2.6)by wv+w1vtand integrating over Rn,we have the equality

where the weighted energy

and

Different conditions are needed for the damping weights w1,w to ensure that F(vt,∇v)+G(v)>0,and hence the weighted energy E(vt,∇v,v)is bounded.

Lemma 2.1 Let a(x)and b(x)satisfy conditions(1.7)and(1.8).There exists a large number t0>0 such that for t≥t0the following conditions hold:

(i)Q≥0,Qt≤0,

(ii)−∂tw1+w ≥0,

(iii)(−∂tw1+2(a+2w−1wt)w1−2w)(−∂tw1+2w)≥ b(∇w1−2w1w−1∇w)2.

If u is a solution of(1.1)–(1.2),where t∈ (t0,Tm),we have

Proof The proof of conditions(i)–(iii)is similar to[19],so we omit it.Notice that conditions(i)and(ii)imply hence G(v)≥0.Condition(iii)and−∂tw1+2w ≥0,which follows from(ii),guarantee that the quadratic form F(vt,∇v)≥ 0.Therefore,after integrating(2.7)over[t0,t],we can obtain the final inequality,where t0

We need estimates(I)and(II)in the right side of(2.12).Now we introduce a new function:

For(I)in the right side of(2.12),we have the following crucial estimate.

Lemma 2.2 Let a(x)and b(x)satisfy conditions(1.7)–(1.8).If p>pcr(n,α,β),then there exists a number ρ >0,which depends on p,n,α,β and δ such that

where

Proof Using definitions of w(t,x)and w1(t,x),we have

and

We add the above two estimates and integrate it over Rn,then get

By setting

we can rewrite(2.14)in the form

To estimate the weighted norm ke−ηψ(t,·)vkp+1,we use the Gagliardo-Nirenberg inequality

where

We estimate the first term(A)in the right side of inequality(2.16),beginning with following decomposition

Furthermore,there exists a constant C>0,such that for any x>0,it is true that

Thus,

this inequality combined with(2.17)implies

To estimate the second term(B)in the right of inequality(2.16),notice that

Integrating(B2)by parts

combining(B1)–(B3)and integrating over Rn,we obtain

To estimate the first term(B4)in the right side of inequality(2.19),we use

So,(B4)of inequality(2.19)will be

Since the exponential term satisfies

(B4)is estimated as follows

To proceed,we estimate the second term(B5)in the right side of inequality(2.19).From the definition of w1and(2.20),we see that

where 0 ≤ β <2,C>0,x 7→ (6+x)e−kx(k>0)is bounded above.Then the final estimate of(B5)is

Therefore,by using(2.18)and(2.21)–(2.22),we rewrite inequality(2.16)as

This inequality together with(2.15)gives

where

Finally,from the assumption pcr(n,α,β)

0,which implies the desired estimate.

Now using the result of Lemma 2.2,we are able to estimate the second term(II)to the right side of(2.12).

Lemma 2.3 Under the assumptions in Lemma 2.2,we have

with some constant C>0,where ρ>0 is the constant determined in Lemma 2.2.

Proof It follows from the definitions of w1and w that

It is easy to see that

moreover,

Hence,we obtain

which implies

Integrating over[t0,t],one has

Thus,by using Lemma 2.2 and(2.24),we derive the estimate in Lemma 2.3.

From Lemmas 2.2–2.3 and the weighted energy inequality(2.12),we get the following estimate

Using w1>0 and Q≥0,which follows from Lemma 2.1(iii),we obtain Qw1≥0.Sincewe can rewrite(2.25)as follows

We need one more preparation.

Lemma 2.4 Let γ∈[0,1],c0>0,E0>0 be given real numbers,and let f∈C([t0,Tm))be a monotone increasing function.If a function h∈C1([t0,Tm))satisfies the inequality

then the following estimate holds

with some constant C>0.

Proof The proof is similar to[7],so we omit it.

Let

Note that the function t 7→M(t)is monotone increasing.Under these preparations one can prove the following lemma.

Lemma 2.5 Let γ∈[0,1],then the following bound holds

for t∈[t0,Tm)with large t0>0.

ProofSince a(x)≥ g(t)≥ g0t−γas it was presented in Proposition 2.2,it follows from(2.26)that

Note that the function

is monotone increasing.We can apply Lemma 2.4 with

and obtain the desired estimate.

Denote by

where u ∈ X1[0,Tm)is the weak solutions of(1.1)–(1.2).We are in need of the following lemmas.

Lemma 2.6 For each t∈[0,Tm),it is true that Z

with some t-dependent constant CR(t)satisfying

ProofThe proof is omitted since it elementary follows from the fact that v=w−1u,the compact support of the data and the Poincarinequality.

The standard energy inequality associated with the problem(1.1)–(1.2)is

Lemma 2.7 Let t0>0 be the time defined in Lemmas 2.1–2.6,then there exists T ∈(t0,Tm),which depends on ε >0,such that for all t∈ [0,T]

Proof With a simple modification for[7],the lemma can be easily proved,so we omit it here.

3 Global Existence

In this section,we are going to prove the main theorem and corollaries.However,we need to state several lemmas in order to find the decay estimates for the energy,L2and Lp+1norms.

Lemma 3.1 Assume that the a(x)and b(x)satisfy(1.7)–(1.8),and γif the weights w and w1satisfy conditions(i)–(iii)and

then

for all t≥t0.

Proof For the proof of conditions(iv)–(v)(see[19]),Lemma 2.7 shows that we can consider the case t0

For any ǫ∈ (0,1),since

then

Hence,(3.1)becomes

That is,

conditions(iv)and(v)yield

Having this together with Lemma 2.5,we obtain

where C>0 is a constant independent of ǫ.By taking ǫ>0 sufficiently small,one has

for all t∈[t0,Tm).From this inequality and Lemma 2.6,we obtain the final estimate of W(t)

Applying Lemma 2.7,we have

for sufficiently small ε,then continuous non-decreasing function M(t)must remain bounded.

that is

which is false.Therefore

This implies that Tm=∞,in other words,we have global solutions.

Lemma 3.2 Let a(x)and b(x)satisfy(1.7)–(1.8),assume that the weights w and w1satisfy conditions(i)–(v)together with

where C is a constant.Then,the solution u of(1.1)–(1.2)satisfies

for all t≥t0.

ProofFor the proof of condition(vi),see[19].To prove the first estimate,we use the second estimate of Lemma 3.1 with v=w−1u.It is left to prove.

We have the second estimate

and

These equalities imply

Integrating this inequality over Rnand using(vi),Lemma 3.1,we obtain

Here we are in a position to proof the main theorem and corollaries.

Proof of Theorem 1.1 Using the first estimate of Lemma 3.2,we have the following weighted estimates

Further by using the bounds for A(x),namely,

together with(1.7)we get the estimate

where C>0 and t≥t0is sufficiently large.Substituting this lower bound for a(x)into inequality(3.3),we have the decay estimate for the L2-norm of solution

To prove the decay estimate for the energy of solution u,we use the second estimate of Lemma 3.2,

which is equivalent to

To simplify the estimate,we notice that

with some C0>0 depending on δ.Hence,

Finally,we can show the decay estimate for the Lp+1norm of the solution u.From Lemma 2.2 and(3.2),we find that

Applying v=w−1u and the weights w,w1,one obtains

Using(3.4)again,we have

These yield the estimates in Theorem 1.1 with a loss of decay δ.To obtain the final form we only replace 2δ by δ.

Proof of Corollary 1.1 To obtain the weighted energy estimate,we combine Theorem 1.1 with the lower bound of A(x)from Proposition 1.1,namely,

We complete the proof by substituting this lower bound of A(x)into the result of Theorem 1.1.

Proof of Corollary 1.2 The result is similar to but more precise than the first corollary.Using properties(A3)and(A4)in Proposition 1.1,we write the main decay estimates as

where

Hence there exists a C>0,such that

The remain part is a lower bound of A(x)which is similar to(3.5).

Proof of Corollary 1.3 It can be concluded from Corollary 1.2 easily,and so we omit it.

Proof of Corollary 1.4 For the energy estimate in Theorem 1.1,we restrict the integration to{x:A(x)≥ t1+κ}and complete the proof.

4 Blow-up

In this section,we prove the blow-up part of Theorem 1.2.We adopt on the method of test functions developed by Zhang[21].

Proof of Theorem 1.2 First we find a non-negativesuch that

The function φ also satisfies the addition condition

where D=(∂t,∇)and C>0 is some constant,see[21]for the existence of such function.Then the test function φTis defined by

where T is some large parameter.Let PTbe the subset of R × Rn,where φT=1 and

that is,QTis the support of derivatives restricted to t≥0.It is easy to see that

Assume that a global solution u exists.To obtain a contradiction,we multiply the equation(1.1)bywith q=First using integration by part over[0,+∞]× Rn,it is easy to see that

Here we use φT(0,x)=1, ∂φT(0,x)=1 and the initial conditions on u to evaluate boundary integral at t=0.

Next,we estimate the integral on the left side of(4.2)and compare it with the integrals on the right side of(4.2).A straightforward calulation yields

By Holder’s inequality,we have

where I(T)=I1(T)+I2(T)+I3(T)+I4(T),Ii(T)(i=1,2,3,4)will be given as follows,and we will estimate Ii(T)(i=1,2,3,4)separately.

where we use 2(2−α−β)≥2−β from the assumption 2α+β≤2.

To proceed,we estimate the I3(T)and I4(T).

Using 2−α−β<1,which follows from the assumption α+β>1,we obtain

thus an upper bound is

With the same reason of estimating I3(T),we have

Therefore,by using(4.4)–(4.7),we derive the final estimate of I(T),

Combing with(4.2)–(4.3)and the assumption of initial data,we have

where C is indepent of T.Finally,we show that the above inequality cannot hold as T→∞.If p ≤ p2(n,α,β),then

and(4.8)shows that

Letting T→∞and using(4.1),we conclude that u∈Lp([0,+∞)×Rn).Hence(4.1)also implies that kukLp(QT)→ 0 as T → ∞.Passing to the limit in(4.8),we obtain kukLp([0,+∞)×Rn)≤ 0 for any 1

AcknowledgementWe are grateful to the anonymous referees for a number of valuable comments and suggestions.