We study some properties of the remotely almost periodic functions. This
paper studies viscosity solutions of general Hamilton-Jacobi equations in the time remotely
almost periodic case. Existence and uniqueness results are presented under usual hypotheses.

1. Introduction

In this paper we consider the viscosity solutions of first-order Hamilton-Jacobi equations of the form∂tu+H(x,u,Du)=f(t),(x,t)∈RN×R.
This problem was studied in [1] in the time periodic and almost periodic cases. And papers by Crandall and Lions (see [2–5]) proved the uniqueness and stability of viscosity solutions for a large class of equations, in particular for the initial value problem∂tu+H(x,t,u,Du)=0,(x,t)∈RN×]0,T[,u(x,0)=u0(x),x∈RN
and also for the stationary problemH(x,u,Du)=0,x∈RN.
These results were extended by several papers, for example [6, 7].

Now in this paper we study this problem in a more regular condition, that is, in the time remotely almost periodic case. That is, we will look for such viscosity solutions when the Hamiltonian H and f are continuous functions f is remotely almost periodic in t. The definition of remotely almost periodic was introduced by Sarason in 1984 in [8]. And Zhang and Yang in [9] and Zhang and Jiang in [10] gave such functions' applications.

This paper is structured as follows. In Section 2, we study a new type of almost periodic function—remotely almost periodic function. We present the definitions and prove some properties of such functions. Section 3 proves the uniqueness and existence of time remotely almost periodic viscosity solutions. In Section 3.1, we list some usual hypotheses used for the existence and uniqueness results and present two properties of viscosity solutions. In Section 3.2, we get some theorems for the uniqueness and existence of time remotely almost periodic viscosity solutions. And for the proof of the theorem we give two lemmas which play an important part. In Section 3.3, we concentrate on the asymptotic behaviour of time remotely almost periodic solutions for large frequencies.

In this paper, there are some abbreviations, like BUC, u.s.c, l.s.c, they stand for bounded uniformly continuous, upper semicontinuous, and lower semicontinuous, respectively. For the definition of viscosity subsolution and supersolution the reader can refer to [11].

2. Remotely Almost Periodic Function

It is well known that Bohr almost periodic function space is a Banach space, in which the distance is the supremum of the function. In [8], the author uses the superior limit as the distance in the space and defines a new type of almost periodic function, that is, remotely almost periodic function.

Definition 2.1.

Let f be a bounded uniformly continuous function on ℝ=(-∞,+∞). We say that f is remotely almost periodic if and only if for all ɛ>0T(f,ɛ)={τ∈R:limsup|t|→∞|f(t+τ)-f(t)|<ɛ}
is relatively dense on ℝ. The number τ∈T(f,ɛ) is called ɛ remotely almost period.

And RAP(ℝ) denotes all these functions.

Definition 2.2.

Let f be a bounded uniformly continuous function on ℝ. We say that f oscillates slowly if and only if for every τ∈ℝlim|t|→∞|f(t+τ)-f(t)|=0.
And SO(ℝ) denotes all these functions.

Next we will prove two propositions.

Proposition 2.3.

Assume that f(t) is remotely almost periodic and denote by F(t)=∫0tf(s)ds a primitive of f(t). Then F(t) is remotely almost periodic if and only if F(t) is bounded.

Proof.

When F(t) is remotely almost periodic, F(t) is certainly bounded. For the converse, let F(t) be bounded, without losing general, and assume that F(t) is a real function. For any ɛ>0, there exists t0>0 large enough; we have
G=sup|t|>t0F(t)>g=inf|t|>t0F(t);
take fixed t1 and t2, |t1|>t0, |t2|>t0, and assume that t1<t2, satisfying
F(t1)<g+ɛ6,F(t2)>G-ɛ6.
Assume that l=l(ɛ1) is an interval length of T(f,ɛ1), where ɛ1=ɛ/6d, d=|t1-t2|. For every α∈R, take τ∈T(f,ɛ1)∩[α-t1,α-t1+l].

As we already know that f(t) is remotely almost periodic, then we have
limsup|t|→∞|f(t+τ)-f(t)|<ɛ1;
that is, for ɛ1>0, there exists t0>0, and when |t|>t0, there is
|f(t+τ)-f(t)|<ɛ1,-ɛ1<f(t+τ)-f(t)<ɛ1.
Now take si=ti+τ(i=1,2),L=l+d. So s1,s2∈[α,α+L], and
F(s2)-F(s1)=F(t2)-F(t1)-∫t1t2f(t)dt+∫t1+τt2+τf(t)dt=F(t2)-F(t1)+∫t1t2[f(t+τ)-f(t)]dt>G-g-ɛ3-ɛ1d=G-g-ɛ2,
that is,
(F(s1)-g)+(G-F(s2))<ɛ2;
as the formulas in two brackets of previous inequality are both nonnegative, so there are two numbers s1 and s2 in any interval of length L satisfying simultaneously
F(s1)<g+ɛ2,F(s2)>G-ɛ2.
Now take ɛ2=ɛ/2L, and we will prove that when τ∈T(f,ɛ2), there is τ∈T(f,ɛ). In fact for every t∈R, we can choose s1 and s2 in the interval [t,t+L] satisfying F(s1)<g+(ɛ/2) and F(s2)>G-(ɛ/2). Hence for τ∈T(f,ɛ2), there are, respectively,
limsup|t|→∞(F(t+τ)-F(t))=limsup|t|→∞[F(s1+τ)-F(s1)+∫ts1f(t)dt-∫t+τs1+τf(t)dt]>g-(g+ɛ2)-ɛ2L=-ɛ,limsup|t|→∞(F(t+τ)-F(t))=limsup|t|→∞[F(s2+τ)-F(s2)+∫ts2f(t)dt-∫t+τs2+τf(t)dt]<G-(G-ɛ2)+ɛ2L=ɛ.
So for τ∈T(f,ɛ2), we have τ∈T(f,ɛ); hence F(t) is remotely almost periodic.

Proposition 2.4.

Assume that f(t) is remotely almost periodic. Then (1/T)∫aa+Tf(t)dt converges as T→+∞ uniformly with respect to a∈R. Moreover the limit does not depend on a, and it is called the average of f∃〈f〉≔limT→+∞1T∫aa+Tf(t)dt,uniformly with respect toa∈R.

Proof.

As f(t)∈RAP(R), then f(t) is bounded, and for allɛ>0, for all τ∈T(f,ɛ), there exists s0>0, when |t|>s0,|f(t+τ)-f(t)|<ɛ. Let G=supt∈R|f(t)|, take ɛ>0, and assume that l=l(ɛ/4) is an interval length of T(f,ɛ/4). Take τ∈T(f,ɛ/4)∩[a,a+l]; then for any a,s∈R|∫aa+sf(t)dt-∫0sf(t)dt|=|(∫ττ+s-∫0s+∫τ+sa+s+∫aτ)f(t)dt|≤∫0s|f(t+τ)-f(t)|dt+∫τ+sa+s|f(t)|dt+∫0τ|f(t)|dt=∫0s0|f(t+τ)-f(t)|dt+∫s0s|f(t+τ)-f(t)|dt+∫τ+sa+s|f(t)|dt+∫0τ|f(t)|dt≤sup[s0,s]|f(t+τ)-f(t)|⋅(s-s0)+2G(l+s0)<ε4(s-s0)+2G(l+s0),
so
|1T∫aa+Tf(t)dt-1T∫0Tf(t)dt|≤ɛ4T(T-T0)+2G(l+T0)T,|1nT∫0nTf(t)dt-1T∫0Tf(t)dt|=1n|∑k=1n1T[∫(k-1)TkTf(t)dt-∫0Tf(t)dt]|≤ɛ4T(T-T0)+2G(l+T0)T.
By passing n→+∞ in (2.14), we get
|〈f〉-1T∫0Tf(t)dt|≤ɛ4T(T-T0)+2G(l+T0)T.
Using triangle inequality from (2.13) and (2.15) we deduce
|1T∫aa+Tf(t)dt-〈f〉|≤ɛ2T(T-T0)+4G(l+T0)T<ɛ,
if only T>(8G(l+T0)/ɛ)-T0. That is, when T→∞,(1/T)∫aa+Tf(t)dt converges at 〈f〉 uniformly with respect to a∈R. Moreover notice the identical equation
1T∫aa+Tf(t)dt=1T∫0Tf(t+a)dt.
This means that the limit does not depend on a.

3. Remotely Almost Periodic Viscosity Solutions

In this section we get some results for remotely almost periodic viscosity solutions.

Definition 3.1.

One says that u:ℝN×ℝ→ℝ is remotely almost periodic in t uniformly with respect to x if u is bounded and uniformly continuous in t uniformly with respect to x and for all ε>0, and there existsl(ε)>0 such that all intervals of length l(ɛ) contain a number τ which is ɛ remotely almost periodic for u(x,·), for allx∈ℝNlimsup|t|→∞|u(x,t+τ)-u(x,t)|<ɛ,∀(x,t)∈RN×R.

3.1. Some Hypotheses and Theorems

In this section we list some usual hypotheses used for the uniqueness and existence results and present two properties of viscosity solutions.

First let us list some hypotheses in the stationary case:∀0<R<+∞,∃γR>0:H(x,u,p)-H(x,v,p)≥γR(u-v),∀x∈RN,-R≤v≤u≤R,p∈RN,∀R>0,∃mR,limz→0mR(z)=0:|H(x,u,p)-H(y,u,p)|≤mR(|x-y|⋅(1+|p|)),∀x,y∈RN,-R≤u≤R,p∈RN,∀0<R<+∞,lim|p|→+∞H(x,u,p)=+∞,uniformly for(x,u)∈RN×[-R,R],∀0<R<+∞,His uniformly continuous onRN×[-R,R]×B¯R,∃M>0:H(x,-M,0)≤0≤H(x,M,0),∀x∈RN.
From [1] we know that hypotheses (3.2), (3.3) or (3.4), (3.5), (3.6) ensure the existence of a unique solution for the stationary equation (1.3). And more regularly (3.2) can be replaced byH(x,u,p)-H(x,v,p)≥0,∀x∈RN,v≤u,p∈RN
(which comes to taking γR=0 in (3.2)).

When the Hamiltonian is time dependent the corresponding assumptions are∀0<R<+∞,∃γR>0:H(x,t,u,p)-H(x,t,v,p)≥γR(u-v),∀x∈RN,0≤t≤T,-R≤v≤u≤R,p∈RN,∀R>0,∃mR:|H(x,t,u,p)-H(y,t,u,p)|≤mR(|x-y|⋅(1+|p|)),∀x,y∈RN,t∈[0,T],-R≤u≤R,p∈RN,wherelimz→0mR(z)=0,∀0<R<+∞,His uniformly continuous onRN×[0,T]×[-R,R]×B¯R,∃M>0:H(x,t,-M,0)≤0≤H(x,t,M,0),∀x∈RN,t∈[0,T].
Now we present two results of viscosity solutions (see [1, 6, 7]).

Theorem 3.2.

Assume that (3.8), (3.9), (3.10), and (3.11) hold (with γR∈ℝ, for all R>0). Then for every u0∈BUC(ℝN) there is a unique viscosity solution u∈BUC(ℝN×[0,T]) of (1.2), for all T>0.

Theorem 3.3.

Let u be a bounded time periodic viscosity u.s.c. subsolution of ∂tu+H(x,t,u,Du)=f(x,t) in ℝN×ℝ and v a bounded time periodic viscosity l.s.c. supersolution of ∂tv+H(x,t,v,Dv)=g(x,t) in ℝN×ℝ, where f,g∈BUC(ℝN×ℝ) and H are T periodic such that (3.8), (3.9), and (3.10) hold. Then one has
supx∈RN(u(x,t)-v(x,t))≤sups≤t∫stsupx∈RN(f(x,σ)-g(x,σ))dσ.
Moreover, the hypothesis (3.9) can be replaced by u∈W1,∞(ℝN×ℝ) or v∈W1,∞(ℝN×ℝ).

3.2. Uniqueness and Existence of Time Remotely Almost Periodic Viscosity Solutions

In this section we establish uniqueness and existence results for time remotely almost periodic viscosity solutions. For the uniqueness we have the more general result.

Proposition 3.4.

Let u a bounded u.s.c. viscosity subsolution of ∂tu+H(x,t,u,Du)=f(x,t), in ℝN×ℝ and v a bounded l.s.c. viscosity supersolution of ∂tv+H(x,t,v,Dv)=g(x,t), in ℝN×ℝ where f,g∈BUC(ℝN×ℝ) and (3.8), (3.9), (3.10) hold uniformly for t∈ℝ. Then one has for all t∈ℝsupx∈RN(u(x,t)-v(x,t))+≤e-γt∫-∞tsupx∈RN(f(x,σ)-g(x,σ))+dσ.
Moreover hypotheses (3.9) can be replaced by u∈W1,∞(ℝN×ℝ) or v∈W1,∞(ℝN×ℝ).

The proof of this proposition is similar to Proposition 6.5 in [1]. Hence we do not prove it here.

Before we concentrate on the existence part, let us see two important lemmas first. Now take h(t)=∫-∞teγ(σ-t)f(σ)dσ, where γ>0 is a constant, t∈ℝ.

Lemma 3.5.

If f(t)∈SO(ℝ), then h(t)∈SO(ℝ).

Proof.

As f(t)∈SO(ℝ), so for every τ∈ℝlim|t|→∞|f(t+τ)-f(t)|=0.
Now for every τ∈ℝ|h(t+τ)-h(t)|=|∫-∞t+τeγ(σ-t-τ)f(σ)dσ-∫-∞teγ(σ-t)f(σ)dσ|=|∫-∞0eγσf(t+σ+τ)dσ-∫-∞0eγσf(t+σ)dσ|=|∫-∞0eγσ[f(t+σ+τ)-f(t+σ)]dσ|≤∫-∞0eγσ|f(t+σ+τ)-f(t+σ)|dσ≤supσ|f(t+σ+τ)-f(t+σ)|⋅1γ,
hence
lim|t|→∞|h(t+τ)-h(t)|≤limsup|t|→∞|f(t+σ+τ)-f(t+σ)|⋅1γ=0.
Since we already know that f(t)∈BUC(ℝ), we deduce also that h(t)∈BUC(ℝ). That is, h(t)∈SO(ℝ).

Lemma 3.6.

If f(t)∈RAP(ℝ), then h(t)∈RAP(ℝ).

Proof.

The main result in [8] proved that f(t)∈RAP(ℝ) is the closed subalgebra in C(ℝ) created by AP(ℝ) and SO(ℝ). Hence, if f(t)∈RAP(ℝ), for every ɛ>0, take ɛ1=γ·ɛ, there exists g1,g2∈AP(ℝ) and φ1,φ2∈SO(ℝ); hence
‖f-[g1+φ1+g2φ2]‖<ɛ14.
If φ2=0, consider a number τ which is an ɛ1/2 remotely almost period of g1:
|h(t+τ)-h(t)|=|∫-∞0eγσ[f(t+σ+τ)-f(t+σ)]dσ|≤∫-∞0eγσ|f(t+σ+τ)-f(t+σ)|dσ≤∫-∞0eγσ|f(t+σ+τ)-[g1(t+σ+τ)+φ1(t+σ+τ)]|dσ+∫-∞0eγσ|g1(t+σ+τ)-g1(t+σ)|dσ+∫-∞0eγσ|φ1(t+σ+τ)-φ1(t+σ)|dσ+∫-∞0eγσ|f(t+σ)-[g1(t+σ)+φ1(t+σ)]|dσ<ɛ14γ+ɛ12γ+∫-∞0eγσ|φ1(t+σ+τ)-φ1(t+σ)|dσ+ɛ14γ.
By using Lemma 3.5 we deduce
limsup|t|→∞|h(t+τ)-h(t)|<ɛ14γ+ɛ12γ+ɛ14γ=ɛ.
Thus this proves that any ɛ1/2 remotely almost period of g1 is an ɛ remotely almost period of h.

If φ2≠0, assume that δ=min{ɛ1/4,ɛ1/(4·∥φ2∥)}, and take number τ which is a common δ remotely almost period of g1 and g2. We will prove that τ is an ɛ1/2 remotely almost period of (g1+φ1+g2φ2), and an ɛ remotely almost period of h:
|g2(t+σ+τ)φ2(t+σ+τ)-g2(t+σ)φ2(t+σ)|≤|g2(t+σ+τ)φ2(t+σ+τ)-g2(t+σ+τ)φ2(t+σ)|+|g2(t+σ+τ)φ2(t+σ)-g2(t+σ)φ2(t+σ)|≤‖g2‖⋅|φ2(t+σ+τ)-φ2(t+σ)|+‖φ2‖⋅|g2(t+σ+τ)-g2(t+σ)|.
We have
limsup|t|→∞|g2(t+σ+τ)φ2(t+σ+τ)-g2(t+σ)φ2(t+σ)|<ɛ14.
Hence
limsup|t|→∞|[g1(t+σ+τ)+φ1(t+σ+τ)+g2(t+σ+τ)φ2(t+σ+τ)]-[g1(t+σ)+φ1(t+σ)+g2(t+σ)φ2(t+σ)]|≤limsup|t|→∞|g1(t+σ+τ)-g1(t+σ)|+limsup|t|→∞|φ1(t+σ+τ)-φ1(t+σ)|+limsup|t|→∞|g2(t+σ+τ)φ2(t+σ+τ)-g2(t+σ)φ2(t+σ)|<ɛ14+ɛ14=ɛ12,|f(t+σ+τ)-f(t+σ)|≤|f(t+σ+τ)-[g1(t+σ+τ)+φ1(t+σ+τ)+g2(t+σ+τ)φ2(t+σ+τ)]|+|[g1(t+σ+τ)+φ1(t+σ+τ)+g2(t+σ+τ)φ2(t+σ+τ)]-[g1(t+σ)+φ1(t+σ)+g2(t+σ)φ2(t+σ)]|+|[g1(t+σ)+φ1(t+σ)+g2(t+σ)φ2(t+σ)]-f(t+σ)|.
So we have
limsup|t|→∞|h(t+τ)-h(t)|=limsup|t|→∞|∫-∞0eγσ[f(t+σ+τ)-f(t+σ)]dσ|≤limsup|t|→∞|f(t+σ+τ)-f(t+σ)|⋅1γ<(ɛ14+ɛ12+ɛ14)⋅1γ=ɛ.
Since we already know that f(t)∈BUC(ℝ), we deduce also that h(t)∈BUC(ℝ). So this proves that h(t)∈RAP(ℝ).

Now we concentrate on the existence part.

Proposition 3.7.

Assume that f:ℝ→ℝ is remotely almost periodic and that the Hamiltonian H=H(x,z,p) satisfying the hypotheses (3.2), (3.3), (3.5), and there existsM>0 such that H(x,-M,0)≤f(t)≤H(x,M,0), for all (x,t)∈ℝN×ℝ. Then there is a time remotely almost periodic viscosity solution in BUC(ℝN×ℝ) of ∂tu+H(x,u,Du)=f(t), in ℝN×ℝ.

Proof.

We consider the unique viscosity solution of the problem
∂tun+H(x,un,Dun)=f(t),(x,t)∈RN×]-n,+∞[,un(x,-n)=0,x∈RN
for all n≥1. Such a solution exists by Theorem 3.2. Next we will prove that for all t∈ℝ, (un(t))n≥-t converges to a remotely almost periodic viscosity solution of ∂tu+H(x,u,Du)=f(t), in ℝN×ℝ. Similar to the proof of Proposition 6.6 in [1], we obtain by fixing t∈ℝ and n large enough
|un(x,t)-un(x,t+τ)|≤2M⋅e-γ(t-tn)+e-γt∫tnteγσ|f(σ+τ)-f(σ)|dσ.
By passing n→+∞ we have tn→-∞, and therefore
|u(x,t)-u(x,t+τ)|≤∫-∞te-γ(t-σ)|f(σ+τ)-f(σ)|dσ.
As f is remotely almost periodic, using Lemma 3.6 we deduce
limsup|t|→∞|u(x,t)-u(x,t+τ)|≤limsup|t|→∞∫-∞teγ(σ-t)|f(σ+τ)-f(σ)|dσ<ɛ.
Since we already know that u∈BUC(ℝN×[a,b]), for all a,b∈ℝ,a≤b, by time remotely almost periodicity we deduce also that u∈BUC(ℝN×ℝ).

Now we will study the time remotely almost periodic viscosity solutions of∂tu+H(x,u,Du)=f(t),(x,t)∈RN×R,
for Hamiltonians satisfying (3.7). We introduce also the stationary equationH(x,u,Du)=〈f〉:=1T∫0Tf(t)dt,x∈RN.
We have the following theorem for the existence of time remotely almost periodic viscosity solution.

Theorem 3.8.

Assume that Hamiltonian H=H(x,z,p) satisfies hypotheses (3.7), (3.4), (3.5), sup{|H(x,0,0)|:x∈ℝ}=C<+∞ and f is a time remotely almost periodic function such that F(t)=∫0t{f(σ)-〈f〉}dσ is bounded on ℝ. Then there is a bounded Lipschitz time remotely almost periodic viscosity solution of (3.28) if and only if there is a bounded viscosity solution of (3.29).

Proof.

Assume that V is a bounded viscosity of (3.29). We deduce that V is a Lipschitz function as the Hamiltonian satisfies (3.4). For any α>0, take Mα=∥V∥L∞(ℝN)+(1/α)(C+∥f∥L∞(ℝ)). By Propositions 3.4 and 3.7 we can construct the family of time remotely almost periodic solutions vα for
α(vα-V(x))+∂tvα+H(x,vα,Dvα)=f(t),(x,t)∈RN×R.
Similar to Theorems 4.1 and 6.1 in [1], we can extract a sequence which converges uniformly on compact sets of ℝN×ℝ towards a bounded Lipschitz solution v of (3.28). Next we will prove that v is remotely almost periodic. By the hypotheses and Proposition 2.3 we deduce that F is remotely almost periodic, and thus, for all ɛ>0, there is l(ɛ/2) such that any interval of length l(ɛ/2) contains an ɛ/2 remotely almost period of F. Take an interval of length l(ɛ/2) and τ an ɛ/2 remotely almost period of F in this interval. We have for all α>0,(x,t)∈ℝN×ℝ|vα(x,t+τ)-vα(x,t)|≤|sups≤t∫st{f(σ+τ)-f(σ)}dσ|=|sups≤t{∫s+τt+τ(f(σ)-〈f〉)dσ-∫st(f(σ)-〈f〉)dσ}|=|sups≤t{(F(t+τ)-F(t))-(F(s+τ)-F(s))}|≤2supt|F(t+τ)-F(t)|.
After passing to the limit for α↘0 one gets |v(x,t+τ)-v(x,t)|≤2supt|F(t+τ)-F(t)|, and hence
limsup|t|→∞|v(x,t+τ)-v(x,t)|≤2limsup|t|→∞|F(t+τ)-F(t)|≤ɛ.
By using the uniform continuity of F, we can prove exactly in the same manner that v is continuous in t uniformly with respect to x. The converse implication follows similarlyTheorem 4.1 in [1]; here we do not prove it.

3.3. Asymptotic Behaviour for Large Frequencies

In this section we study the asymptotic behaviour of time remotely almost periodic viscosity solutions of∂tun+H(x,un,Dun)=fn(t),(x,t)∈RN×R,
where f:ℝ→ℝ is a remotely almost periodic function. For all n≥1 notice that fn(t)=f(nt), for all t∈ℝ is remotely almost periodic and has the same average as f. Now suppose that such a hypothesis exists∃M>0such thatH(x,-M,0)≤f(t),∀(x,t)∈RN×R.

Theorem 3.9.

Let H=H(x,z,p) be a Hamiltonian satisfying (3.7), (3.3), (3.5), (3.34) where f is remotely almost periodic function. Suppose also that there is a bounded l.s.c viscosity supersolution Ṽ≥-M of (3.29), that t→F(t)=∫0t{f(s)-〈f〉}ds is bounded, and denote by V the minimal stationary l.s.c. viscosity supersolution of (3.29), vn the time remotely almost periodic l.s.c. viscosity supersolution of (3.33). Then the sequence (vn)n converges uniformly on ℝN×ℝ towards V and ∥vn-V∥L∞(ℝN×ℝ)≤(2/n)∥F∥L∞(ℝ), for all n≥1.

Proof.

As vn=supα>0vn,α is remotely almost periodic, we introduce wn,α(x,t)=vn,α(x,t/n), (x,t)∈ℝN×ℝ, which is also remotely almost periodic. Similar to Theorem 5.1 in [1] and by using Theorem 3.3 we deduce that
wn,α(x,t)-Vα(x)≤sups≤t1n∫st(f(σ)-〈f〉)dσ≤2n∥F∥L∞(R),
and similarly Vα(x)-wn,α(x,t)≤(2/n)∥F∥L∞(ℝ), for all n≥1. We have for all n≤1|wn,α(x,t)-Vα(x)|≤2n‖F‖L∞(R),
and after passing to the limit for α↘0 one gets for all (x,t)∈ℝN×ℝ|wn(x,t)-V(x)|≤2n‖F‖L∞(R).
Finally we deduce that ∥vn-V∥L∞(ℝN×ℝ)≤(2/n)∥F∥L∞(ℝN×ℝ) for all n≥1.

Acknowledgment

This work was supported by National Science Foundation of China (Grant no. 11001152).

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