On a viscous Hamilton–Jacobi equation with an unbounded potential term

Nonlinear Analysis 73 (2010) 1802–1811

Contents lists available at ScienceDirect

Nonlinear Analysis journal homepage: www.elsevier.com/locate/na

On a viscous Hamilton–Jacobi equation with an unbounded potential term Thomas Strömberg ∗ Department of Mathematics, Luleå University of Technology, SE-971 87 Luleå, Sweden

article

abstract

info

Article history: Received 17 April 2009 Accepted 6 May 2010

We present comparison, uniqueness and existence results for unbounded solutions of a viscous Hamilton–Jacobi or eikonal equation. The equation includes an unbounded potential term V (x) subject to a quadratic upper bound. The results are obtained through a tailor-made change of variables in combination with the Hopf–Cole transformation. An integral representation formula for the solution of the Cauchy problem is derived in the case where V (x) = ω2 |x|2 /2. © 2010 Elsevier Ltd. All rights reserved.

Keywords: Viscous Hamilton–Jacobi equation Unbounded potential Integral formula

1. Introduction This paper is devoted to the viscous Hamilton–Jacobi or eikonal equation 1

|∇ S |2 + V (x) − ε 1S = 0 in QT = (0, T ] × Rn . 2 The Cauchy problem consists in finding a classical solution S of (1) fulfilling the initial condition St +

S (0, x) = S0 (x) for all x ∈ Rn ,

(1)

(2)

where S0 is a prescribed function. We may view (1) as a regularized version of the Hamilton–Jacobi equation arising in classical mechanics [1]. We allow V and S0 to be unbounded functions and avoid restrictions on the growth of solutions S to the extent it is possible. The data induce, in general, blow-up of solutions in finite time thus imposing restrictions on T . We are particularly interested in the existence-uniqueness problem for (1) and (2) in the space of all classical solutions as well as in comparison between viscosity sub- and supersolutions of (1). In particular, we demonstrate that uniqueness can hold even though the comparison principle fails. Of importance throughout the paper is the interplay between the respective growth properties of V and S0 . Our standing assumptions on the potential V and the initial datum S0 are the following. (H1) The potential V is a locally Hölder continuous function on Rn . There exist constants B ∈ R and ω > 0 such that V (x) ≤ B +

ω2 |x|2

, x ∈ Rn . 2 (H2) The initial function S0 is a continuous function on Rn . There exist constants A ∈ R and a ≥ 0 such that S0 (x) ≥ A −

a|x|2 2

,

x ∈ Rn .

On replacing the unknown function S (t , x) by S˜ (t , x) = S (t , x) − A + Bt, problem (1) and (2) becomes S˜t +

∗

1 2

|∇ S˜ |2 + (V (x) − B) − ε 1S˜ = 0

Fax: +46 920 491073. E-mail addresses: [email protected], [email protected].

0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.05.015

(3)

T. Strömberg / Nonlinear Analysis 73 (2010) 1802–1811

1803

supplemented with the initial condition S˜ (0, x) = S0 (x)− A. Therefore, we may without loss of generality assume A = B = 0 in conditions (H1), (H2), i.e., V (x) ≤

ω2 |x|2 2

and S0 (x) ≥ −

a|x|2 2

,

x ∈ Rn .

(4)

Our approach is based upon a change of variables in combination with the classical Hopf–Cole transformation. In effect, on applying these transformations to the problem at hand, (1) is converted into a linear parabolic equation with the potential V replaced by a nonpositive function W (see Lemmas 2 and 3 below). The change of variables is tailor-made for V (x) = ω2 |x|2 /2 and arises as follows. Let ω > 0. In investigating the inviscid equation St +

1 2

|∇ S |2 +

ω2 |x|2 2

= 0 in QT

(5)

it is classical and fruitful to consider the associated characteristics, i.e., the extremals of the corresponding Lagrangian, which is given by L(x, v) = 21 |v|2 − 21 ω2 |x|2 . These extremals are the solutions of x¨ (t ) + ω2 x(t ) = 0, i.e., x(t ) = C cos ωt + Dω−1 sin ωt. We obtain trivial dynamics in the sense of d2 ξ/dτ 2 = 0 (i.e., ξ(τ ) = C + τ D) in new variables τ ∈ R and ξ ∈ Rn , therefore, by setting

 tan ωt  τ = , ω x  ξ = , cos ωt

 arctan ωτ   , t = ω ξ   . x = p 1 + (ωτ )2

or

(6)

It is assumed that −π /2 < ωt < π /2. In effect, on applying (6) in tandem with F (τ , ξ ) = S (t , x) +

ω|x|2 tan ωt 2

,

the Hamilton–Jacobi equation (5) is converted into Fτ + 12 |∇ξ F |2 = 0, i.e., the potential term vanishes. It is shown below that this transformation, appropriately modified, retains its advantages in the viscid case where ε > 0. There exists considerable literature on viscous Hamilton–Jacobi equations. Some recent developments in various directions appear in, e.g., [2–10]. The motivation for the present paper is to allow unbounded V and S0 , which is favorable in various applications. The rest of the paper consists of three sections. As a preparatory step, the transformations are defined and investigated in Section 2. An integral formula is given in Section 3 for the special case where V (x) = ω2 |x|2 /2. Section 4 furnishes comparison, uniqueness and existence results for general potentials satisfying (H1). 2. The change of variables It will be advantageous to modify (6) by a translation of time. Suppose that (H1) is in force. Change of variables. For a specific t0 such that −π /2 < ωt0 < π /2, we set

 tan ω(t + t0 )  τ = , ω x  ξ = , cos ω(t + t0 )

(7)

in conjunction with F (τ , ξ ) = S (t , x) +

ω|x|2 tan ω(t + t0 ) 2

− nε ln(cos ω(t + t0 )),

(8)

for

−π /2 < ω(t + t0 ) < π /2,

x ∈ Rn ; τ ∈ R, ξ ∈ Rn .

We let τ = τ0 and τ = τT correspond to t = 0 and t = T , respectively, i.e., we put

τ0 =

tan ωt0

,

τT =

ω assuming ω(T + t0 ) < π /2.

tan ω(T + t0 )

ω

,

(9)

Assuming the growth hypotheses of (H1) and (H2) in the reduced form (4), a favorable choice for t0 is t0 =

arctan(a/ω)

ω

.

(10)

1804

T. Strömberg / Nonlinear Analysis 73 (2010) 1802–1811

This choice ensures F (τ0 , ξ ) ≥ 0 where τ0 = ω−1 tan ωt0 = aω−2 . Indeed, (10) implies

ω|x|2 tan ωt0

F (τ0 , ξ ) = S (0, x) +

≥−

a|x|2 2

+

2 ω|x|2 tan ωt0 2

− nε ln(cos ωt0 ) = 0.

It turns out that F solves the same kind of equation as S but with a non-positive potential term, which simplifies the investigation. Lemma 1. On employing the transformation defined by (7) and (8), Eq. (1) is converted into the following equivalent one: Fτ +

1 2

|∇ξ F |2 + W (τ , ξ ) − ε 1ξ F = 0 in (τ0 , τT ] × Rn ,

(11)

where W is defined by W (τ , ξ ) =

V

!

ξ p

1 + (ωτ )

2

ω2 |ξ |2 − 2(1 + (ωτ )2 )

!

1 1 + (ωτ )2

for all (τ , ξ ) ∈ R × Rn .

(12)

In particular, if V (x) = ω2 |x|2 /2, then the potential term W in (11) vanishes yielding Fτ +

1 2

|∇ξ F |2 − ε 1ξ F = 0 in (τ0 , τT ] × Rn .

(13)

Furthermore, if V (x) ≤ ω2 |x|2 /2, then W ≤ 0. Proof. The time derivative St becomes St =

1 cos2 ω(t + t0 )

Fτ +

ω sin ω(t + t0 ) ω2 |x|2 F − x · ∇ − nεω tan ω(t + t0 ), ξ cos2 ω(t + t0 ) 2 cos2 ω(t + t0 )

while

∂S 1 ∂F = − ωxj tan ω(t + t0 ), ∂ xj cos ω(t + t0 ) ∂ξj 1 ∂ 2S ∂ 2F = − ω tan ω(t + t0 ). 2 2 cos ω(t + t0 ) ∂ξj2 ∂ xj Inserting these expressions into (1) yields (mixing old and new variables for the moment)

ω sin ω(t + t0 ) ω2 |x|2 x · ∇ξ F − − nεω tan ω(t + t0 ) 2 ω(t + t0 ) cos ω(t + t0 ) 2 cos2 ω(t + t0 ) 2   1 1 1 + ∇ξ F − ωx tan ω(t + t0 ) + V (x) − ε ∆ F − n ω tan ω( t + t ) = 0, ξ 0 2 cos ω(t + t ) cos2 ω(t + t ) 1

Fτ +

cos2

0

0

which simplifies to Fτ +

1 2

  ω2 2 |∇ξ F |2 + V (x) − |x| cos2 ω(t + t0 ) − ε 1ξ F = 0 2

or to (11). If V (x) = ω2 |x|2 /2, then (14) reduces to (13).

(14)



We shall use (7) and (8) in tandem with the Hopf–Cole transformation (HC) according to the following diagram. S (tx, x)

←→

 yHC s(t , x)

F (τx, ξ )

(15)

 yHC f (τ , ξ )

The Hopf–Cole transformation. We let



s(t , x) = exp −

S (t , x) 2ε





and f (τ , ξ ) = exp −

F (τ , ξ )



2ε

for (t , x) ∈ QT = [0, T ] × Rn and (τ , ξ ) ∈ [τ0 , τT ] × Rn , respectively.

(16)

T. Strömberg / Nonlinear Analysis 73 (2010) 1802–1811

1805

In the sequel, we tacitly assume the relation between S, s, F and f imposed by (7), (8), (15) and (16). Likewise, we assume relation (12) between V and W and keep (9) in mind. The Hopf–Cole transformation carries the viscous Hamilton–Jacobi equations (1) and (11) to linear parabolic partial differential equations. In our context, the unknowns s and f of these equations are positive functions. Lemma 2. Under the Hopf–Cole transformation, Eqs. (1) and (11) are equivalent to 1

st −

2ε

fτ −

2ε

V (x)s − ε 1x s = 0 and

s > 0 in QT ,

(17)

and 1

W (τ , ξ )f − ε ∆ξ f = 0

and

f > 0 in (τ0 , τT ] × Rn ,

(18)

respectively. In fact, Eqs. (1), (11), (17) and (18) are mutually equivalent. If V (x) = ω2 |x|2 /2, then (18) reduces to the heat equation f τ − ε ∆ξ f = 0

in (τ0 , τT ] × Rn .

The proof of the lemma is omitted as it is straightforward with the aid of Lemma 1. However, we prove a version for viscosity solutions. We refer to the survey article [11] for the definition and basic properties of this concept of a generalized solution. Lemma 3. The following statements are equivalent: (i) (ii) (iii) (iv)

S is a viscosity subsolution (supersolution, solution, respectively) of (1); F is a viscosity subsolution (supersolution, solution, respectively) of (11); s > 0 is a viscosity supersolution (subsolution, solution, respectively) of (17); f > 0 is a viscosity supersolution (subsolution, solution, respectively) of (18).

Proof. The equivalence between (i) and (ii) follows from the fact that the transformation connecting S and F is a diffeomorphism. We prove only (iii) ⇒ (i), the remaining implications (i.e., (i) ⇒ (iii) and (ii) ⇔ (iv)) being analogous. The relation between S and s is s = h ◦ S where h(α) = e−α/2ε . Notice that S is upper (lower, respectively) semicontinuous exactly when s is lower (upper, respectively) semicontinuous. Assume the supersolution assertion of (iii). Suppose Ψ ∈ C 2 (QT ) and S − Ψ has a local maximum at (tˆ, xˆ ) ∈ QT . Then s − ψ has a local minimum at the same point if ψ = h ◦ Ψ , since h0 < 0. Since we are assuming that s is a viscosity supersolution of (17), we have therefore

ψt (tˆ, xˆ ) −

1 2ε

V (ˆx)ψ(tˆ, xˆ ) − ε 1ψ(tˆ, xˆ ) ≥ 0.

(19)

In terms of Ψ , carrying out the differentiation, (19) reads as the following inequality at (tˆ, xˆ ):

− e−Ψ /2ε

1 Ψt − V (ˆx)e−Ψ /2ε − ε e−Ψ /2ε 2ε 2ε



|∇ Ψ |2 1Ψ − 2 4ε 2ε



≥ 0.

(20)

On dividing through in (20) by the negative factor −e−Ψ /2ε /2ε we deduce

Ψt (tˆ, xˆ ) +

1 2

|∇ Ψ (tˆ, xˆ )|2 + V (ˆx) − ε1Ψ (tˆ, xˆ ) ≤ 0.

We conclude that S is a subsolution of (1) provided that s is a supersolution of (17).



3. An integral formula In our first theorem we are concerned with the Cauchy problem St +

1 2

|∇ S |2 +

S (0, x) = S0 (x)

ω2 |x|2 2

− ε 1S = 0 in QT ,

in Rn .

(21) (22)

The potential is here V (x) = ω |x| /2 corresponding to an isotropic harmonic oscillator. Up to a constant, it constitutes the maximal choice for V (x) permitted by (H1). By means of the changes of variables introduced above we can derive an integral formula for the solution of (21) and (22). The uniqueness follows from extensions of the classical Widder uniqueness theorem [12], which says that each nonnegative solution of the heat equation on R is uniquely determined by its initial value. The extension obtained in [13] to more general linear parabolic equations set in Rn with unbounded coefficients is sufficient for our purposes. 2

2

1806

T. Strömberg / Nonlinear Analysis 73 (2010) 1802–1811

Theorem 4. Assume (H2) and ω > 0. Let T > 0 satisfy 0 < ω(t0 + T ) < π /2 where t0 is given by (10), i.e., 0 < ωT <

π 2

− arctan

a

ω

.

Let ε > 0. Then the Cauchy problem (21) and (22) has a unique classical solution S ∈ C (QT ) ∩ C 1,2 (QT ) given by S (t , x) = F (τ , ξ ) −

ω|x|2 tan ωt

+ nε ln(cos ωt ),

2

(t , x) ∈ QT ,

where



F (τ , ξ ) = −2ε ln (4π ετ )−n/2



Z

exp − Rn

1 2ε



S0 (η) +

1 2τ

|ξ − η|2



dη



and

 tan ωt  τ = , ω x  ξ = . cos ωt

(23)

Furthermore, S (t , x) ≥ A −

ω|x|2 tan ω(t + t0 ) 2

+ nε ln(cos ω(t + t0 ))

(24)

for all (t , x) ∈ QT . Proof. We may assume A = 0 in condition (H2) by replacing the unknown function S (t , x) by S˜ (t , x) = S (t , x) − A. Choose t0 as (10). By Lemma 2, solving (21) and (22) is equivalent to finding a positive solution of fτ −ε ∆ξ f = 0 in (τ0 , τT ]× Rn with initial value f (τ0 , ·) = exp(−F (τ0 , ·)/2ε). Also, by the paragraph following (10), F (τ0 , ·) is a nonnegative function whence 0 < f (τ0 , ξ ) ≤ 1 for all ξ ∈ Rn . Hence, a solution f (τ , ξ ) > 0 exists for all τ0 < τ < ∞ and all ξ ∈ Rn and, moreover, is unique. One may consult, e.g., the articles [13,14] which cover the general case later treated in this paper. This existence and uniqueness assertion translates to a corresponding assertion for S: there exists a unique solution S of the Cauchy problem (21) and (22) for the stated value of T . Returning to f , we easily see that 0 < f (τ , ξ ) ≤ 1 for all (τ , ξ ) ∈ [τ0 , τT ] × Rn since 0 < f (τ0 , ·) ≤ 1. It ensues that S admits the lower bound (24). In order to derive the representation formula for the solution S, we choose now t0 = 0, i.e., we assume (23). The unique positive solution is given by integral convolution of the initial function f (0, ·) = exp(−S0 (·)/2ε) with the heat kernel 2 Kε (τ , ξ ) = (4π ετ )−n/2 e−|ξ | /4ετ ,

i.e., by f (τ , ξ ) = (4π ετ )

−n/2

   1 1 2 S0 (η) + |ξ − η| dη. exp − 2ε 2τ Rn

Z

(25)

Since (H2) is assumed, the Poisson formula (25) furnishes the solution so long as τ < 1/a. Summarizing, the solution S exists and is given by the stated formulas when 0 < t ≤ T . Finally, when the assumption A = 0 is removed, only (24) is affected by the appearance of the term ‘‘A’’ in the right-hand side.  4. Comparison, uniqueness and existence We proceed to the question of the well-posedness of the initial-value problem (1) and (2). We present first a comparison result in the context of viscosity sub- and supersolutions. Theorem 5. Let V ∈ C (Rn ) satisfy the upper bound (3). Let 0 < ωT < π /2 and ε > 0. Let S1 and S2 be a viscosity subsolution and viscosity supersolution of (1), respectively. Suppose that S2 fulfills for some constants A ∈ R and a ≥ 0 the quadratic lower bound S2 (t , x) ≥ A −

a|x|2 2

,

(t , x) ∈ QT .

Assume also that S1 (0, x) ≤ S2 (0, x) for all x ∈ Rn . Then S1 ≤ S2 in QT . Proof. We may without loss of generality assume A = B = 0 since Sj (t , x) can be replaced by S˜j (t , x) = Sj (t , x) − A + t max{0, B}. We switch interchangeably between the functions Fj (τ , ξ ) and fj (τ , ξ ) corresponding to Sj (t , x) (j = 1, 2). It suffices to demonstrate that comparison holds on some time interval [0, T˜ ] ⊆ [0, T ], since Eq. (1) is autonomous. Therefore,

T. Strömberg / Nonlinear Analysis 73 (2010) 1802–1811

1807

we may decrease the value of T to have, for some δ > 0, 0 < ωT <

π 2

− arctan

a+δ

ω

.

Choosing t0 as

ωt0 = arctan

a+δ

ω

,

we may estimate F2 from below as F2 (τ , ξ ) = S2 (t , x) +

≥−

a|x|2 2

+

ω|x|2 tan ω(t + t0 ) 2 ω|x|2 tan ωt0 2

=

− nε ln(cos ω(t + t0 ))

δ|x|2 2

δ|ξ |2 δ|ξ |2 ≥ = 2 2(1 + (ωτ ) ) 2(1 + (ωτT )2 ) 2

when (t , x) ∈ [0, T ]× Rn or (τ , ξ ) ∈ [τ0 , τT ]× Rn . Thus the bound 0 < f2 (τ , ξ ) ≤ e−c |ξ | holds true for some constant c > 0. As opposed to f2 , f1 is not subject to an upper bound but still f1 > 0. According to Lemma 3, f1 and f2 are a supersolution and a subsolution, respectively, of (18) with f2 (τ0 , ·) ≤ f1 (τ0 , ·). We write (18) as fτ + L(f , D2 f ) = 0

in (τ0 , τT ] × Rn ,

where, for (τ , ξ , r , E ) ∈ [τ0 , τT ] × Rn × R × Sn , L(r , E ) = L(τ , ξ , r , E ) = −

1 2ε

W (τ , ξ )r − ε tr E .

Here, the variable E is a symmetric n × n matrix, in symbols E ∈ Sn . Since W ≤ 0, L is elliptic and proper in standard viscosity solution terminology, for which we refer to the User’s Guide [11]. Let us prove that f2 ≤ f1 in [τ0 , τT ] × Rn . We argue by contradiction and assume sup(f2 − f1 ) > 0. Then, for some sufficiently small µ > 0, also M := sup{f2 (τ , ξ ) − f1 (τ , ξ ) − µτ : (τ , ξ ) ∈ [τ0 , τT ] × Rn } > 0. 2 Since (f2 − f1 )(τ , ξ ) ≤ e−c |ξ | , this supremum is attained and for some sufficiently large ρ > 0 we have

M = max{f2 (τ , ξ ) − f1 (τ , ξ ) − µτ : (τ , ξ ) ∈ [τ0 , τT ] × Bρ } and f2 (τ , ξ ) − f1 (τ , ξ ) − µτ < M if |ξ | ≥ ρ . Setting, for λ > 0,

Φλ (τ , ξ , η) = f2 (τ , ξ ) − f1 (τ , η) − µτ −

λ 2

|ξ − η|2 ,

it holds that max{Φλ (τ , ξ , η): τ ∈ [τ0 , τT ], ξ , η ∈ Bρ } ≥ M > 0

(26)

for all λ > 0. Let (τλ , ξλ , ηλ ) be a maximizing point of (26). As λ → ∞, by the compactness, passing to a subsequence if necessary, we may assume that (τλ , ξλ , ηλ ) converges to a limit which necessarily has the form (τ¯ , ξ¯ , ξ¯ ). Moreover, (τ¯ , ξ¯ ) is a maximizing point of f2 (τ , ξ ) − f1 (τ , ξ ) − µτ . The properties of (f2 − f1 )(τ , ξ ) − µτ at τ = τ0 and when |ξ | ≥ ρ force τ¯ > τ0 and |ξ¯ | < ρ . Hence τλ > τ0 , |ξλ | < ρ and |ηλ | < ρ for large λ. We apply Theorem 8.3 of [11] to find the existence of aλ , bλ ∈ R and Xλ , Yλ ∈ Sn such that

(aλ , λ(ξλ − ηλ ), Xλ ) ∈ P

2,+

f2 (τλ , ξλ ),

(bλ , λ(ξλ − ηλ ), Yλ ) ∈ P

2,−

f1 (τλ , ηλ ),

and fulfilling the conditions



I −3λ 0

aλ − bλ = µ, 2,+

2,−



0 I

 ≤

Xλ 0

0 −Y λ



≤ 3λ



I

−I

−I

I



.

(27)

(Here, P f2 and P f1 denote the closures of the second-order superjet of f2 and subjet of f1 relative to (τ0 , τT ] × Rn .) We recall that it follows from the rightmost matrix inequality in (27) that Xλ ≤ Yλ and so tr(Xλ − Yλ ) ≤ 0. Now, taking into

1808

T. Strömberg / Nonlinear Analysis 73 (2010) 1802–1811

account that L is proper and f2 (τλ , ξλ ) > f1 (τλ , ηλ ) (which is a consequence of (26)), we find that aλ + L(τλ , ξλ , f2 (τλ , ξλ ), Xλ ) ≤ 0,

(28)

bλ + L(τλ , ηλ , f2 (τλ , ξλ ), Yλ ) ≥ 0.

(29)

Subtracting inequality (29) from (28) and using aλ − bλ = µ gives

µ + L(τλ , ξλ , f2 (τλ , ξλ ), Xλ ) − L(τλ , ηλ , f2 (τλ , ξλ ), Yλ ) ≤ 0, which can be written as

µ−

1 2ε

(W (τλ , ξλ ) − W (τλ , ηλ ))f2 (τλ , ξλ ) − ε tr(Xλ − Yλ ) ≤ 0.

(30)

Sending λ → ∞, owing to the continuity of W , the boundedness of f2 on [τ0 , τT ] × Bρ , and tr(Xλ − Yλ ) ≤ 0, it follows from (30) that µ ≤ 0. This conclusion contradicts the choice of µ as a positive number above and completes the proof of f2 ≤ f1 . We conclude that S1 ≤ S2 throughout QT .  An immediate consequence of the theorem is the uniqueness of viscosity solutions of (1) having quadratic lower bounds. Corollary 6. Let V ∈ C (Rn ) satisfy the lower bound (3). Let 0 < ωT < π /2 and ε > 0. Let S1 and S2 be viscosity solutions of (1). Suppose that S1 and S2 fulfill for some constants A ∈ R and a ≥ 0 the quadratic lower bound Sj (t , x) ≥ A −

a|x|2 2

,

(t , x) ∈ QT , j = 1, 2.

Then S1 = S2 in QT if S1 (0, x) = S2 (0, x) for all x ∈ Rn . Theorem 5 is sharp in the sense that the comparison principle fails in general if the minorization condition is relaxed to S2 (t , x) ≥ A − a|x|α for some exponent α > 2. This is witnessed by the following modification of the counterexample presented at the end of [15]. Example 1. Let us for simplicity choose n = 1 and V ≡ 0. We set z (t , x) = −t |x|α + |x|β

and S2 (t , x) = min{0, z (t , x)}

for all (t , x) ∈ [0, ∞) × R. The exponents α and β fulfill 2β − 2 > α > β > 2.

(31)

We may choose α > 2 arbitrarily close to 2 and then select β so as to satisfy (31). We consider the set Ω = {S2 = z < 0}:

Ω = {(t , x) ∈ [0, ∞) × Rn : t |x|α−β > 1}. We claim that, for some T > 0, both (i) zt + 12 (zx )2 ≥ 0 and (ii) zxx ≤ 0 when (t , x) ∈ Ω and t ∈ (0, T ]. To confirm these claims we restrict (t , x) to Ω and calculate. As regards (i) we find that zt +

1

1

(zx )2 = −|x|α + 2  2 = |x|α −1 +  ≥ |x|α −1 +

α t |x|α−1 − β|x|β−1 1

−α+2β−2

α−β


2

−β  1 (α−2β+2)/(α−β) t (α − β)2 . 2

|x|

α t |x|


2 

2

(To justify the inequality, note that |x|−α+2β−2 has a positive exponent and that α t |x|α−β − β ≥ α − β > 0.) The expression within brackets on the last line is independent of x and has the form

− 1 + ct γ

where γ =

α − 2β + 2 . α−β

(32)

Thanks to (31), the exponent γ < 0 while the coefficient c > 0 ensuring that (32) is nonnegative for all t ∈ (0, T ] provided that T is sufficiently small. Thus claim (i) has been verified. To demonstrate (ii), for any (t , x) ∈ Ω , zxx = β(β − 1)|x|β−2 − α(α − 1)t |x|α−2 β−2

= β|x|



α t |x|α−β (β − 1) − (α − 1) β

≤ β|x|β−2 [(β − 1) − (α − 1)] ≤ 0 because α t |x|α−β /β ≥ 1.



T. Strömberg / Nonlinear Analysis 73 (2010) 1802–1811

1809

An immediate consequence of inequalities (i) and (ii) is that zt +

1 2

(zx )2 − ε zxx ≥ 0 in Ω ,

for any ε ≥ 0. It is now easy to see that S2 is a supersolution of St + 12 (Sx )2 − ε Sxx = 0 in QT fulfilling S2 (0, x) = 0 for all x ∈ R. Evidently, the comparison principle fails for S1 ≡ 0 and S2 . Although the comparison principle fails for general sub- and supersolutions of (1), it is nevertheless possible to prove that classical solutions of (1) are unique when the potential term V has at most quadratic growth. Theorem 7. Assume (H1) . Let T and ε be positive numbers. Let S1 , S2 ∈ C (QT ) ∩ C 1,2 (QT ) be two solutions of (1) such that S1 (0, x) = S2 (0, x) for all x ∈ Rn . Then S1 = S2 in QT provided that |V (x)| ≤ C (1 + |x|2 ) for some constant C and all x ∈ Rn . Proof. Under these assumptions on V , it is known that classical solutions s of (17) (or f of (18)), being positive, are uniquely determined by their initial data. One may consult [13] for a sufficiently general uniqueness theorem. The two solutions sj (t , x) = exp(−Sj (t , x)/2ε), j = 1, 2, of (17) coincide at t = 0 and, hence, are equal everywhere. We conclude that S1 = S2 .  Remark 8. While undoubtedly a sufficient condition, the comparison principle stated in the standard way for viscosity suband supersolutions is not, we emphasize, a necessary condition for the uniqueness of classical solutions of (1). Indeed, we draw this conclusion from Example 1 and Theorem 7. A similar remark can be made in the context of viscosity solutions of first-order Hamilton–Jacobi equations St + H (x, ∇ S ) = 0

in QT ,

(33)

where, for each x ∈ Rn , H (x, ·) is the Legendre–Fenchel transform of a Lagrangian L(x, ·) and L(x, v) is a superliner convex function of v . Uniqueness results for (33) have been obtained in [16–19] applying to: (i) Lagrangians L(x, v) ≡ L(v) that are independent of x [17]; (ii) L(x, v) = 12 |v|2 − V (x) where V satisfies (H1) [19] (see also Theorem 11 at the end of this paper); and (iii) Lagrangians fulfilling on the one hand L(x, v) ≥ `(|v|) where lims→∞ `(s)/s = ∞ and, on the other, certain vital structure conditions [16,18] ensuring well-behaved variational problems. These theorems state that if S is any viscosity solution of (33), then S is necessarily the value function that is associated with S0 (x) := limt →0 S (t , x) and L, i.e.,



S (t , x) = inf S0 (x(0)) +

t

Z

L(x(s), x˙ (s)) ds: x(·) is AC, x(t ) = x

 (34)

0

when (t , x) ∈ QT . However, the comparison principle is not fulfilled for these equations. Indeed, it fails for St + 21 |∇ S |2 = 0 (again by Example 1), yet uniqueness holds for the initial-value problem because each viscosity solution in QT is given by the Hopf–Lax formula



S (t , x) = inf S0 (y) + y

1 2t

 |x − y|2 ,

(t , x) ∈ QT ,

where S0 is the pointwise limit of S (t , ·) as t → 0; see [17]. Still, comparison actually holds between viscosity solutions if the value function is the unique viscosity solution. This can be observed directly from (34). As regards the existence of solutions one observes that solutions may collapse in finite time. Example 2. We choose for V (x) a quadratic form in diagonal form. Let λ1 ≥ λ2 ≥ · · · ≥ λn . The function S ( t , x) =

 n  X 1 − g (t , λj )x2j + ε h(t , λj ) j =1

2

where

√ √    if λ < 0,  −λ tanh t −λ 0,  g (t , λ) = 0 if λ = √ √    λ tan t λ if λ > 0,    √    if λ < 0, − ln cosh t −λ = 0,  h(t , λ) = 0 if λ  √   ln cos t λ if λ > 0,

1810

T. Strömberg / Nonlinear Analysis 73 (2010) 1802–1811

is, for sufficiently small values of T , a classical solution of St +

1 2

|∇ S |2 +

n 1X

2 j =1

λj x2j − ε 1S = 0 in QT ,

S (0, x) = 0

in Rn .

If no eigenvalue is √ positive, then the solution exists for all times, i.e., T can be chosen as ∞. By contrast, if λ1 > 0, then S (t , x) → −∞ as t λ1 → π /2. Choosing all λj = ω2 , ω > 0, the solution S, which is given by 1 S (t , x) = − ω|x|2 tan ωt + nε ln(cos ωt ), 2 blows up as ωt → π /2. The translate S t0 (t , x) = S (t + t0 , x) is also a solution of (1) which satisfies 1 S t0 (0, x) = − ω|x|2 tan ωt0 + nε ln(cos ωt0 ) 2 and blows up as ω(t + t0 ) → π /2. Next we present well-posedness results for (1) and (2). Theorem 9. Let V and S0 satisfy (H1) and (H2), respectively. Let T > 0 satisfy 0 < ω(t0 + T ) < π /2 where t0 is given by (10). Let ε > 0. Then Cauchy problem (1) and (2) has a classical solution S ∈ C (QT ) ∩ C 1,2 (QT ) satisfying the lower bound S (t , x) ≥ A − Bt −

ω|x|2 tan ω(t + t0 ) 2

+ nε ln(cos ω(t + t0 ))

(35)

for all (t , x) ∈ QT . Furthermore, S is the only viscosity solution satisfying in QT a lower bound of the form S (t , x) ≥ −K (1 + |x|2 ) for some constant K ≥ 0. Proof. We assume again A = B = 0 and (10). By Lemmas 1 and 2, the Cauchy problem (1) and (2) amounts to finding a positive solution of (18) with a certain prescribed value of f (τ0 , ·) = exp(−F (τ0 , ·)/2ε). By Lemma 1, W is a nonpositive function. Also, by the paragraph following (10), F (τ0 , ·) is a nonnegative function whence 0 < f (τ0 , ξ ) ≤ 1. Hence, there exists a bounded positive solution f (τ , ξ ) of (18). This solution f can be represented by means of a positive fundamental solution Γ (τ , ξ ; θ , η) as f (τ , ξ ) =

Z Rn

Γ (τ , ξ ; τ0 , η)f (τ0 , η) dη,

(τ , ξ ) ∈ (τ0 , τT ] × Rn .

For proofs of these facts see, e.g., [13,14]. This existence assertion translates to the existence statement about S formulated in the theorem. In fact, the solution f does not exceed 1, i.e., 0 < f ≤ 1 implying the lower bound (35) with A = B = 0. This can be seen by comparing the bounded solution f with the supersolution 1 or, equivalently by comparing the solution S with the subsolution S1 (t , x) = −

ω|x|2 tan ω(t + t0 ) 2

+ nε ln(cos ω(t + t0 )),

(t , x) ∈ QT .

According to Theorem 5, S1 ≤ S since this inequality holds initially and S has a quadratic minorant as f is bounded from above. Moreover, by Corollary 6 the solution S is unique among all viscosity solutions having a quadratic minorant of the form −K (1 + |x|2 ), K ≥ 0. We leave to the reader to verify that when the assumption A = B = 0 is dropped, only (35) is affected by the appearance of the term ‘‘A − Bt’’ in the right-hand side.  In view of Example 2, the assumed inequality for T in Theorem 9 gives the correct general lifespan of the solution. The minorization assumption can be removed when V grows at most quadratically. Theorem 10. Let V and S0 satisfy (H1) and (H2), respectively. Assume also that V grows at most quadratically, i.e., |V (x)| ≤ C (1 + |x|2 ) for some constant C . Let T > 0 satisfy 0 < ω(t0 + T ) < π /2 where t0 is given by (10). Let ε > 0. Then the solution of Cauchy problem (1) and (2) furnished by Theorem 9 is unique among all classical solutions S ∈ C (QT ) ∩ C 1,2 (QT ). Proof. The classical solution f (τ , ξ ) of (18) is unique by Theorem 7. Therefore the Cauchy problem (1) and (2) possesses a unique solution.  One expects that the solution S ε of (1) and (2), as ε → 0, converges to the viscosity solution S of the Hamilton–Jacobi equation 1

|∇ S |2 + V (x) = 0 in QT , S (0, x) = S0 (x) in Rn . (36) 2 We shall not deal with the vanishing viscosity limit here but instead state the existence and uniqueness result for (36) obtained in [19]. One observes the similarities with Theorem 10. St +

T. Strömberg / Nonlinear Analysis 73 (2010) 1802–1811

1811

Theorem 11. Let V and S0 satisfy (H1) and (H2), respectively. Assume also that V grows at most quadratically, i.e., |V (x)| ≤ C (1 + |x|2 ) for some constant C . Let T > 0 satisfy 0 < ω(t0 + T ) < π /2 where t0 is given by (10). Then Cauchy problem (36) admits a viscosity solution S given by (34) with L(x, v) = 12 |v|2 − V (x). It is unique in the class of all continuous viscosity solutions. References [1] V.I. Arnold, Mathematical methods of classical mechanics, Graduate Texts in Mathematics, vol. 60, Springer-Verlag, New York, 1989. [2] S. Benachour, M. Ben-Artzi, P. Laurençot, Sharp decay estimates and vanishing viscosity for diffusive Hamilton–Jacobi equations, Adv. Differential Equations 14 (2009) 1–25. [3] S. Benachour, S. Dăbuleanu-Hapca, P. Laurençot, Decay estimates for a viscous Hamilton–Jacobi equation with homogeneous Dirichlet boundary conditions, Asymptot. Anal. 51 (2007) 209–229. [4] Y. Fujita, H. Ishii, P. Loreti, Asymptotic solutions of viscous Hamilton–Jacobi equations with Ornstein–Uhlenbeck operator, Comm. Partial Differential Equations 31 (2006) 827–848. [5] T. Gallay, P. Laurençot, Asymptotic behavior for a viscous Hamilton–Jacobi equation with critical exponent, Indiana Univ. Math. J. 56 (2007) 459–479. [6] B.H. Gilding, The Cauchy problem for ut = 1u + |∇ u|q , large-time behaviour, J. Math. Pures Appl. 84 (9) (2005) 753–785. [7] P. Laurençot, Convergence to steady states for a one-dimensional viscous Hamilton–Jacobi equation with Dirichlet boundary conditions, Pacific J. Math. 230 (2007) 347–364. [8] P. Laurençot, P. Souplet, Optimal growth rates for a viscous Hamilton–Jacobi equation, J. Evol. Equ. 5 (2005) 123–135. [9] P. Quittner, P. Souplet, Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States, Birkhäuser Verlag, Basel, 2007. [10] P. Souplet, Q.S. Zhang, Global solutions of inhomogeneous Hamilton–Jacobi equations, J. Anal. Math. 99 (2006) 355–396. [11] M.G. Crandall, H. Ishii, P.-L. Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (NS) 27 (1992) 1–67. [12] D.V. Widder, Positive temperatures on an infinite rod, Trans. Amer. Math. Soc. 55 (1944) 85–95. [13] D.G. Aronson, P. Besala, Uniqueness of positive solutions of parabolic equations with unbounded coefficients, Colloq. Math. 18 (1967) 125–135. [14] D.G. Aronson, P. Besala, Parabolic equations with unbounded coefficients, J. Differential Equations 3 (1967) 1–14. [15] G. Barles, Uniqueness for first-order Hamilton–Jacobi equations and Hopf formula, J. Differential Equations 69 (1987) 346–367. [16] T. Strömberg, On viscosity solutions of the Hamilton–Jacobi equation, Hokkaido Math. J. 28 (1999) 475–506. [17] T. Strömberg, The Hopf–Lax formula gives the unique viscosity solution, Differential Integral Equations 15 (2002) 47–52. [18] T. Strömberg, Hamilton–Jacobi equations having only action functions as solutions, Arch. Math. (Basel) 83 (2004) 437–449. [19] T. Strömberg, Well-posedness for the system of the Hamilton–Jacobi and the continuity equations, J. Evol. Equ. 7 (2007) 669–700.