Open Access
 Issue Wuhan Univ. J. Nat. Sci. Volume 27, Number 3, June 2022 189 - 194 https://doi.org/10.1051/wujns/2022273189 24 August 2022

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## 0 Introduction

In this paper, we consider the Cauchy problem for the evolutionary Hamilton-Jacobi equation

Here is an unknown function on , is a given function, and , . We study the long-time asymptotic behavior of the viscosity solution to (CP) and furthermore, discuss the relation between the limit of the viscosity solution of (CP) and the viscosity solution of the stationary Hamilton-Jacobi equation

There has been much study about the long-time behavior of the viscosity solutions of Hamilton-Jacobi equations either by means of dynamical techniques or by PDE methods. We occasionally suppress "viscosity" for simplicity.

The dynamical approach is based on the weak KAM theory initiated by Fathi[1,2]. It needs strong regularity assumptions on the Hamiltonian (-regularity, strict convexity and superlinearity in ) because it is based on the analysis of the associated Hamiltonian flow. Such flow is connected with the visc.solution of through the Lax-Oleinik formula. The dynamical approach has been later modified by Roquejoffre[3], Davini and Siconolfi in Ref. [4], and others.

The PDE approach is initiated by the work of Namah and Roquejoffre[5]. It does not depend on the Lax-Oleinik formula, so it is possible to be applied to more general cases. Barles and Souganidis have obtained in Ref. [6] more general results in the case , for possible non-convex Hamiltonians. We refer to Ref. [7] for a recent view on this approach.

In this paper, we will explore the visc. solutions'long time behavior of the Hamilton-Jacobi equation of contact type, in which the Hamiltonian explicitly depends on the unknown function . The contact Hamiltonian system is a natural extension to Hamiltonian system. Various applications of contact Hamiltonian dynamics has been found in many fields such as classic mechanics of dissipative system[8,9], mesoscopic dynamics[10], equilibrium statistical mechanics[11], and thermodynamics[12,13], etc. Su, Wang, and Yan first studied visc.solutions' long-time behavior of the contact Hamilton-Jacobi equation with implicit variational principle in Ref. [14], under Tonelli assumptions ( is , strict convexity and superlinear growth in for every , uniform Lipschitzity and monotonicity with respect to ). Their series of work are aimed at building the variational frame in the contact Hamiltonian system[15-17]. In the recent paper, the author has studied the long-time behavior of solutions of the contact Hamilton-Jacobi equations with the method combining the PDE-viscosity solutions approach and dynamical approach under more general conditions ( is , strict convexity and coercive in for every , monotonicity with respect to )[18].

Motivated by above-mentioned results, we will continue this direction of research on the long-time behavior of visc. solutions and we want to discuss if the conditions are necessary for the convergence in this paper. We mainly use and slightly modify the PDE approach which has been introduced by Barles, Ishii and Mitake (see Ref. [7]). The main difference is that we deal with the contact Hamiltonian-Jacobi equation for the consideration of the effect of in the proof.

We assume that

(C)

, for all (CER)

The function is non-decreasing on , for all (MON)

(EP) with has a visc. solution (Z)

There exist positive constants , and a positive constant with , such that for all ,, if and , then .

()

There exist positive constants , and a positive constant with , such that for all , , if and , then .

()

Condition ()(()) means some kind of strict convexity of in . Indeed, if is strictly convex in , then

A condition similar to () has appeared in Ref. [1]. The difference is the strict convexity about in this paper and the strict convexity about both in the -independent case in Ref. [7] and in the -dependent case in Ref. [18]. The convexity in is the necessary condition for using this PDE approach, bcause we cannot fix to as we have done with the PDE approach in Ref. [18].

In this case, we can deal with the convergence problem of the Hamiltonian which is not strictly convex in in contrast to what happens in Ref.[14, 18]. Our main result is:

Theorem 1   Assume (C), (CER), (MON), (Z) and ()(()). Let , and let be the visc.solution of (CP). Then there exists such that

uniformly on .

Moreover, is a visc. solution of (EP), with .

The paper is organized as follows. In Section 1, we will give some classical results about visc.solution theory which are needed for the next proof. Based on the comparison theorem and the Perron method, we get the existence theorem of (CP) (Theorem 3). Assuming moreover (Z), is bounded and uniformly continuous on . In Section 2, we will give the proof of Theorem 1 with the condition ()(()) instead of the condition (CON) in the Ref.[18].

## 1 The Preliminary Results

As the basis of the existence theorem and the uniqueness theorem, we first introduce the comparison theorem.

Theorem 2[18] Assume (C), (CER) and (MON). Let and be a visc. subsolution and a visc.supersolution of (CP), respectively, where . Then

for all .

Corollary 1[18] If, in addition, are both visc. solutions of (CP), then we have

Theorem 3[18] Assume (C), (CER) and (MON). Let . Then there exists a (unique) solution of (CP).

Theorem 4 [18] Assume (C), (CER), (MON) and (Z). Let be a visc. solution of (CP). Then is bounded and uniformly continuous on .

We will give some stability results concerning viscosity solutions.

Theorem 5[18] Let be locally compact. is a family of viscosity subsolutions of (EP). Assume that is locally bounded in , then is also a visc. subsolution of (EP).

The theorems above are classical results in viscosity solution theory. We can find the proof in Refs. [19-22].

## 2 The Main Result

In this section, we want to prove our main result.

Theorem 6   Assume (C), (CER), (MON), (Z) and ()(()). Let , and let be a visc.solution of (CP). Then there exists such that

uniformly on .

Moreover, is a visc. solution of (EP), with .

First, we reduce the result to the case . Indeed, we have

Lemma 1   If the result of Theorem 6 holds for any , then it holds for any .

This is an easy consequence of Theorem 2 and the reader can find a proof of the lemma above in Ref. [7].

Lemma 2   There exists a viscosity subsolution of (EP), with , such that

for all .

Proof   Due to (Z), there exists a solution of (EP), with . Since the function is a solution of (CP), by Theorem 2 we obtain

for all , which can be written as

for all

with . If we set and , then we have

for all ,

and, by (MON), the function is a subsolution of (EP), with .

For , we define the function on , by

(1)

where is the function given by Lemma 2. We define the functions , with , by

Lemma 3   We have

for all .

Proof   According to Lemma 2, for all ,

and

Theorem 7   The function is a subsolution of

(2)

where is the constant from (DSTC+), and .

Proof   Noting that and , then .

Fix any and , and assume that

If , then we have finished the proof. Therefore, we may assume that . We choose an so that

If , we get , and we are done. We may thus assume that .

Define the function by

Note that the function

(3)

on attains a strict maximum at , and that , and .

Now, if is an open ball of centered at with its closure contained in , we consider the function on given by

where is a large constant.

Let be a maximum point of . We can get

(4)

Next, set

We observe that

We have by the definition of , and by sending along an appropriate sequence, we can find points such that

(5)

(6)

(7)

where stands for the closure of , for instance, denotes the set of points for which there are sequences and such that and for all . Recall that and so that we have

from (5) and (6). Therefore, () ensures

(8)

Since , we have

(9)

Because of the assumption (MON), (8) and (9),

The second inequality holds since and because of (7). Therefore, we get

i.e.,

We set

for all .

Lemma 4   We have

for all .

Moreover, the convergence

is uniform in .

Proof   If the convergence does not hold uniformly in , we can choose a sequence such that and for all and some constant . We may assume that for some . In view of the Ascoli-Arzela theorem, we may assume by passing to a subsequence of if needed that

uniformly in ,

for some bounded function and .

By the stability of the subsolution property under uniform convergence, we see that is a subsolution of

in . Since and is bounded on , for every , the function attains a maximum at a point , then we have

Therefore, we know that

In particular, we have . Then, as usual in the viscosity solutions theory, we get

which, in the limits as , yields , a contradiction.

Proof of Theorem 6 under condition (   )

Let be the function defined by (1), with arbitrary .

Fix any . Because of (1), we may choose a constant so that for any , for all .

From the above, for any , we have

Thus, for any , we have

(10)

Now, since is bounded and Lipschitz continuous in , in view of the Ascoli-Arzela theorem, we may choose a sequence and a bounded function so that

(11)

By (10) we get

(12)

for all . This is valid for all . Hence we obtain

(13)

Thus we find that the function is nondecreasing in for all .

From this we conclude that

(14)

for some function . Since is a viscosity solution of (CP), and is bounded on , we get from Theorem 5 that is a solution of (CP), and moreover, is a solution of (EP).

Fix any . By (14) there is a constant such that

and by (11) there is a such that

Therefore,

By the contraction property, we see that for any ,

which completes the proof.

Proof of Theorem 6 under condition (   )

where is chosen arbitrarily in and the constants and are from ().

Theorem 8   The function is a subsolution of

where is the constant from ( ), and .

We can prove

for all .

On the other hand, for any and ,

Hence, for any we have

We fix any test function and assume that attains a strict maximum at a point .

We can get the following conclusion like the proof of ():

for some .

Similarly, we can assume . According to assumption (MON), we have

Since , we get

by the condition ().

Also, we can prove

(15)

We can get is nonincreasing for all and enough large from (15). Since is bounded and Lipschitz continuous in , we can choose a sequence such that for some function ,

locally uniformly on .

is nonincreasing in for all , then there exists the function such that

uniformly on .

Furthermore

uniformly on .

Thus we complete the proof.

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