aa r X i v : . [ m a t h . A P ] A p r WELL-POSEDNESS FOR CONSTRAINED HAMILTON-JACOBIEQUATIONS
YEONEUNG KIM
Abstract.
The goal of this paper is to study a Hamilton-Jacobi equation ( u t = H ( Du ) + R ( x, I ( t )) in R n × (0 , ∞ ) , sup R n u ( · , t ) = 0 on [0 , ∞ ) , with initial conditions I (0) = 0, u ( x,
0) = u ( x ) on R n . Here ( u, I ) is a pair ofunknowns and the Hamiltonian H and the reaction R are given. And I ( t ) is anunknown constraint (Lagrange multiplier) that forces supremum of u to be alwayszero. We construct a solution in the viscosity setting using a fixed point argumentwhen the reaction term R ( x, I ) is strictly decreasing in I . We also discuss bothuniqueness and nonuniqueness. For uniqueness, a certain structural assumptionon R ( x, I ) is needed. We also provide an example with infinitely many solutionswhen the reaction term is not strictly decreasing in I . Contents
1. Introduction 12. Construction of a solution of relaxed problem via a fixed point argument 33. Limiting equation 104. Uniqueness for a certain birth rate 145. Nonuniqueness result 156. Conclusion 16References 171.
Introduction
Darwin’s theory of evolution suggests that biological individuals evolve underthe competition between natural selection and mutation. The mathematical modelbased on such theory has been studied in literatures (see [7, 8, 9, 10]). In the model,we usually denote traits, density of population and net birth rate by x ∈ R n , n ( x, t ), R ( x, I ), respectively, where I ( t ) represents the total consumptions of the resourcesof the environment at the time t . We can take mutation into account using diffusion ε ∆ for some small ε > Date : April 13, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Hamilton-Jacobi equation with constraint, selection-mutation model.Supported in part by NSF grant DMS-1664424. equation ( n εt − ε ∆ n ε = n ε ε R ( x, I ε ( t )) in R n × (0 , ∞ ) ,n ε ( x,
0) = n ε ∈ L ( R n ) on R n , where we denote the total population associated with the rate ε by n ε ( x, t ) and n ε ≥ I ε ( t ) is defined as I ε ( t ) = Z R n ψ ( x ) n ε ( t, x ) dx, where ψ is a given smooth, non-negative compactly supported kernel representingconsumption rate of resources.It was studied by G. Barles, S. Mirrahimi, B. Perthame [3] that after taking Hopf-Cole transformation n ε ( x, t ) = e u ε ( x,t ) /ε , as mutation rate ε vanishes, u ε convergeslocally uniformly to u in R n × [0 , ∞ ) which is a solution of the constrained Hamilton-Jacobi equation u t = | Du | + R ( x, I ( t )) in R n × (0 , ∞ ) , max R n u ( · , t ) = 0 on [0 , ∞ ) ,u ( x,
0) = u ( x ) on R n . Motivated from this, we will discuss well-posedness of viscosity solutions for anequation with a general Hamiltonian H ( p ) and an unknown constraint I ( t ). Theequation we consider is the following u t = H ( Du ) + R ( x, I ( t )) in R n × [0 , T ] , sup R n u ( · , t ) = 0 on [0 , T ] ,I (0) = 0 ,u ( x,
0) = u ( x ) on R n . (1.1) Main Assumptions .
We need assumptions on R ( x, I ) : R n × [ − I M , I M ] → R for I M > , u ( x ) and I ( t ) , some of which are natural but some are technical. (A1) There exist K , K > such that − K ≤ R I ( x, I ) ≤ − K ; (A2) max R n R ( · , I M ) = 0 ; (A3) min R n R ( · ,
0) = 0 ; (A4) sup | I |≤ I M k R ( · , I ) k W , ∞ ( R n ) < ∞ ; (A5) u ( x ) ∈ W , ∞ ( R n ) and sup x ∈ R n u ( x ) = 0 ; (H1) H ∈ C ( R n , [0 , ∞ )) is a nonnegative Hamiltonian with H (0) = 0 and islocally Lipschitz continuous in p . Throughout the paper, the above assumptions are always in force. Additionally, f ∈ W , ∞ ( R n ), that is; k f k L ∞ ( R n ) + k Df k L ∞ ( R n ) < ∞ . Remark 1.
We define R ( x, I ) as R ( x, I M ) − I + 2 I M when I > I M ,R ( x, I ) when − I M ≤ I ≤ I M ,R ( x, − I M ) − I M − I when I < − I M , ELL-POSEDNESS FOR CONSTRAINED HAMILTON-JACOBI EQUATIONS 3 so that R ( x, I ) is continuously extended. Later on, we will see that I ( t ) ∈ [0 , I M ]for t ≥ u ε , I ε ) for a relaxed equation using Banach’s fixed point argument.In Section 3, we show that ( u ε , I ε ) converges to a solution pair ( u, I ) for the originalequation (1.1) up to subsequences. In Section 4, uniqueness result for (1.1) ispresented when the reaction R ( x, I ) is assumed to be separable in variables x and t . In Section 5, we finish this paper by giving an example where uniqueness failswhen R ( x, I ) is not strictly decreasing in I .2. Construction of a solution of relaxed problem via a fixed pointargument
We first provide some regularity properties for ( u t = H ( Du ) + R ( x, I ( t )) in R n × [0 , T ] ,u ( x,
0) = u ( x ) on R n , (2.1)where I ( t ) is a given continuous function on [0 , T ]. It is known that there existsa unique viscosity to u ( x, t ) for (2.1) which is bounded uniformly continuous in R n × [0 , T ]. Theorem 2.1.
Let u be a unique viscosity solution of (2.1) for a given continuousfunction I ( t ) . Set L = sup t ∈ [0 ,T ] k R ( · , I ( t ) k W , ∞ ( R n ) . Then, for T > , k Du k L ∞ ( R n × [0 ,T ]) ≤ ( L + 1) T + k Du k L ∞ ( R n ) . Proof.
We follow arguments presented in [2, 11, 12, 1]. We first define C ( t ) = ( L + 1) t + K where K = k u k W , ∞ ( R n ) and assume the solution u is not Lipschitz continuous inspace x . In other words, there exists σ > x,y ∈ R n ,t ∈ [0 ,T ] ( u ( x, t ) − u ( y, t ) − C ( t ) | x − y | ) = σ. We then define Φ asΦ( x, y, t, s ) := u ( x, t ) − u ( y, s ) − C ( t ) | x − y | − α | t − s | − β ( | x | + | y | ) . for ( x, y, t, s ) ∈ R n × [0 , T ] and α, β >
0. Since u is bounded, we can find( x, y, t, s ) ∈ R n × [0 , T ] such thatmax ( x,y,t,s ) ∈ R n × [0 ,T ] Φ( x, y, t, s ) = Φ( x, y, t, s ) . We can also note thatΦ( x, y, t, s ) ≥ max x,y ∈ R n ,t ∈ [0 ,T ] Φ( x, y, t, t )= sup x,y ∈ R n ,t ∈ [0 ,T ] u ( x, t ) − u ( y, t ) − C ( t ) | x − y | − β | x | , YEONEUNG KIM which yields Φ( x, y, t, s ) > σ β small enough regardless of α . Moreover, x = y for α small enough. If not,max ( x,y,t,s ) ∈ R n × [0 ,T ] Φ( x, y, t, s ) = Φ( x, x, t, s ) = u ( x, t ) − u ( x, s ) − α | t − s | − β | x | ≤ σ α small enough as u ∈ BU C ( R n × [0 , T ]).We use Φ( x, y, t, s ) ≥ Φ(0 , , ,
0) to get1 α | t − s | + β ( | x | + | y | ) ≤ u ( x, t ) − u ( y, s ) − u ( x,
0) + u ( y, − C ( t ) | x − y |≤ u ( x, t ) − u ( y, s ) − u ( x,
0) + u ( y, . The inequality above implies | t − s | = O ( α ), | x | , | y | = O (1 / √ β ) since u is bounded.Moreover, t, s have to be away from 0 since σ < Φ( x, y, t, s ) < u ( x, t ) − u ( y, s ) − C ( t ) | x − y | and u ( x, − u ( y, ≤ K | x − y | where K = C (0).Observing that u ( x, t ) − φ ( x, t ) has maximum at ( x, t ) where φ ( x, t ) := u ( y, s ) + C ( t ) | x − y | + 1 α | t − s | + β ( | x | + | y | ) . By the definition of viscosity subsolutions,( L + 1) | x − y | + 2 α ( t − s ) ≤ H (cid:18) C ( t ) x − y | x − y | + 2 βx (cid:19) + R ( x, I ( t )) . (2.2)Similarly, u ( y, t ) − η ( y, t ) has minimum at ( y, s ) where η ( y, s ) := u ( x, t ) − C ( t ) | x − y | − α | t − s | − β ( | x | + | y | ) . By the definition of viscosity supersolutions,2 α ( t − s ) ≥ H (cid:18) C ( t ) x − y | x − y | − βy (cid:19) + R ( y, I ( s )) (2.3)Subtracting (2.3) from (2.2) gives us( L + 1) | x − y | ≤ H (cid:18) C ( t ) x − y | x − y | + 2 βx (cid:19) − H (cid:18) C ( t ) x − y | x − y | − βy (cid:19) + R ( x, I ( t )) − R ( y, I ( s )) ≤ Aβ | x + y | + R ( x, I ( t )) − R ( y, I ( s ))where A > H ( p ). Here, we canchoose such A since the terms inside of the Hamiltonian are not growing to either ELL-POSEDNESS FOR CONSTRAINED HAMILTON-JACOBI EQUATIONS 5 ∞ or −∞ . We note that R ( x, I ( t )) − R ( y, I ( s )) = R ( x, I ( t )) − R ( x, I ( s )) + R ( x, I ( s )) − R ( y, I ( s )) . ≤ K | I ( t ) − I ( s ) | + k R ( · , I ( t )) k W , ∞ ( R n ) | x − y | Then, taking α to 0 and combining previous two above gives( L + 1) | x − y | ≤ Aβ | x + y | + L | x − y | . Finally, sending β to 0 to give us L + 1 ≤ L, which is a contradiction. Therefore,sup x,y ∈ R n ,t ∈ [0 ,T ] u ( x, t ) − u ( y, t ) − C ( t ) | x − y | ≤ . By symmetry, we get, for x, y ∈ R n , t ∈ [0 , T ], | u ( x, t ) − u ( y, t ) | ≤ C ( t ) | x − y | , and it is consistent with Lipschitz bound for u as C (0) = K . (cid:3) Theorem 2.2.
Assume I is given and let u be a unique viscosity solution of (2.1).Then, for a positive C(T) depending on T, k u t k L ∞ ( R n × [0 ,T ]) < C ( T ) . Proof.
We first show − C ≤ u t ≤ C for ( x, t ) ∈ R n × [0 , T ] in viscosity sense for apositive constant C depending only on T when u is the viscosity solution of (2.1).Let us assume that φ ( x, t ) ∈ C ( R n × (0 , ∞ ) touches u ( x, t ) from above so that u − φ has maximum at ( x , t ) ∈ R n × (0 , T ]. For φ to touch u from above at ( x , t ),lim sup ( y,s ) → ( x ,t ) u ( y, s ) − u ( x , t ) − ( Dφ, φ t ) · ( y − x , s − t ) | ( y, s ) − ( x , t ) | ≤ . Let us take a sequence ( x ′ , t ) converging to ( x , t ) such that − Dφ ( x , t ) | Dφ ( x , t ) | = x ′ − x | x ′ − x | Then we have0 ≥ lim sup ( y,s ) → ( x ,t ) u ( y, s ) − u ( x , t ) − ( Dφ, φ t ) · ( y − x , s − t ) | ( y, s ) − ( x , t ) |≥ lim sup ( x ′ ,t ) → ( x ,t ) u ( x ′ , t ) − u ( x , t ) − Dφ ( x ′ , t ) · ( x ′ − x ) | x ′ − x |≥ − C ( t ) + | Dφ ( x , t ) | where C ( t ) = ( L + 1) t + K from Theorem 2.1. Hence, | Dφ | is bounded dependingonly on T regardless of choice of test functions φ .By the definition of viscosity subsolutions, we obtain φ t ≤ H ( Dφ ) + R ( x , I ( t )) ≤ C YEONEUNG KIM for a positive constant C which depends on T . Therefore, u t ≤ C in viscosity sense.One can show the other inequality similarly.It remains to check that the inequality above in viscosity sense implies Lipschitzcontinuity of u in time. Although elementary, we present the proof of it. Let us fixthe time s and define u and u as u ( x, t ) = u ( x, t + s ) ,u ( x, t ) = u ( x, s ) + Ct, so that u ( x,
0) = u ( x, u is a viscosity subsolution of u t = C while u isa viscosity solution of u t = C . Therefore, we have u ( x, t ) ≤ u ( x, t ) for ( x, t ) ∈ R n × [0 , T ]by comparison principle, which implies u ( x, t + s ) ≤ u ( x, s ) + Ct.
Similarly, we can derive u ( x, s ) − Ct ≤ u ( x, t + s ) , which finishes the proof. (cid:3) Definition 1.
For ε > and I ∈ C ([0 , T ]) such that I (0) = 0 , we define a mapping Σ from W = { I ∈ C ([0 , T ]) , I (0) = 0 } to itself as following Σ : I ( t ) (1 − ε ) I ( t ) + sup R n u ε ( · , t ) where u ε ( x, t ) is a unique bounded uniformly continuous viscosity solution of (2.1)corresponding to I ( t ) . The mapping Σ is well defined as sup R n u ε ( · , t ) is continuous in time t due toLipschitz regularity properties of the viscosity solution u ε ( x, t ). We first prove thefollowing proposition before we show Σ is contraction mapping. Proposition 2.3.
Let ( u , I ) , ( u , I ) be two solution pairs for (2.1). Then, for f ( t ) := k ( u − u )( · , t ) k L ∞ ( R n ) , f ′ ( t ) ≤ K | I ( t ) − I ( t ) | in viscosity sense where K is a constant from (A1) .Proof. Let ( u , I ), ( u , I ) be two viscosity solution pairs satisfying (2.1). We firstprove that ddt k ( u − u )( · , t ) k L ∞ ( R n ) ≤ K | I ( t ) − I ( t ) | in viscosity sense. Clearly f ( t ) is Lipschitz continuous as both u , u are boundedand Lipschitz continuous. Without loss of generality, we may assume there exists φ ( t ) ∈ C ((0 , T ]) for which f ( t ) − φ ( t ) has a strict maximum at t > φ ( t ) = f ( t ) and f ( t ) = sup R n ( u − u )( · , t ) ≥
0. We can also assume that φ ≥ λ > ε > x, y, t, s ) := u ( x, t ) − u ( y, s ) + λ ( t + s ) −
12 ( φ ( t ) + φ ( s )) − ε ( | t − s | + | x − y | ) − ε ( | x | + | y | ) . ELL-POSEDNESS FOR CONSTRAINED HAMILTON-JACOBI EQUATIONS 7
Since u , u are bounded, for λ > x ε , y ε , t ε , s ε ) ∈ R n × [0 , T ]such that max ( x,y,t,s ) ∈ R n × [0 ,T ] Φ( x, y, t, s ) = Φ( x ε , y ε , t ε , s ε ) > λt (2.4)for all ε small enough. To verify this, for λ > δ > λt − δ >
0. There also exists x ′ ∈ R n such that u ( x ′ , t ) − u ( x ′ , t ) − φ ( t ) > − δ . By observingΦ( x ε , y ε , t ε , s ε ) ≥ Φ( x ′ , x ′ , t , t ) > λt − δ − ε | x ′ | , we deduce lim inf ε → Φ( x ε , y ε , t ε , s ε ) ≥ λt since δ is arbitrary. Hence, we can conclude byΦ( x ε , y ε , t ε , s ε ) > λt for all ε small enough, which verifies (2.4).From Φ( x ε , y ε , t ε , s ε ) ≥ Φ(0 , , , u ( x ε , t ε ) − u ( y ε , s ε ) + λ ( t ε + s ε ) −
12 ( φ ( t ε ) + φ ( s ε )) − ε ( | t ε − s ε | + | x ε − y ε | ) − ε ( | x ε | + | y ε | ) ≥ u (0 , − u (0 , − φ (0) , which implies 1 ε ( | t ε − s ε | + | x ε − y ε | ) + ε ( | x ε | + | y ε | ) < ∞ since u , u , φ are all bounded. Hence, we obtain | x ε − y ε | , | t ε − s ε | = O ( ε ) , | x ε | , | y ε | = O (1 / √ ε ) . Similarly, we use Φ( x ε , y ε , t ε , s ε ) ≥ Φ( x ε , x ε , t ε , t ε ) to have1 ε ( | t ε − s ε | + | x ε − y ε | ) ≤ u ( x ε , t ε ) − u ( y ε , s ε ) + λ ( s ε − t ε )+ 12 ( φ ( t ε ) − φ ( s ε )) + ε ( x ε − y ε ) · ( x ε + y ε ) . It follows that | t ε − s ε | , | x ε − y ε | = o ( ε ) since u is uniformly continuous. Addition-ally, from Φ( x ε , y ε , t ε , s ε ) ≥ Φ( x ε , x ε , t ε , s ε ), we obtain u ( x ε , s ε ) − u ( y ε , s ε ) + ε ( x ε − y ε ) · ( x ε + y ε ) ≥ ε | x ε − y ε | and by Lipschitz continuity of u in space, boundedness of ε | x ε − y ε | follows giventhat x = y . Even if x = y , we can claim the same bound. Moreover, from (2.4), wehave 0 < λt < u ( x ε , t ε ) − u ( y ε , s ε ) + λ ( t ε + s ε )for all ε small enough, we deduce that there exists µ > t ε , s ε ≥ µ . YEONEUNG KIM
Noticing that u ( x, t ) − η ( x, t ) achieves maximum at ( x ε , t ε ) where η ( x, t ) := u ( y ε , s ε ) − λ ( t + s ε )+ 12 ( φ ( t )+ φ ( s ε ))+ 1 ε ( | t − s ε | + | x − y ε | )+ ε ( | x | + | y ε | ) , we can use the viscosity subsolution test to get − λ + 2 ε ( t ε − s ε ) + 12 φ ′ ( t ε ) ≤ H (cid:18) ε ( x ε − y ε ) + 2 εx ε (cid:19) + R ( x ε , I ( t ε )) . (2.5)Similarly, u ( y, s ) − η ( y, s ) achieves minimum at ( y ε , s ε ) where η ( y, s ) = u ( x ε , t ε )+ λ ( t ε + s ) −
12 ( φ ( t ε )+ φ ( s )) − ε ( | t ε − s | + | x ε − y | ) − ε ( | x ε | + | y | ) . Again by the viscosity supersolution test, we get λ + 2 ε ( t ε − s ε ) − φ ′ ( s ε ) ≥ H (cid:18) ε ( x ε − y ε ) − εy ε (cid:19) + R ( y ε , I ( s ε )) . (2.6)Subtracting (2.6) from (2.5) results in − λ + 12 ( φ ′ ( t ε ) + φ ′ ( s ε )) ≤ H (cid:18) ε ( x ε − y ε ) + 2 εx ε (cid:19) − H (cid:18) ε ( x ε − y ε ) − εy ε (cid:19) + R ( x ε , I ( t ε )) − R ( y ε , I ( s ε )) . Using the fact that ε ( x ε − y ε ) is bounded. Because of Theorem 2.1 and H ( p ) locallyLipschitz continuous, we have φ ′ ( t ) ≤ K | I ( t ) − I ( t ) | if t ε and s ε converge to t as ε vanishes as λ > t ε , s ε actually converge to t as ε goes to 0 up to subsequence for agiven λ >
0. Let us suppose that t ε , s ε converge to t ′ = t up to subsequence withrespect to ε . Then there exists γ > x ∈ R n ( u − u )( · , t ′ ) − φ ( t ′ ) ≤ − γ since f ( t ) − φ ( t ) obtains a strict maximum 0 at t .We now observe thatΦ( x ε , y ε , t ε , s ε ) ≤ u ( x ε , t ε ) − u ( y ε , s ε ) + λ ( t ε + s ε ) −
12 ( φ ( t ε ) + φ ( s ε ))= ( u ( x ε , t ε ) − u ( x ε , t ′ )) − ( u ( y ε , s ε ) − u ( y ε , t ′ )) + ( λ ( t ε + s ε ) − λt ′ ) −
12 ( φ ( t ε ) + φ ( s ε )) + φ ( t ′ ) + u ( x ε , t ′ ) − u ( y ε , t ′ ) + 2 λt ′ − φ ( t ′ ) . We then take lim inf ε → on both sides, the following is obtained; λt ≤ − γ + 2 λt ′ . However, taking λ to 0 yields that 0 ≤ − γ , which is a contradiction. Therefore, t ε , s ε must converge to t . (cid:3) ELL-POSEDNESS FOR CONSTRAINED HAMILTON-JACOBI EQUATIONS 9
Proposition 2.4.
The map
Σ : W W is a contraction mapping for a short time T ′ = εK > , and there exists a viscosity solution of u εt = H ( Du ε ) + R ( x, I ε ( t )) in R n × [0 , T ] ,εI ε ( t ) = sup R n u ε ( · , t ) on [0 , T ] ,I ε (0) = 0 ,u ε ( x,
0) = u ( x ) on R n for I ε ∈ W . We call this a relaxed equation.Proof. First of all, the proposition (2.3) results in k ( u − u )( · , t ) k W , ∞ ( R n ) ≤ K k I − I k L ∞ ([0 ,T ]) . (2.7)Now, let I and I be in W and T ′ = εK . ObservingΣ( I ) − Σ( I ) ≤ (1 − ε )( I ( t ) − I ( t )) + sup R n u ( · , t ) − sup R n u ( · , t ) ≤ (1 − ε )( I ( t ) − I ( t )) + | sup R n u ( · , t ) − sup R n u ( · , t ) | , and combining with (2.7), we have k Σ( I ) − Σ( I ) k L ∞ ([0 ,T ′ ]) ≤ (1 − ε + K T ′ ) k I − I k L ∞ ([0 ,T ′ ]) = (1 − ε k I − I k L ∞ ([0 ,T ′ ]) , which implies Σ is a contraction mapping.For a fixed ε > T ′ , by Banach’s fixed point theorem, there exists aunique fixed point I ε ∈ W such that I ε = Σ( I ε ) = (1 − ε ) I ε + sup R n u ε ( · , t ) ⇔ εI ε = sup R n u ε ( · , t )Therefore, for the short time T ′ = εK , we have a solution pair ( u ε , I ε ) for u εt = H ( Du ε ) + R ( x, I ε ( t )) in R n × [0 , T ′ ] ,εI ε ( t ) = sup R n u ε ( · , t ) on [0 , T ′ ] ,I ε (0) = 0 ,u ε ( x,
0) = u ( x ) on R n . We also notice that that T ′ depends only on K and ε , applying the argumentabove successively on time intervals [0 , T ′ ], [ T ′ , T ′ ], [2 T ′ , T ′ ] , · · · , one can obtaina solution which is valid for whole time interval [0 , T ]. (cid:3) Limiting equation
In the previous section, for each ε >
0, we have constructed a solution pair ( u ε , I ε )to u εt = H ( Du ε ) + R ( x, I ε ( t )) in R n × [0 , T ] ,εI ε ( t ) = sup R n u ε ( · , t ) on [0 , T ] ,I ε (0) = 0 ,u ε ( x,
0) = u ( x ) on R n . (3.1)By Theorems 2.1 and 2.2, u ε is Lipschitz continuous in both time and space butLipschitz constants are not uniform in ε . The constants rather depend on thebound of I ( t ) as we can see in the proofs. In this section, we first prove I ε isnondecreasing and uniformly bounded by I M regardless of ε . Then it follows thatLipschitz constants in time and space for u ε are uniform so that u ε converges locallyuniformly to a bounded Lipschitz continuous function u up to subsequence of ε bythe Arzela-Ascoli theorem. We finish this section by noting that a limit function u actually solves the original constrained problem with I ( t ) using the stability resultfor discontinuous Hamilton-Jacobi equations. Proposition 3.1.
Let ( u ε , I ε ) be a solution of (3.1). Then I ε ( t ) is nondecreasingin t .Proof. We first claim that I ε ( t ) cannot have an interior strict local maximum. Letus suppose I ε ( t ) obtains a strict local maximum at t ∈ (0 , T ) so it satisfies εI ε ( t ) = sup R n u ε ( · , t ) . (3.2)For β >
0, let us define f ( x, t ) := u ε ( x, t ) − εI ε ( t ) − β p | x | so that we can find ( x β , t β ) ∈ R n × [0 , T ] such thatmax ( x,t ) ∈ R n × [0 ,T ] f ( x, t ) = f ( x β , t β )Clearly, t β = t as f ( x, t ) ≤ f ( x, t ). For any positive δ > y ∈ R n , such that εI ε ( t ) − δ < u ε ( y, t ) < εI ε ( t ) , which yields, f ( x β , t ) ≥ f ( y, t ) = u ( y, t ) − εI ε ( t ) − β p | y | ≥ − δ − β p | y | . Now we take lim inf on both sides, we get lim inf β → f ( x β , t ) ≥ − δ for any δ > f ( x β , t ) ≤
0. Combining these two, we getlim β → f ( x β , t ) = 0 . Additionally, one can derive lim β → β q | x β | = 0 ELL-POSEDNESS FOR CONSTRAINED HAMILTON-JACOBI EQUATIONS 11 since 0 ≥ − β q | x β | ≥ f ( x β , t )and f ( x β , t ) → β goes to 0. Therefore, we can conclude withlim β → u ε ( x β , t ) = εI ε ( t ) . We can also observe that u ( x, t ) − φ ( x, t ) obtains a local maximum at ( x β , t ) ∈ R n × (0 , T ) where φ ( x, t ) := εI ε ( t ) − β p | x | . Hence, we have0 ≤ H β x β p | x β | ! + R ( x β , I ε ( t ))by the definition of viscosity subsolutions. Taking lim inf with respect to β , we havelim inf β → R ( x β , I ε ( t )) ≥ x β √ | x β | is bounded. From (3.1), we get the following inequalities for h > εI ε ( t + h ) − u ε ( x β , t ) ≥ u ε ( x β , t + h ) − u ε ( x β , t )= Z t + ht u εt ( x β , t ) dt ≥ Z t + ht R ( x β , I ε ( t )) dt ≥ Z t + ht R ( x β , I ε ( t )) dt = hR ( x β , I ε ( t )) . Consequently, taking lim inf with respect to β yields ε ( I ε ( t + h ) − I ε ( t )) ≥ lim inf β → hR ( x β , I ε ( t )) ≥ , which contradicts the fact that I ε ( t ) achieves a strict local maximum at t . Hence, I ε ( t ) cannot have an interior strict local maximum.It remains to prove I ε ( t ) is nonnegative and is even nondecreasing on [0 , T ]. Letus first assume that there exists t ∈ (0 , T ) such that I ε ( t ) <
0. If I ε ( t ) is negativefor all t ∈ (0 , t ), we get a contradiction using the similar argument above with t replaced by 0 since R ( x, I ε ( t )) > I ε ( t ) < I ε (0) = 0. Else if thereexists t ∈ (0 , t ) such that I ε ( t ) >
0, then I ε ( t ) has an interior local maximum,which cannot happen. Therefore, I ε ( t ) is nonnegative. Now it is easy to see I ε isnondecreasing on [0 , T ] by the following argument; if we can find 0 < t < t in(0 , T ) such that I ε ( t ) > I ε ( t ) >
0, then I ε ( t ) achieves interior local maximum aswell because I ε (0) = 0. Hence, we finish the proof. (cid:3) Proposition 3.2.
Let ( u ε , I ε ) be a solution of (3.1). Then ≤ I ε ( t ) ≤ I M for t ≥ . Proof.
We may assume that there exists t ∈ (0 , T ) at which I ε ( t ) is differentiableand I ε ( t ) > I M . It follows that R ( x, I ε ( t )) < . (3.4)since max x ∈ R n R ( x, I M ) = 0. We can also find φ ( t ) ∈ C ( R + ) such that εI ε ( t ) − φ ( t )has a local maximum at t and φ ′ ( t ) >
0. For β > x, t ) = u ε ( x, t ) − φ ( t ) − β p | x | , which has a maximum at ( x β , t ). By the definition of viscosity subsolutions, wehave φ ′ ( t ) ≤ H β x β p | x β | ! + R ( x β , I ε ( t )) . Therefore, we have 0 < φ ′ ( t ) ≤ lim inf β → R ( x β , I ε ( t )) . However, this contradicts (3.4). Therefore, 0 ≤ I ε ( t ) ≤ I M since I ε is nondecreasingand I ε (0) = 0. (cid:3) For a family of locally uniformly bounded functions { u α } α ∈ R , we define up-per(lower) half-relaxed limit u (or u ) as u = lim sup α →∞ ⋆ u α ( x ) := lim α →∞ sup { u δ ( y ) : | x − y | ≤ /β where δ, β ≥ α } and u = lim inf α →∞ ⋆ u α ( x ) := lim α →∞ inf { u δ ( y ) : | x − y | ≤ /β where δ, β ≥ α } Lemma 3.3.
Upper half-relaxed limit I ( t ) and lower-half relaxed limit I ( t ) of I ε ( t ) agree almost everywhere for { I ε } ⊂ C ([0 , T ]) nondecreasing.Proof. By Helly’s compactness theorem, we may assume that I ε ( t ) → I ( t ) everywhere up to passing to a subsequencewhere I ( t ) is nondecreasing on [0 , T ]. Moreover, I ( t ) has only countably many jumpdiscontinuities. We claim that I ( t ) ≤ I ( t +)and I ( t ) ≥ I ( t − ) . For simplicity, we may assume t = 0 and it is enough to prove the first case asproving the second case is pretty much similar.By the definition of upper half-relaxed limit and the property that I ε is nonde-creasing, we have I (0) = lim γ → sup β { I β ( γ ) : β ≤ γ } . ELL-POSEDNESS FOR CONSTRAINED HAMILTON-JACOBI EQUATIONS 13
We note that sup β { I β ( γ ) : β ≤ γ } is decreasing in γ . For δ > γ > β ≤ γ such that | I β ( γ ) − I ( γ ) | < δ for all β ≤ β . Therefore, for γ > I (0) ≤ sup β { I β ( β ) : β ≤ β }≤ sup β { I β ( γ ) : β ≤ β } = sup β { I β ( γ ) − I ( γ ) : β ≤ β } + I ( γ ) ≤ δ + I ( γ ) . Since, δ is arbitrary, taking γ to 0 yields I (0) ≤ I (0+) . Similarly, one can prove the other inequality, hence, we can conclude that I ( t ) = I ( t ) a.e. (cid:3) Theorem 3.4.
There exists a pair ( u, I ) solving (1.1).Proof. By the stability result for discontinuous Hamiltonian in [2, 11], u is a subso-lution of u t = H ( Du ) + R ( x, I ( t ))and u is a supersolution of u t = H ( Du ) + R ( x, I ( t )) . Since 0 ≤ I ε ≤ I M , there exists a subsequence { u ε j } j ∈ N such that u ε j → u locally uniformly on R n × [0 , T ]Therefore, we can let u = u = u . Moreover, I ( t ) = I ( t ) almost everywhere, we let I ( t ) = I ( t ) = I ( t ) . With this new I ( t ), u is a viscosity solution of ( u t = H ( Du ) + R ( x, I ( t )) in R n × [0 , T ] ,u ( x,
0) = u ( x ) on R n . Moreover, we obtain sup R n u ( · , t ) = 0from the relation εI ε ( t ) = sup R n u ε ( · , t )together with locally uniform convergence of u ε . (cid:3) Uniqueness for a certain birth rate
In this section, we deal with uniqueness of a pair ( u , I ) where u ( x, t ) is a boundeduniformly continuous viscosity solution of equation (1.1). We provide uniquenessresult when R ( x, I ) has certain structures. Here, we follow structural conditions in[13]. We have not been able to obtain full unconditional uniqueness result when R ( x, I ) is not decoupled. Theorem 4.1.
For the equation (1.1), there exists a unique viscosity solution u ( x, t ) and nonnegative increasing function I ( t ) when the birth R ( x, I ) can be written aseither R ( x, I ( t )) = b ( x ) − d ( x ) Q ( I ) , with Q ( I ) > increasing,or R ( x, I ( t )) = b ( x ) Q ( I ) − d ( x ) , with Q ( I ) > decreasing,where b ( x ) , d ( x ) ∈ W , ∞ ( R n ) and there exists b m > such that b ( x ) > b m .Proof. We only need to deal with the first case as handling the second case issimilar. Again, we follow the argument by B. Perthame and G. Barles in [13]. Letus assume that there are two viscosity solutions u and u corresponding to I ( t )and I ( t ) respectively. In other words, u ′ i s satisfy ( u i ) t = H ( Du i ) + R ( x, I i ( t )) in R n × [0 , T ] , sup R n u i ( · , t ) = 0 on [0 , T ] ,I (0) = 0 ,u i ( x,
0) = u ( x ) on R n , for i = 1 , i = u i ( x, t ) − b ( x )Σ i ( t )Σ i ( t ) = Z t Q ( I i ( s )) ds and they satisfy (Ψ i ) t = − d ( x ) + H ( D (Ψ i + b ( x )Σ i ( t )))for i = 1 , ddt k Ψ − Ψ ( · , t ) k L ∞ ( R n ) ≤ C | Σ ( t ) − Σ ( t ) | (4.1)in viscosity sense for a positive C .Noting that sup x ∈ R n u ( x, t ) = 0, for any δ > y ∈ R n suchthat − δ ≤ u ( y, t ) ≤ . ELL-POSEDNESS FOR CONSTRAINED HAMILTON-JACOBI EQUATIONS 15
By the following inequalities, − δ ≤ u ( y, t ) − sup R n u ( · , t ) ≤ u ( y, t ) − u ( y, t )= b ( y )[Σ ( t ) − Σ ( t )] + Ψ ( y, t ) − Ψ ( y, t ) ≤ b ( y )[Σ ( t ) − Σ ( t )] + sup R n [Ψ ( · , t ) − Ψ ( · , t )] , we get b ( y )[Σ ( t ) − Σ ( t )] ≤ sup R n [Ψ ( · , t ) − Ψ ( · , t )] + δ. Here we may assume Σ ( t ) − Σ ( t ) is positive. Then b m | Σ ( t ) − Σ ( t ) | = b m [Σ ( t ) − Σ ( t )] ≤ b ( y )[Σ ( t ) − Σ ( t )] ≤ sup x ∈ R n | Ψ ( · , t ) − Ψ ( · , t ) | + δ. Since δ > b m | Σ ( t ) − Σ ( t ) | ≤ sup R n | Ψ ( · , t ) − Ψ ( · , t ) | by switching the role of Σ and Σ if necessary. Combining with (4.1) yields ddt k (Ψ − Ψ )( · , t ) k L ∞ ( R n ) ≤ C k (Ψ − Ψ )( · , t ) k L ∞ ( R n ) . Consequently, uniqueness follows by Gronwall’s inequality. (cid:3) Nonuniqueness result If R ( x, I ) is not strictly decreasing in I , we can give an example of nonuniqueness. Theorem 5.1.
Let I ( t ) be a nondecreasing continuous function such that I (0) = 0 and I ( t ) ≥ . Assume R ( x, I ) is defined as ( R ( x, I ) = 0 | x | ≥ ,R ( x, I ) = (1 − | x | )(1 − I ( t )) | x | ≤ , and u ( x ) satisfies | x | ≥ , − ( x − ) ≤ x ≤ , − ( x +22 ) − ≤ x ≤ − , x − | x | ≤ . Then, for
T > small enough, there exist infinitely many viscosity solutions to ( u t = | Du | + R ( x, I ( t )) in R n × [0 , T ] ,u ( x,
0) = u ( x ) on R n , (5.1) satisfying sup R n u ( · , t ) = 0 on [0 , T ] Proof.
Let I ( t ) be a continuous nondecreasing function such that I (0) = 0 and c > c ≥ − I ( t ). Also we denote a unique viscosity solution of (5.1) by u ( x, t ).Now let us define v c ( x, t ) as v c ( x, t ) = | x | ≥ , − ( x − ) ct ≤ x ≤ , − + ct | x | ≤ ct, − ( x +22 ) − ≤ x ≤ − − ct, and w c ( x, t ) as w c ( x, t ) = | x | ≥ , − ( x − ) − √ − ct ≤ x ≤ , − − ct | x | ≤ − √ − ct, − ( x +22 ) − ≤ x ≤ − (2 − √
3) + ct.
It is straightforward to check w c is a viscosity subsolution to (5.1) since there isno test function touching from above at | x | = 1 + ct . On the other hand, as(5.1) is a concave Hamilton-Jacobi equation and a Lipschitz continuous function v c solves (5.1) almost everywhere sence. By Proposition 7.25 in [11], v c is a viscositysupersolution. Therefore, we have w c ( x, ≤ u ( x, ≤ v c ( x, . It follows that w c ( x, t ) ≤ u ( x, t ) ≤ v c ( x, t )by comparison principle. Moreover, for T > R n v c ( · , t ) = sup R n w c ( · , t ) = 0 for 0 ≤ t ≤ T. Therefore, u satisfies the constraint condition (5.1). As we can repeat that processfor any choice of I ( t ) and c , infinitely many solution pairs ( u, I ) are generated. (cid:3) Conclusion
We presented a new way of building a viscosity solution of Hamilton-Jacobi equa-tion with an unknown function I ( t ) and a supremum constraint via a fixed pointargument. We also provided that when R ( x, I ) is separable in x and t , the solutionis unique for any nonnegative, locally Lipschitz continuous Hamiltonian H ( p ) satis-fying H (0) = 0. Here, we do not need convexity of the Hamiltonian. On the otherhand, many solutions can be generated when the reaction R ( x, I ) fails to strictlydecreasing with respect to resource, I . Up to now, uniqueness of a solution pair( u, I ) corresponding to a general reaction R ( x, I ) is still open. In the recent work byMirrahimi and Roquejoffre [12], it was proved that the solution is unique under re-strictive assumptions that Hamiltonian and initial condition are uniformly concaveand satisfy some further structural assumptions using optimal control formulation.We plan to investigate this matter in the near future. ELL-POSEDNESS FOR CONSTRAINED HAMILTON-JACOBI EQUATIONS 17
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