Well Posedness and Control in a NonLocal SIR Model
aa r X i v : . [ m a t h . A P ] O c t Well Posedness and Control in a NonLocal SIR Model
Rinaldo M. Colombo Mauro Garavello October 17, 2019
Abstract
SIR models, also with age structure, can be used to describe the evolution of an infectivedisease. A vaccination campaign influences this dynamics immunizing part of the susceptibleindividuals, essentially turning them into recovered individuals. We assume that vaccina-tions are dosed at prescribed times or ages which introduce discontinuities in the evolutionsof the S and R populations. It is then natural to seek the “best” vaccination strategiesin terms of costs and/or effectiveness. This paper provides the basic well posedness andstability results on the SIR model with vaccination campaigns, thus ensuring the existenceof optimal dosing strategies.
Keywords:
Vaccination; Optimal Control of Balance Laws; Control in Age-StructuredPopulations Models.
Aim of this paper is to provide a rigorous analytic environment where different vaccinationstrategies can be described, tested and optimized.Our starting point is the following age–structured
Susceptible – Infected – Recovered ( SIR )model, which originated in [10], see also [9, Chapter 6], [14, Chapter 19] or [15, § ∂ t S + ∂ a S = − d S ( t, a ) S − R + ∞ λ ( a, a ′ ) I ( t, a ′ ) d a ′ S∂ t I + ∂ a I = − d I ( t, a ) I + R + ∞ λ ( a, a ′ ) I ( t, a ′ ) d a ′ S − r I ( t, a ) I∂ t R + ∂ a R = − d R ( t, a ) R | {z } | {z } + r I ( t, a ) I | {z } . mortality disease transmission recovery (1.1)As usual, S = S ( t, a ) is the density of individuals at time t of age a susceptible to the disease; I = I ( t, a ) is the density of infected individuals at time t and of age a and the density of individualsthat can not be infected by the disease is R = R ( t, a ), comprising individuals that recoveredfrom the disease as well as those that are immune. The death rates of the three portions of thepopulations are d S , d I and d R . Above, λ ( a, a ′ ) quantifies the susceptible individuals of age a thatare infected by individuals of age a ′ . Thus, the nonlocal term R + ∞ λ ( a, a ′ ) I ( t, a ′ ) d a ′ S ( t, a ) inthe former two right hand sides represents the total number of susceptible individuals of age a INdAM Unit, University of Brescia. [email protected] Department of Mathematics and its Applications, University of Milano - Bicocca. [email protected] t . Finally, r I ( t, a ) is the fraction of infected individuals of age a that recover at time t , independently from the vaccination campaign.We now introduce a vaccination campaign in (1.1). To this aim, differently from variouspaper in the literature, e.g. [7, 8, 12, 13, 18, 19], we do not introduce any source term in theright hand sides of (1.1). We consider two different approaches.In a first policy, vaccinations are dosed at a, possibly time dependent, percentage of thepopulation of the prescribed ages ¯ a , ¯ a , . . . , ¯ a N , with ¯ a j − < ¯ a j . Call η j ( t ), with η j ( t ) ∈ [0 , S population of age ¯ a j that is dosed a vaccine at time t . Then, assuming thatvaccination has an immediate effect, the evolution described by (1.1) has to be supplementedby the vaccination conditions S ( t, ¯ a j +) = (cid:0) − η j ( t ) (cid:1) S ( t, ¯ a j − ) [ ∀ t , S ( t, ¯ a j ) decreases due to vaccination] I ( t, ¯ a j +) = I ( t, ¯ a j − ) [the infected population is unaltered] R ( t, ¯ a j +) = R ( t, ¯ a j − ) + η j ( t ) S ( t, ¯ a j − ) [vaccinated individuals are immunized] (1.2)for a.e. t > j ∈ { , · · · , N } . Whenever vaccinations can be dosed only tosusceptible individuals, the total cost of the vaccination campaign (1.2) at all ages ¯ a , . . . , ¯ a N is proportional to the total number of vaccinations dosed, say C ( η ) = N X i =1 Z I η i ( t ) S ( t, ¯ a i − ) d t , (1.3) I being the time interval under consideration and S depending on η through (1.2). However, itis reasonable to consider also the case of vaccinations dosed to the η j ( t ) portion of the whole population at time t , that is also to infected and immune individuals, in which case (1.3) hasto be substituted by C ( η ) = N X i =1 Z I η i ( t ) (cid:0) S ( t, ¯ a i − ) + I ( t, ¯ a i − ) + R ( t, ¯ a i − ) (cid:1) d t , (1.4)where S , I and R depend on η through (1.2). Indeed, not always individuals belonging to the R or even I population can be easily distinguished from those in the S population. Remarkthat in both cases (1.3) and (1.4), the dynamics of the disease is described by (1.1)–(1.2), sincevaccination is assumed to have no effects on R or I individuals.Alternatively, in a second policy, a vaccination campaign may aim at immunizing an agedependent portion, say ν ( a ) , . . . , ν N ( a ), of the S population at given times ¯ t , . . . , ¯ t N . Thisamounts to substitute (1.2) with S (¯ t k + , a ) = (cid:0) − ν k ( a ) (cid:1) S (¯ t k − , a ) [ ∀ a , S (¯ t k , a ) decreases due to vaccination] I (¯ t k + , a ) = I (¯ t k − , a ) [the infected population is unaltered] R (¯ t k + , a ) = R (¯ t k − , a ) + ν k ( a ) S (¯ t k − , a ) [vaccinated individuals are immunized]. (1.5)Now, a reasonable cost due to this campaign is C ( ν ) = N X k =1 Z R + ν k ( a ) S (¯ t k − , a ) d a (1.6)2henever vaccination is dosed only to susceptible individuals. On the other hand, vaccinationscan be dosed to all individuals, in which case we replace (1.6) with C ( ν ) = N X k =1 Z R + ν k ( a ) (cid:0) S (¯ t k − , a ) + I (¯ t k − , a ) + R (¯ t k − , a ) (cid:1) d a . (1.7)As above, in both cases (1.6) and (1.7), the dynamics of the disease is described by (1.1)–(1.5),since vaccination is assumed to have no effects on R or I individuals.The most natural way to evaluate the effect of a vaccination campaign is to compute the,possibly weighted, number of infected individuals, namely E = Z I Z R + ϕ ( t, a ) I ( t, a ) d a d t , (1.8) E being a function of η in case (1.2) and a function of ν in case (1.5). The dependence ofthe weight ϕ on time t may account for a possible targeting a decrease in the total numberof infected individuals after an initial period, while the dependence of ϕ on a may account fordifferent degrees of danger of the disease at the different ages.Once the cost C and the effect E are selected, we are left with two modeling choices: “Theoptimization problem in this framework is to find the strategy with minimal costs at a given levelfor the effect or to find the strategy with the best effect at given costs.” , from [13, Introduction].In more formal terms, we are lead to tackle the problemsminimize C subject to E ≤ E ∗ or minimize E subject to C ≤ C ∗ (1.9)for assigned positive E ∗ and C ∗ , with time dependent controls η i in cases (1.3) or (1.4), or elsewith age dependent controls ν k in cases (1.6) or (1.7). The analytic results presented belowprovide a framework, consisting of well posedness results and stability estimates, where theseproblems can be rigorously addressed, see [6] for soem specific examples.The current literature offers a variety of alternative approaches to similar modeling situa-tions. For instance, in the recent [7], the vaccination control enters an equation for S similar tothat in (1.1) through a term − uS in the right hand side, meaning that vaccination takes placeuniformly at all ages. A similar approach is followed also in [12, 13].From the analytic point of view, below we prove well posedness and stability for (1.1)–(1.2)and for (1.1)–(1.5) which, in turn, ensure the existence of optimal vaccination strategies. Toachieve this, we prove well posedness and stability of a more general IBVP, see (3.1).The next section presents solutions to problems (1.9), as a consequence of the analyticframework developed in Section 3. All analytic proofs are collected in Section 4. Denote by I the time interval [0 , T ], for a positive T , or [0 , + ∞ [.Throughout, we supplement (1.1) with the initial and boundary conditions ( S ( t,
0) = S b ( t ) , I ( t,
0) = I b ( t ) , R ( t,
0) = R b ( t ) , t ∈ I ,S (0 , a ) = S o ( a ) , I (0 , a ) = I o ( a ) , R (0 , a ) = R o ( a ) , a ∈ R + ; (2.1)3e require below the following assumptions on the functions defining (1.1)–(2.1) : ( λ ) λ ∈ C ( R + × R + ; R ) admits positive constants Λ ∞ , Λ L such that for all a , a , a ′ ∈ R + k λ k L ∞ ( R + × R + ; R ) + TV( λ ( · , a ′ ); R + ) ≤ Λ ∞ (2.2) (cid:12)(cid:12) λ ( a , a ′ ) − λ ( a , a ′ ) (cid:12)(cid:12) ≤ Λ L | a − a | . (2.3) (dr) The maps d S , d I , d R , r I : I × R + → R are Caratheodory functions, in the sense of Defini-tion 4.2, and there exist positive R L , R , R ∞ such that for ϕ = d S , d I , d R , r I , t ∈ I and a , a ∈ R + , k ϕ k L ∞ ( I × R + ; R ) + TV( ϕ ( t, · ); R + ) ≤ R ∞ (2.4) (cid:12)(cid:12) ϕ ( t, a ) − ϕ ( t, a ) (cid:12)(cid:12) ≤ R L | a − a | (2.5) k ϕ k C ( I ; L ( R + ; R )) ≤ R . (2.6) (IB) The initial and boundary data satisfy S o , I o , R o ∈ ( L ∩ BV )( R + ; R + ) and S b , I b , R b ∈ ( L ∩ BV )( I ; R + ) . (2.7)First, we provide the basic well posedness result for the model presented above, based on thenonlocal renewal equations (1.1), in the case of the vaccination policy (1.2). Theorem 2.1.
Under hypotheses ( λ ) and (dr) , for any initial and boundary data satis-fying (IB) , for any choice of the vaccination ages ¯ a , . . . , ¯ a N and of the control function η ∈ BV ( I ; [0 , N ) , problem (1.1) – (1.2) – (2.1) admits a unique solution ( S, I, R ) ∈ C , (cid:16) I ; L ( R + ; R ) (cid:17) with S ( t, a ) ≥ I ( t, a ) ≥ R ( t, a ) ≥ for all ( t, a ) ∈ I × R + , (2.8) depending Lipschitz continuously on the initial datum, through its L norm, and on η , throughits L ∞ norm. The proof, deferred to § Theorem 2.2.
Under hypotheses ( λ ) and (dr) , for any initial and boundary data satis-fying (IB) , for any choice of the vaccination times ¯ t , . . . , ¯ t N and of the control function ν ∈ BV ( I ; [0 , N ) , problem (1.1) – (1.5) – (2.1) admits a unique solution as in (2.8) , depend-ing Lipschitz continuously on the initial datum, through the L norm, and on ν , through the L ∞ norm. The proof is deferred to § C and E are easily shown to bestrongly continuous functions of the control η . The existence of an optimal strategy then followsthrough an application of Weierstraß Theorem, as soon as the choice of η or ν is restricted toa suitable strongly compact set. We refer to [6] for a selection of control problems based onTheorem 2.1 or Theorem 2.2. Throughout, we strive to have dimensionally correct expressions at the cost of distinguishing the variousconstants whenever they are dimensionally different. Analytic Results
The proofs of Theorem 2.1 and of Theorem 2.2 follow from a slightly more general statement.
Theorem 3.1.
Consider the following Initial – Boundary Value Problem (IBVP) ∂ t u i + ∂ x ( g i ( t, x ) u i ) = (cid:0) α i [ u ( t )]( x ) + γ i ( t, x ) (cid:1) · uu i (0 , x ) = u o,i ( x ) g i ( t, u i ( t, β i (cid:0) t, u ( t, ¯ x − ) , . . . , u n ( t, ¯ x n − ) (cid:1) i = 1 , . . . , n (3.1) where (IBVP.1) g , · · · , g n ∈ C , ( I × R + ; [ˇ g, ˆ g ]) and for all t ∈ I , x ∈ R + , i = 1 , . . . , n TV( g i ( · , x ); I ) + TV( g i ( t, · ); R + ) ≤ G ∞ (3.2)TV( ∂ x g i ( t, · ); R + ) + (cid:13)(cid:13) ∂ x g i ( t, · ) (cid:13)(cid:13) L ∞ ( R + ; R ) ≤ G . (3.3) (IBVP.2) α , . . . , α n : L ( R + ; R n ) → C ( R + ; R n ) are linear and continuous maps andthere exist positive constants A L and A such that (cid:13)(cid:13) α i [ w ] (cid:13)(cid:13) L ∞ ( R + ; R n ) + TV( α i [ w ]; R + ) ≤ A L k w k L ( R + ; R n ) (3.4) (cid:13)(cid:13) α i [ w ] (cid:13)(cid:13) L ( R + ; R n ) ≤ A k w k L ( R + ; R n ) (3.5) for every i ∈ { , · · · , n } , w ∈ L ( R + ; R n ) . Moreover, for every w ∈ ( L ∩ L ∞ )( R + ; R ) ,there exists a positive A such that for all x , x ∈ R + (cid:13)(cid:13) α i [ w ]( x ) − α i [ w ]( x ) (cid:13)(cid:13) ≤ A | x − x | . (3.6) (IBVP.3) γ , . . . , γ n ∈ C ( I ; L ( R + ; R n )) are Caratheodory functions, in the sense of Def-inition 4.2, and there exist positive constants C L , C ∞ such that (cid:13)(cid:13) γ i ( t, x ) − γ i ( t, x ) (cid:13)(cid:13) ≤ C L | x − x | (3.7) (cid:13)(cid:13) γ i ( t, · ) (cid:13)(cid:13) L ∞ ( R + ; R ) + TV( γ i ( t, · ); R + ) ≤ C ∞ (3.8) for every i ∈ { , · · · , n } , t ∈ I and x , x ∈ R + . (IBVP.4) β , . . . , β n are Caratheodory functions in the sense of Definition 4.2; for all t , β ( t ) , . . . , β n ( t ) ∈ C ( R n ; R ) and, for all ( t, u ) ∈ I × R n , ∂ u j β i ( t, u ) = 0 whenever j ≥ i .Moreover, there exist constants B , B ∞ and B L such that (cid:12)(cid:12) β i ( t, u ) − β i ( t, u ) (cid:12)(cid:12) ≤ B L k u − u k (3.9) (cid:13)(cid:13) β i ( · , (cid:13)(cid:13) L ( I ; R ) ≤ B (3.10) (cid:13)(cid:13) β i ( · , (cid:13)(cid:13) L ∞ ( I ; R ) ≤ B ∞ (3.11)TV( β i ( · , u ( · )); I ) ≤ B ∞ + B L TV( u ; I ) (3.12) for every i ∈ { , · · · , n } , t ∈ I u , u ∈ R n and u ∈ BV ( I ; R n ) . (IBVP.5) u o ∈ L ( R + ; R n ) . hen, there exists constants K and K ∞ , a positive time t ∗ and a constant L dependent only on t ∗ , k u o k L ( R + ; R n ) , TV( u o ; R + ) and on the parameters in (IBVP.1) – (IBVP.5) such that (3.1) admits a unique solution u ∗ ∈ C ([0 , t ∗ ]; L ( R + ; R n )) , satisfying, for all t, t ′ ∈ [0 , t ∗ ] , (cid:13)(cid:13) u ∗ ( t ) (cid:13)(cid:13) L ( R + ; R n ) ≤ K , (cid:13)(cid:13) u ∗ ( t ) − u ∗ ( t ′ ) (cid:13)(cid:13) L ( R + ; R n ) ≤ L (cid:12)(cid:12) t − t ′ (cid:12)(cid:12) , (cid:13)(cid:13) u ∗ ( t ) (cid:13)(cid:13) L ∞ ( R + ; R n ) ≤ K ∞ , TV (cid:0) u ∗ ( t ); R + (cid:1) ≤ K ∞ . (3.13) Moreover, if u ′∗ and u ′′∗ are the solutions to (3.1) corresponding to initial data u ′ o and u ′′ o and toboundary data β ′ and β ′′ , then the following estimate holds for all t ∈ [0 , t ∗ ] : (cid:13)(cid:13) u ′∗ ( t ) − u ′′∗ ( t ) (cid:13)(cid:13) L ( R + ; R n ) ≤ L (cid:16)(cid:13)(cid:13) u ′ o − u ′′ o (cid:13)(cid:13) L ( R + ; R n ) + (cid:13)(cid:13) β ′ − β ′′ (cid:13)(cid:13) L ∞ ([0 ,t ∗ ] × [0 ,K ∞ ] n ; R n ) (cid:17) . (3.14)The proof is detailed in § K and K ∞ are in (4.29). Above, solutions to (3.1) are intended essentially in the sense of Definition 4.3: note indeed that foreach i = 1 , . . . , n , problem (3.1) fits into (4.8), refer to (4.32) for the details. Remark 3.2.
The assumptions in Theorem 3.1 do not rule out a finite time blow-up in u . Infact, consider the IBVP ∂ t u + ∂ x u = R R + u ( t, ξ ) d ξ uu (0 , x ) = e − x u ( t,
0) = 0 that clearly fits into (IBVP.1) – (IBVP.5) with α [ w ] = R R + w ( ξ ) d ξ and I = R + . Its (strong)solution for ( t, x ) ∈ [0 , ln 2[ × R + is u ( t, x ) = e t − x − e t , which blows up as t → ln 2 . Motivated by Remark 3.2, we now strengthen the assumptions in Theorem 3.1 to ensuretwo properties of key interest in the vaccination model (1.1)–(2.1), namely that the solution u attains positive values and that it is defined on all I . Corollary 3.3.
Assume that, besides all assumptions (IBVP.1) – (IBVP.5) in Theorem 3.1,we also have that (POS) For all i , u oi ≥ and β i ≥ . (NEG) For all u ∈ L ( R + ; ( R + ) n ) , t ∈ I and x ∈ R + , P ni =1 (cid:0) α i [ u ]( x ) + γ i ( t, x ) (cid:1) · u ( x ) ≤ . (EQ) For all i, j = 1 , . . . , n , t ∈ I and x ∈ R + , g i ( t, x ) = g j ( t, x ) .Then, each component of the solution u ∗ constructed in Theorem 3.1 attains non negative values.Moreover, u ∗ can be uniquely extended to all I . The proof is deferred to § Let J denote a (non empty) real interval. We use throughout the norms k f k C ( J ; L ( R + ; R )) = sup t ∈ J k f k L ( R + ; R ) ; k f k L ( R + ; R ) = R R + (cid:12)(cid:12) f ( x ) (cid:12)(cid:12) d x ; k f k C ( J ; L ∞ ( R + ; R )) = sup t ∈ J k f k L ∞ ( R + ; R ) ; k f k L ∞ ( R + ; R ) = ess sup x ∈ R + (cid:12)(cid:12) f ( x ) (cid:12)(cid:12) ; k f k L ( R + ; R n ) = P ni =1 k f i k L ( R + ; R ) . (4.1)6 .1 Preliminary Properties of BV Functions Recall the following elementary estimates on BV functions, see also [5, Section 4] or [1]: u ∈ BV ( R + ; R ) w ∈ BV ( R + ; R ) ) ⇒ TV( u w ) ≤ TV( u ) k w k L ∞ ( R + ; R ) + k u k L ∞ ( R + ; R ) TV( w ) ; (4.2) ϕ ∈ C , ( R n ; R ) u ∈ BV ( R + ; R n ) ) ⇒ TV( ϕ ◦ u ) ≤ Lip ( ϕ ) TV( u ) ; (4.3) u ∈ BV ( R + ; R ) w ∈ BV ( R + ; R ) w ( x ) ≥ ˇ w > ⇒ TV (cid:18) uw (cid:19) ≤ w TV( u ) + 1ˇ w TV( w ) k u k L ∞ ( R + ; R ) ; (4.4) u ∈ L ( J ; L ( R + ; R )) u ( t ) ∈ BV ( R + ; R ) ) ⇒ TV Z t u ( τ, · ) d τ ! ≤ Z t TV (cid:0) u ( τ ) (cid:1) d τ ; (4.5) u ∈ BV ( R + ; R ) δ ∈ L ∞ ( R ; R + ) ) ⇒ Z R + (cid:12)(cid:12)(cid:12) u (cid:0) x + δ ( x ) (cid:1) − u ( x ) (cid:12)(cid:12)(cid:12) d x ≤ TV( u ) k δ k L ∞ ( R + ; R ) . (4.6)Inequality (4.2) follows from [1, Formula (3.10)]. The definition of total variation directlyimplies (4.3), (4.4) and (4.5). For a proof of (4.6) see for instance [3, Lemma 2.3]. We supplementthe estimates above with the following one. Lemma 4.1.
Let J = [0 , t ∗ ] , with t ∗ > . Assume that u ∈ L ∞ ( J × J ; R ) is such that sup τ ∈ J TV( u ( τ, · ); J ) < + ∞ . Then, setting U ( t ) = R t u ( τ, t ) d τ , TV( U ; J ) ≤ k u k L ∞ ( J × J ; R ) t ∗ + Z J TV u ( τ, · ) d τ . (4.7) Proof.
Fix t , t , . . . , t n in J with t i − < t i for i = 1 , . . . , n . Using [1, Theorem 3.27 and (3.24)], n X i =1 (cid:12)(cid:12) U ( t i ) − U ( t i − ) (cid:12)(cid:12) = n X i =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z t i u ( τ, t i ) d τ − Z t i − u ( τ, t i − ) d τ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = n X i =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z t i t i − u ( τ, t i ) d τ + Z t i − (cid:0) u ( τ, t i ) − u ( τ, t i − ) (cid:1) d τ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ n X i =1 Z t i t i − (cid:12)(cid:12) u ( τ, t i ) (cid:12)(cid:12) d τ + n X i =1 Z t i − (cid:12)(cid:12) u ( τ, t i ) − u ( τ, t i − ) (cid:12)(cid:12) d τ ≤ n X i =1 Z t i t i − k u k L ∞ ( J × J ; R ) d τ + n X i =1 Z J (cid:12)(cid:12) u ( τ, t i ) − u ( τ, t i − ) (cid:12)(cid:12) d τ ≤ k u k L ∞ ( J × J ; R ) t ∗ + Z J TV u ( τ, · ) d τ , which completes the proof. 7 .2 A Scalar Renewal Equation We consider the following initial–boundary value problem for a linear scalar balance law, or renewal equation , see also [15, Chapter 3], of the form ∂ t u + ∂ x (cid:0) g ( t, x ) u (cid:1) + m ( t, x ) u = f ( t, x ) ( t, x ) ∈ J × R + ,u (0 , x ) = u o ( x ) x ∈ R + ,g ( t, u ( t, b ( t ) t ∈ J . (4.8)Let F , F ∞ , G , G ∞ , M, ˆ g, ˇ g be positive with ˇ g < ˆ g . We require the following conditions: (f ) f ∈ C (cid:0) J ; L ( R + ; R ) (cid:1) and for all t ∈ J ( (cid:13)(cid:13) f ( t, · ) (cid:13)(cid:13) L ( R + ; R ) ≤ F ; (cid:13)(cid:13) f ( t, · ) (cid:13)(cid:13) L ∞ ( R + ; R ) + TV( f ( t, · ); R + ) ≤ F ∞ . (g) g ∈ C , ( J × R + ; [ˇ g, ˆ g ]), for ( t, x ) ∈ J × R + ( TV( g ( t, · ); R + ) + TV( g ( · , x ); J ) ≤ G ∞ ; (cid:13)(cid:13) ∂ x g ( t, · ) (cid:13)(cid:13) L ∞ ( R + ; R ) + TV( ∂ x g ( t, · ); R + ) ≤ G . (m) m is a Caratheodory function and for all t ∈ J , (cid:13)(cid:13) m ( t, · ) (cid:13)(cid:13) L ∞ ( R + ; R ) + TV( m ( t, · ); R + ) ≤ M ; (b) b ∈ BV loc ( J ; R ).Above, we refer to the usual definition of Caratheodory function , namely:
Definition 4.2 ([4, (A) and (B) in § . The map m : J × R + → R m is a Caratheodoryfunction if1. For all x ∈ R + , the map m x : J → R m defined by m x ( t ) = m ( t, x ) is measurable.2. For a.e. t ∈ J , the map m t : R + → R m defined by m t ( x ) = m ( t, x ) is continuous.3. For all compact K ⊂ J × R + , there exist constants M ∞ , M L > such that for a.e. t ∈ J and for all x , x ∈ R + , (cid:13)(cid:13) m ( t, x ) (cid:13)(cid:13) ≤ M ∞ and (cid:13)(cid:13) m ( t, x ) − m ( t, x ) (cid:13)(cid:13) ≤ M L · | x − x | . Recall the following definition of solution to (4.8), see also [2, 3, 11, 15, 16, 17].
Definition 4.3.
Assume that (f ) , (g) , (m) and (b) hold. Choose an initial datum u o ∈ L ( R + ; R ) . The function u ∈ C (cid:0) J ; L ( R + ; R ) (cid:1) is a solution to (4.8) if1. for all ϕ ∈ C ( ˚ J × ˚ R + ; R ) , R R + R J (cid:2) u ∂ t ϕ + g u ∂ x ϕ + ( f − m u ) ϕ (cid:3) d t d x = 0 ;2. u (0 , x ) = u o ( x ) for a.e. x ∈ R + ;3. for a.e. t ∈ J , lim x → g ( t, x ) u ( t, x ) = b ( t ) . As shown below, problem (4.8) admits as unique solution in the sense of Definition 4.3 the map u ( t, x ) = u o ( X (0; t, x )) E (0 , t, x ) + Z t f (cid:0) τ, X ( τ ; t, x ) (cid:1) E ( τ, t, x ) d τ x > σ ( t ) b (cid:0) T (0; t, x ) (cid:1) g (cid:0) T (0; t, x ) , (cid:1) E ( T (0; t, x ) , t, x ) + Z tT (0; t,x ) f (cid:0) τ, X ( τ ; t, x ) (cid:1) E ( τ, t, x ) d τ x < σ ( t )(4.9)8here E ( τ, t, x ) = exp " − Z tτ (cid:16) m ( s, X ( s ; t, x )) + ∂ x g (cid:0) s, X ( s ; t, x ) (cid:1)(cid:17) d s (4.10)and, for t o , t ∈ J , x o , x ∈ R + , t → X ( t ; t o , x o ) solves ( ˙ x = g ( t, x ) x ( t o ) = x o and x → T ( x ; t o , x o ) solves t ′ = 1 g ( t, x ) t ( x o ) = t o (4.11)and we also set σ ( t ) = X ( t ; 0 , x ) = T ( x ; 0 , Lemma 4.4.
Let (g) and (m) hold. Then, E defined in (4.10) satisfies the following estimates,for x ∈ R + and τ, t ∈ J with τ ≤ t : E ( τ, t, x ) ≤ e ( G + M )( t − τ ) , (4.12)TV( E ( τ, t, · ); R + ) ≤ ( G + M ) ( t − τ ) e ( G + M )( t − τ ) , (4.13)TV (cid:0) E ( τ, · , x ); [0 , t ] (cid:1) ≤ ( G + M ) ( t − τ ) e ( G + M )( t − τ ) , (4.14)TV (cid:0) E ( · , t, x ); [0 , t ] (cid:1) ≤ ( G + M ) t e ( G + M ) t . (4.15) Proof.
The bound (4.12) directly follows from (g) , (m) , and (4.10). Consider the total variationestimate (4.13). For τ ≤ t , by (4.3) we haveTV (cid:16) E ( τ, t, · ); R + (cid:17) ≤ e ( G + M )( t − τ ) Z tτ (cid:18) TV (cid:16) m ( s, · ); R + (cid:17) + TV (cid:16) ∂ x g ( s, · ); R + (cid:17)(cid:19) d s which implies (4.13). Now, consider the total variation estimate (4.14). For τ ≤ t we deduceTV (cid:0) E ( τ, · , x ); [0 , t ] (cid:1) [by (4.3)] ≤ e ( G + M )( t − τ ) TV (cid:18)Z · τ h m (cid:0) s, X ( s ; · , x ) (cid:1) + ∂ x g (cid:0) s, X ( s ; · , x ) (cid:1)i d s (cid:19) [by (4.7)] ≤ e ( G + M )( t − τ ) (cid:20) k m k L ∞ ( [0 ,t ] × R + ; R ) + k ∂ x g k L ∞ ( [0 ,t ] × R + ; R ) (cid:21) ( t − τ )+ e ( G + M )( t − τ ) " sup s ∈ [0 ,t ] TV (cid:16) m (cid:0) s, X ( s ; · , x ) (cid:1) + ∂ x g (cid:0) s, X ( s ; · , x ) (cid:1)(cid:17) ( t − τ )and using (g) and (m) we deduce (4.14). Finally, consider the estimate (4.15). We haveTV (cid:0) E ( · , t, x ); [0 , t ] (cid:1) [by (4.3)] ≤ e ( G + M ) t TV Z t · h m (cid:0) s, X ( s ; t, x ) (cid:1) + ∂ x g (cid:0) s, X ( s ; t, x ) (cid:1)i d s ! [by the definition of TV] ≤ e ( G + M ) t (cid:20) k m k L ∞ ( [0 ,t ] × R + ; R ) + k ∂ x g k L ∞ ( [0 ,t ] × R + ; R ) (cid:21) t, concluding the proof.The following Lemma summarizes various properties of the solution to (4.8), see also [15].9 emma 4.5. Let (f ) , (g) and (m) hold. Then, with reference to the scalar problem (4.8) , (SP.1) For any u o ∈ ( L ∩ BV )( R + ; R ) and for any b satisfying (b) , the map u : J × R + → R defined by (4.9) solves (4.8) in the sense of Definition 4.3. (SP.2) For every t ∈ J , the following a priori estimates hold: sup τ ∈ [0 ,t ] (cid:13)(cid:13) u ( τ ) (cid:13)(cid:13) L ∞ ( R + ; R ) ≤ (cid:18) k u o k L ∞ ( R + ; R ) + 1ˇ g k b k L ∞ ([0 ,t ]; R ) + F ∞ t (cid:19) e ( G + M ) t , sup τ ∈ [0 ,t ] (cid:13)(cid:13) u ( τ ) (cid:13)(cid:13) L ( R + ; R ) ≤ (cid:16) k u o k L ( R + ; R ) + k b k L ([0 ,t ]; R ) + F t (cid:17) e Mt . (SP.3) For every t ∈ J , the following total variation estimate holds TV( u ( t ); R + ) ≤ H ( t ) F ∞ t + k b k L ∞ ([0 ,t ]; R ) + TV( b ; [0 , t ])ˇ g + k u o k L ∞ ( R + ; R ) + TV( u o ; R + ) ! (4.16) where H ( t ) is a non decreasing continuous function of t , depending also on ˇ g, G , G ∞ and M , satisfying H (0) ≤ G ∞ / ˇ g . (SP.4) Fix t ∈ J and x ∈ R + . If x > σ ( t ) , then TV (cid:0) u ( · , x ); [0 , t ] (cid:1) ≤ (cid:20) TV( u o ; R + ) + 2( G + M ) k u o k L ∞ ( R + ; R ) t (cid:21) e ( G + M ) t + 4 (cid:2) G + M ) t (cid:3) F ∞ t e ( G + M ) t . (4.17) If x < σ ( t ) , then TV (cid:0) u ( · , x ); [0 , t ] (cid:1) ≤ (cid:20) TV( u o ; R + ) + 1ˇ g TV (cid:0) b ( · ); [0 , t ] (cid:1)(cid:21) e ( G + M ) t + 2 (cid:2) G + M ) t (cid:3) k u o k L ∞ ( R + ; R ) e ( G + M ) t + 1ˇ g (cid:20) G + M ) t + G ∞ ˇ g (cid:21) k b k L ∞ ( [0 ,t ]; R ) e ( G + M ) t + 2 (cid:0) G + M ) t (cid:1) F ∞ t e ( G + M ) t . (4.18) (SP.5) Fix a positive W . For any w ∈ ( C ∩ BV )( J ; [ − W, W ]) , TV (cid:18)Z R + w ( · , x ) u ( · , x ) d x ; [0 , t ] (cid:19) ≤ k u k L ∞ ([0 ,t ] × R + ; R ) Z R + TV (cid:0) w ( · , x ); [0 , t ] (cid:1) d x + W Z R + TV (cid:0) u ( · , x ); [0 , t ] (cid:1) d x (SP.6) For every t ∈ J , there exists a positive L dependent on k u o k L ( R + ; R ) and on theconstants in (f ) , (g) , (m) and (b) , such that, for t ′ , t ′′ ∈ [0 , t ] , (cid:13)(cid:13) u ( t ′ ) − u ( t ′′ ) (cid:13)(cid:13) L ( R + ; R ) ≤ L (cid:12)(cid:12) t ′′ − t ′ (cid:12)(cid:12) . (4.19) (SP.7) If u o ≥ , f ≥ and b ≥ , then u ( t ) ≥ for all t . emark 4.6. The boundedness of the space variation of f required in (f ) is necessary. Indeed,consider Problem (4.8) with g ( t, x ) = 1 , m ( t, x ) = 0 , f ( t, x ) = sin 1 x − t , u o ( x ) = 0 and b ( t ) = 0 .The solution is u ( t, x ) = t sin 1 x − t which has unbounded total variation in space for all t > . Proof of Lemma 4.5.
We prove the different items separately. (SP.1):
A standard integration along characteristics is sufficient to prove it. (SP.2):
These bounds are an immediate consequence of (f ) , (g) , (m) and (4.9). (SP.3): We clearly haveTV (cid:0) u ( t ) (cid:1) = TV (cid:16) u ( t, · ) , (cid:2) , σ ( t ) (cid:2)(cid:17) + (cid:12)(cid:12)(cid:12) u (cid:0) t, σ ( t )+ (cid:1) − u (cid:0) t, σ ( t ) − (cid:1)(cid:12)(cid:12)(cid:12) + TV (cid:16) u ( t, · ) , (cid:3) σ ( t ) , + ∞ (cid:2)(cid:17) (4.20)and we estimate the three terms in the right hand side of (4.20) separately. Begin with the firstone, using the second expression in (4.9):TV (cid:16) u ( t, · ) , (cid:2) , σ ( t ) (cid:2)(cid:17) ≤ TV (cid:18) b ( · ) g ( · ,
0) ; [0 , t ] (cid:19) e ( G + M ) t + ( G + M ) t k b k L ∞ ([0 ,t ]; R ) ˇ g e ( G + M ) t + Z t TV (cid:16) f (cid:0) τ, X ( τ ; t, · ) (cid:1) E ( τ, t, · ) (cid:17) d τ + k f k L ∞ ([0 ,t ] × R + ; R ) e ( G + M ) t TV (cid:0) T (0; t, · ) (cid:1) ≤ g TV( b ; [0 , t ]) + 1ˇ g G ∞ k b k L ∞ ([0 ,t ]; R ) + ( G + M ) t k b k L ∞ ([0 ,t ]; R ) ˇ g ! e ( G + M ) t +( G + M ) F ∞ t e ( G + M ) t + sup t ∈ J TV( f ( t, · )) t e ( G + M ) t + F ∞ t e ( G + M ) t ≤ g TV( b ; [0 , t ]) + (cid:18) G ∞ ˇ g + ( G + M ) t (cid:19) k b k L ∞ ([0 ,t ]; R ) ! e ( G + M ) t + (cid:0) ( G + M ) t + 2 (cid:1) F ∞ t e ( G + M ) t . Concerning the second term in (4.20), the following rough estimate is sufficient for later use: (cid:12)(cid:12) u ( t, σ ( t )+) − u ( t, σ ( t ) − ) (cid:12)(cid:12) ≤ (cid:13)(cid:13) u ( t ) (cid:13)(cid:13) L ∞ ( R + ; R ) [By (SP.2) ] ≤ (cid:18) k u o k L ∞ ( R + ; R ) + 1ˇ g k b k L ∞ ([0 ,t ]; R ) + F ∞ t (cid:19) e ( G + M ) t . The latter term in (4.20) readsTV (cid:16) u ( t, · ) , (cid:3) σ ( t ) , + ∞ (cid:2)(cid:17) ≤ ( G + M ) k u o k L ∞ ( R + ; R ) t e ( G + M ) t + TV( u o ) e ( G + M ) t + Z t TV (cid:16) f (cid:0) τ, X ( τ ; t, · ) (cid:1) E ( τ, t, · ) (cid:17) d τ (cid:16) ( G + M ) k u o k L ∞ ( R + ; R ) t + TV( u o ) (cid:17) e ( G + M ) t +( G + M ) k f k L ∞ ([0 ,t ] × R + ; R ) t e ( G + M ) t + sup t ∈ J TV( f ( t, · )) t e ( G + M ) t ≤ (cid:18) ( G + M ) (cid:16) k u o k L ∞ ( R + ; R ) + F ∞ t (cid:17) t + TV( u o ) + F ∞ t (cid:19) e ( G + M ) t . Using now (4.20),TV (cid:16) u ( t, · ); R + (cid:17) ≤ (cid:20) g TV( b ; [0 , t ]) + 1ˇ g (cid:18) G ∞ ˇ g + ( G + M ) t (cid:19) k b k L ∞ ([0 ,t ]; R ) + (cid:0) ( G + M ) t + 2 (cid:1) F ∞ t + 2 (cid:18) k u o k L ∞ ( R + ; R ) + 1ˇ g k b k L ∞ ([0 ,t ]; R ) + F ∞ t (cid:19) +( G + M ) t (cid:16) k u o k L ∞ ( R + ; R ) + F ∞ t (cid:17) + TV( u o ) + F ∞ t (cid:21) e ( G + M ) t ≤ (cid:20) g (cid:18) G ∞ ˇ g + ( G + M ) t (cid:19) k b k L ∞ ([0 ,t ]; R ) + 1ˇ g TV( b ; [0 , t ]) + (cid:0) ( G + M ) t + 5 (cid:1) F ∞ t + (cid:0) G + M ) t (cid:1) k u o k L ∞ ( R + ; R ) + TV( u o ) (cid:21) e ( G + M ) t ≤ H ( t ) (cid:18) F ∞ t + 1ˇ g k b k L ∞ ([0 ,t ]; R ) + 1ˇ g TV( b ; [0 , t ]) + k u o k L ∞ ( R + ; R ) + TV( u o ) (cid:19) we prove (4.16). (SP.4): First, in view of an application of Lemma 4.1, computeTV (cid:16) f (cid:0) τ, X ( τ ; · , x ) (cid:1) E ( τ, · , x ); [0 , t ] (cid:17) [Use (4.2)] ≤ TV (cid:16) f (cid:0) τ, X ( τ ; · , x ) (cid:1) ; [0 , t ] (cid:17) (cid:13)(cid:13) E ( τ, · , x ) (cid:13)(cid:13) L ∞ ([0 ,t ]; R ) [Use (4.3) and (4.12)]+ (cid:13)(cid:13)(cid:13) f (cid:0) τ, X ( τ ; · , x ) (cid:1)(cid:13)(cid:13)(cid:13) L ∞ ([0 ,t ]; R ) TV (cid:0) E ( τ, · , x ); [0 , t ] (cid:1) [Use (4.14)] ≤ e ( G + M ) t TV (cid:16) f ( τ, · ); R + (cid:17) + 2( G + M ) t e ( G + M ) t k f k L ∞ ([0 ,t ] × R ; R ) . Thus, using (f ) , we deduce thatTV (cid:16) f (cid:0) τ, X ( τ ; · , x ) (cid:1) E ( τ, · , x ); [0 , t ] (cid:17) ≤ (cid:0) G + M ) t (cid:1) F ∞ e ( G + M ) t . (4.21)First consider the simple case x > σ ( t ); we haveTV (cid:0) u ( · , x ); [0 , t ] (cid:1) [Use (4.9)] ≤ TV (cid:16) u o (cid:0) X (0; · , x ) (cid:1) E (0 , · , x ); [0 , t ] (cid:17) [Use (4.2)]+ TV (cid:18)Z · f (cid:0) τ, X ( τ ; · , x ) (cid:1) E ( τ, · , x ) d τ ; [0 , t ] (cid:19) [use Lemma 4.1]12 TV (cid:16) u o (cid:0) X (0; · , x ) (cid:1) ; [0 , t ] (cid:17) (cid:13)(cid:13) E (0 , · , x ) (cid:13)(cid:13) L ∞ ([0 ,t ]; R + ) [Use (4.12)]+ (cid:13)(cid:13)(cid:13) u o (cid:0) X (0; · , x ) (cid:1)(cid:13)(cid:13)(cid:13) L ∞ ([0 ,t ]; R + ) TV (cid:0) E (0 , · , x ); [0 , t ] (cid:1) [Use (4.14)]+ 2 sup τ ∈ [0 ,t ] TV (cid:16) f (cid:0) τ, X ( τ ; · , x ) (cid:1) E ( τ, · , x ); [0 , t ] (cid:17) t [Use (4.21)]+ 2 k f k L ∞ ( [0 ,t ] × R + ; R ) kEk L ∞ ( [0 ,t ] × R + ; R ) t [Use (f ) and (4.12)] ≤ TV (cid:16) u o ; R + (cid:17) e ( G + M ) t + 2( G + M ) k u o k L ∞ ( R + ; R ) t e ( G + M ) t + 2 (cid:0) G + M ) t (cid:1) F ∞ t e ( G + M ) t + 2 F ∞ t e ( G + M ) t . Now consider the case x < σ ( t ), i.e. Σ( x ) < t . We clearly haveTV (cid:0) u ( · , x ); [0 , t ] (cid:1) ≤ TV (cid:0) u ( · , x ); [0 , Σ( x )] (cid:1) + TV (cid:0) u ( · , x ); [Σ( x ) , t ] (cid:1) + 2 k u k L ∞ ( [0 ,t ] × R + ; R ) . (4.22)Let us estimate the second term TV (cid:0) u ( · , x ); [Σ( x ) , t ] (cid:1) .TV (cid:0) u ( · , x ); [Σ( x ) , t ] (cid:1) [Use (4.9)] ≤ TV b (cid:0) T (0; · , x ) (cid:1) g (cid:0) T (0; · , x ) , (cid:1) E ( T (0; · , x ) , · , x ); [Σ( x ) , t ] ! [Use (4.2), (4.4)]+ TV Z · T (0; · ,x ) f (cid:0) τ, X ( τ ; · , x ) (cid:1) E ( τ, · , x ) d τ ; [Σ( x ) , t ] ! [Use Lemma 4.1] ≤ g TV (cid:16) b (cid:0) T (0; · , x ) (cid:1) ; [Σ( x ) , t ] (cid:17) (cid:13)(cid:13) E ( T (0; · , x ) , · , x ) (cid:13)(cid:13) L ∞ ([0 ,t ]; R ) [Use (4.12)]+ 1ˇ g TV( g ( T (0; · , x ) , x ) , t ]) (cid:13)(cid:13) b ( T (0; · , x )) E ( T (0; · , x ) , · , x ) (cid:13)(cid:13) L ∞ ([0 ,t ]; R ) [Use (g) , (4.12)]+ 1ˇ g (cid:13)(cid:13) b ( T (0; · , x )) (cid:13)(cid:13) L ∞ ([0 ,t ]; R ) TV (cid:0) E ( T (0; · , x ) , · , x ); [Σ( x ) , t ] (cid:1) [Use (4.14), (4.15)]+ 4 (cid:0) t − Σ( x ) (cid:1) sup τ ∈ [Σ( x ) ,t ] TV (cid:16) f (cid:0) τ, X ( τ ; · , x ) (cid:1) E ( τ, · , x ); [Σ( x ) , t ] (cid:17) [Use (4.21)]+ 4 (cid:0) t − Σ( x ) (cid:1) (cid:13)(cid:13)(cid:13) f (cid:0) · , X ( · ; · , x ) (cid:1) E ( · , · , x ) (cid:13)(cid:13)(cid:13) L ∞ ( [Σ( x ) ,t ] ; R ) [Use (f ) , (4.12)] ≤ g " TV (cid:0) b ( · ) ; [0 , t ] (cid:1) + (cid:18) G + M ) t + G ∞ ˇ g (cid:19) k b k L ∞ ( [0 ,t ]; R ) e ( G + M ) t + 4 (cid:0) t − Σ( x ) (cid:1) (cid:2) t ( G + M ) (cid:3) F ∞ e ( G + M ) t . Therefore, using (SP.2) , we deduce thatTV (cid:0) u ( · , x ); [0 , t ] (cid:1) ≤ (cid:20) TV (cid:16) u o ; R + (cid:17) + 1ˇ g TV (cid:0) b ( · ); [0 , t ] (cid:1)(cid:21) e ( G + M ) t + 2 (cid:2) G + M ) t (cid:3) k u o k L ∞ ( R + ; R ) e ( G + M ) t + 1ˇ g (cid:20) G + M ) t + G ∞ ˇ g (cid:21) k b k L ∞ ( [0 ,t ]; R ) e ( G + M ) t + 2 (cid:0) G + M ) t (cid:1) F ∞ t e ( G + M ) t . This completes the proof of (SP.4) . 13
SP.5):
Using the already obtained estimates, we have:TV (cid:18)Z J w ( · , x ) u ( · , x ) d x ; [0 , t ] (cid:19) ≤ Z J TV (cid:0) w ( · , x ) u ( · , x ); [0 , t ] (cid:1) d x [By (4.5)] ≤ Z J (cid:16) TV (cid:0) w ( · , x ); [0 , t ] (cid:1) (cid:13)(cid:13) u ( · , x ) (cid:13)(cid:13) L ∞ ([0 ,t ]; R ) + (cid:13)(cid:13) w ( · , x ) (cid:13)(cid:13) L ∞ ([0 ,t ]; R ) TV (cid:0) u ( · , x ); [0 , t ] (cid:1)(cid:17) d x [By (4.2)] ≤ Z J TV (cid:0) w ( · , x ); [0 , t ] (cid:1) d x k u k L ∞ ([0 ,t ] × J ; R ) + W Z J TV (cid:0) u ( · , x ); [0 , t ] (cid:1) d x (SP.6): Fix t ′ , t ′′ ∈ J with t ′ < t ′′ . Then, (cid:13)(cid:13) u ( t ′′ ) − u ( t ′ ) (cid:13)(cid:13) L ( R + ; R ) = Z X ( t ′′ ; t ′ , (cid:12)(cid:12) u ( t ′′ , x ) − u ( t ′ , x ) (cid:12)(cid:12) d x + Z + ∞ X ( t ′′ ; t ′ , (cid:12)(cid:12) u ( t ′′ , x ) − u ( t ′ , x ) (cid:12)(cid:12) d x We estimate the two latter terms above separately: Z X ( t ′′ ; t ′ , (cid:12)(cid:12) u ( t ′′ , x ) − u ( t ′ , x ) (cid:12)(cid:12) d x ≤ (cid:16)(cid:13)(cid:13) u ( t ′′ ) (cid:13)(cid:13) L ∞ ( R + ; R ) + (cid:13)(cid:13) u ( t ′ ) (cid:13)(cid:13) L ∞ ( R + ; R ) (cid:17) X ( t ′′ ; t ′ ,
0) [by (SP.2) and (g) ] ≤ g (cid:18) k u o k L ∞ ( R + ; R ) + 1ˇ g k b k L ∞ ([0 ,t ′′ ]; R ) + k f k L ∞ ([0 ,t ′′ ] × R + ; R ) t ′′ (cid:19) ( t ′′ − t ′ ) e ( G + M ) t ′′ . Passing to the next term, Z + ∞ X ( t ′′ ; t ′ , (cid:12)(cid:12) u ( t ′′ , x ) − u ( t ′ , x ) (cid:12)(cid:12) d x ≤ Z + ∞ X ( t ′′ ; t ′ , (cid:12)(cid:12)(cid:12) u (cid:0) t ′ , X ( t ′ ; t ′′ , x ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E ( t ′ , t ′′ , x ) − (cid:12)(cid:12) d x [Use (4.12)]+ Z + ∞ X ( t ′′ ; t ′ , (cid:12)(cid:12)(cid:12) u (cid:0) t ′ , X ( t ′ ; t ′′ , x ) (cid:1) − u ( t ′ , x ) (cid:12)(cid:12)(cid:12) d x [Use (4.6)]+ Z + ∞ X ( t ′′ ; t ′ , Z t ′′ t ′ (cid:12)(cid:12)(cid:12) f (cid:0) τ, X ( τ ; t ′′ , x ) (cid:1)(cid:12)(cid:12)(cid:12) E ( τ, t ′′ , x ) d τ d x [Use (f ) and (4.12)] ≤ (cid:13)(cid:13) u ( t ′ ) (cid:13)(cid:13) L ( R + ; R ) (cid:16) e ( G + M )( t ′′ − t ′ ) − (cid:17) [Use (SP.2) ]+ TV (cid:0) u ( t ′ ) (cid:1) (cid:13)(cid:13) X ( t ′ ; t ′′ , · ) − · (cid:13)(cid:13) L ∞ ( R + ; R ) [Use (g) and (SP.2) ]+ sup [ t ′ ,t ′′ ] (cid:13)(cid:13) f ( t ) (cid:13)(cid:13) L ( R + ; R ) e ( G + M )( t ′′ − t ′ ) ( t ′′ − t ′ ) [Use (f ) and (4.12)] ≤L ( t ′′ − t ′ ) , completing the proof of (4.19). (SP.7): This bound is a direct consequence of (4.9). (cid:3) emma 4.7. Let (g) holds. Fix u ′ o , u ′′ o ∈ ( L ∩ BV )( R + ; R ) , b ′ , b ′′ satisfying (b) , m ′ , m ′′ satisfying (m) , and f ′ , f ′′ satisfying (f ) . Call u ′ and u ′′ the solutions to ∂ t u + ∂ x (cid:0) g ( t, x ) u (cid:1) + m ′ ( t, x ) u = f ′ ( t, x ) u (0 , x ) = u ′ o ( x ) g ( t, u ( t, b ′ ( t ) and ∂ t u + ∂ x (cid:0) g ( t, x ) u (cid:1) + m ′′ ( t, x ) u = f ′′ ( t, x ) u (0 , x ) = u ′′ o ( x ) g ( t, u ( t, b ′′ ( t ) . (4.23) Then, (SP.8)
The following stability conditions hold: (cid:13)(cid:13) u ′ ( t ) − u ′′ ( t ) (cid:13)(cid:13) L ( R + ; R ) ≤ e Mt (cid:13)(cid:13) u ′ o − u ′′ o (cid:13)(cid:13) L ( R + ; R ) + e G + M ) t (cid:16) (cid:13)(cid:13) f ′ − f ′′ (cid:13)(cid:13) L ( J × R + ; R ) + (cid:13)(cid:13) b ′ − b ′′ (cid:13)(cid:13) L ([0 ,t ]; R ) (cid:17) + e (2 G + M ) t h(cid:13)(cid:13) u ′′ o (cid:13)(cid:13) L ( R + ; R ) + 2 tF + (cid:13)(cid:13) b ′′ (cid:13)(cid:13) L ([0 ,t ]; R ) i t (cid:13)(cid:13) m ′ − m ′′ (cid:13)(cid:13) L ∞ ([0 ,t ] × R + ; R ) (4.24) and (cid:13)(cid:13) u ′ ( t ) − u ′′ ( t ) (cid:13)(cid:13) L ( R + ; R ) ≤ e Mt (cid:13)(cid:13) u ′ o − u ′′ o (cid:13)(cid:13) L ( R + ; R ) + e G + M ) t (cid:16) (cid:13)(cid:13) f ′ − f ′′ (cid:13)(cid:13) L ( J × R + ; R ) + (cid:13)(cid:13) b ′ − b ′′ (cid:13)(cid:13) L ([0 ,t ]; R ) (cid:17) + e (2 G + M ) t (cid:13)(cid:13) u ′′ o (cid:13)(cid:13) L ∞ ( R + ; R ) + 2 tF ∞ + (cid:13)(cid:13) b ′′ (cid:13)(cid:13) L ∞ ([0 ,t ]; R ) ˇ g (cid:13)(cid:13) m ′ − m ′′ (cid:13)(cid:13) L ([0 ,t ] × R + ; R ) . (4.25) (SP.9) The following monotonicity property holds: f ′ ( t, x ) ≤ f ′′ ( t, x ) ∀ ( t, x ) ∈ J × R + u ′ o ( x ) ≤ u ′′ o ( x ) ∀ x ∈ R + b ′ ( t ) ≤ b ′′ ( t ) ∀ t ∈ J ⇒ u ′ ( t, x ) ≤ u ′′ ( t, x ) ∀ ( t, x ) ∈ J × R + . (4.26) (SP.10) If ¯ x > and σ ( t ) < ¯ x , then (cid:13)(cid:13) u ′ ( · , ¯ x ) − u ′′ ( · , ¯ x ) (cid:13)(cid:13) L ([0 ,t ]; R ) ≤ e ( G + M ) t t (cid:20) e G t (cid:13)(cid:13) u ′ o (cid:13)(cid:13) L ( R + ; R ) + t F ∞ (cid:21) (cid:13)(cid:13) m ′ − m ′′ (cid:13)(cid:13) L ∞ ([0 ,t ] × R + ; R ) + e Mt (cid:13)(cid:13) u ′ o − u ′′ o (cid:13)(cid:13) L ( R + ; R ) + e (2 G + M ) t (cid:13)(cid:13) f ′ − f ′′ (cid:13)(cid:13) L ( [0 ,t ] × R + ; R ) . (4.27) and (cid:13)(cid:13) u ′ ( · , ¯ x ) − u ′′ ( · , ¯ x ) (cid:13)(cid:13) L ([0 ,t ]; R ) ≤ e ( G + M ) t G ∞ (cid:20) e G t (cid:13)(cid:13) u ′ o (cid:13)(cid:13) L ∞ ( R + ; R ) + tF ∞ (cid:21) (cid:13)(cid:13) m ′ − m ′′ (cid:13)(cid:13) L ([0 ,t ] × R + ; R ) + e Mt (cid:13)(cid:13) u ′ o − u ′′ o (cid:13)(cid:13) L ( R + ; R ) + e (2 G + M ) t (cid:13)(cid:13) f ′ − f ′′ (cid:13)(cid:13) L ( [0 ,t ] × R + ; R ) . (4.28)15 roof. The stability bounds (4.24) and (4.25) can be easily proved using the explicit for-mula (4.9) and the estimates of Lemma 4.4. Also the monotonicity property (4.26) directlyfollows from (4.9).We pass to the proof of the estimate (4.27). The proof of the estimate (4.28) is completelyanalogous. Using (4.9) with the condition σ ( t ) < ¯ x we deduce that (cid:13)(cid:13) u ′ ( · , ¯ x ) − u ′′ ( · , ¯ x ) (cid:13)(cid:13) L ([0 ,t ]; R ) ≤ Z t (cid:12)(cid:12)(cid:12) u ′ o (cid:0) X (0; s, ¯ x ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E ′ (0 , s, ¯ x ) − E ′′ (0 , s, ¯ x ) (cid:12)(cid:12) d s + Z t (cid:12)(cid:12)(cid:12) u ′ o (cid:0) X (0; s, ¯ x ) (cid:1) − u ′′ o (cid:0) X (0; s, ¯ x ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E ′′ (0 , s, ¯ x ) (cid:12)(cid:12) d s + Z t Z s (cid:12)(cid:12)(cid:12) f ′ (cid:0) τ, X ( τ ; s, ¯ x ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E ′ ( τ, s, ¯ x ) − E ′′ ( τ, s, ¯ x ) (cid:12)(cid:12) d τ d s + Z t Z s (cid:12)(cid:12)(cid:12) f ′ (cid:0) τ, X ( τ ; s, ¯ x ) (cid:1) − f ′′ (cid:0) τ, X ( τ ; s, ¯ x ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E ′′ ( τ, s, ¯ x ) (cid:12)(cid:12) d τ d s . We now estimate all the terms in the previous inequality. Using (4.10), (m) and (g) we have Z t (cid:12)(cid:12)(cid:12) u ′ o (cid:0) X (0; s, ¯ x ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E ′ (0 , s, ¯ x ) − E ′′ (0 , s, ¯ x ) (cid:12)(cid:12) d s ≤ e ( G + M ) t t (cid:13)(cid:13) m ′ − m ′′ (cid:13)(cid:13) L ∞ ([0 ,t ] × R + ; R ) Z t (cid:12)(cid:12)(cid:12) u ′ o (cid:0) X (0; s, ¯ x ) (cid:1)(cid:12)(cid:12)(cid:12) d s ≤ e (2 G + M ) t t (cid:13)(cid:13) m ′ − m ′′ (cid:13)(cid:13) L ∞ ([0 ,t ] × R + ; R ) (cid:13)(cid:13) u ′ o (cid:13)(cid:13) L ( R + ; R ) . Using (4.10), and (m) , we deduce that Z t (cid:12)(cid:12)(cid:12) u ′ o (cid:0) X (0; s, ¯ x ) (cid:1) − u ′′ o (cid:0) X (0; s, ¯ x ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E ′′ (0 , s, ¯ x ) (cid:12)(cid:12) d s ≤ e Mt (cid:13)(cid:13) u ′ o − u ′′ o (cid:13)(cid:13) L ( R + ; R ) . Using (4.10), (m) , (g) and (f ) we have Z t Z s (cid:12)(cid:12)(cid:12) f ′ (cid:0) τ, X ( τ ; s, ¯ x ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E ′ ( τ, s, ¯ x ) − E ′′ ( τ, s, ¯ x ) (cid:12)(cid:12) d τ d s ≤ e ( G + M ) t t (cid:13)(cid:13) m ′ − m ′′ (cid:13)(cid:13) L ∞ ([0 ,t ] × R + ; R ) Z t Z s (cid:12)(cid:12)(cid:12) f ′ (cid:0) τ, X ( τ ; s, ¯ x ) (cid:1)(cid:12)(cid:12)(cid:12) d τ d s ≤ e ( G + M ) t t (cid:13)(cid:13) m ′ − m ′′ (cid:13)(cid:13) L ∞ ([0 ,t ] × R + ; R ) F ∞ . Finally using (4.12) and (g) we get Z t Z s (cid:12)(cid:12)(cid:12) f ′ (cid:0) τ, X ( τ ; s, ¯ x ) (cid:1) − f ′′ (cid:0) τ, X ( τ ; s, ¯ x ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E ′′ ( τ, s, ¯ x ) (cid:12)(cid:12) d τ d s ≤ e ( G + M ) t Z t Z s (cid:12)(cid:12)(cid:12) f ′ (cid:0) τ, X ( τ ; s, ¯ x ) (cid:1) − f ′′ (cid:0) τ, X ( τ ; s, ¯ x ) (cid:1)(cid:12)(cid:12)(cid:12) d τ d s ≤ e ( G + M ) t e G t Z t Z R + (cid:12)(cid:12) f ′ ( τ, x ) − f ′′ ( τ, x ) (cid:12)(cid:12) d x d τ ≤ e (2 G + M ) t (cid:13)(cid:13) f ′ − f ′′ (cid:13)(cid:13) L ( [0 ,t ] × R + ; R ) . (cid:13)(cid:13) u ′ ( · , ¯ x ) − u ′′ ( · , ¯ x ) (cid:13)(cid:13) L ([0 ,t ]; R ) ≤ e (2 G + M ) t t (cid:13)(cid:13) u ′ o (cid:13)(cid:13) L ( R + ; R ) (cid:13)(cid:13) m ′ − m ′′ (cid:13)(cid:13) L ∞ ([0 ,t ] × R + ; R ) + e Mt (cid:13)(cid:13) u ′ o − u ′′ o (cid:13)(cid:13) L ( R + ; R )+ e ( G + M ) t t F ∞ (cid:13)(cid:13) m ′ − m ′′ (cid:13)(cid:13) L ∞ ([0 ,t ] × R + ; R ) + e (2 G + M ) t (cid:13)(cid:13) f ′ − f ′′ (cid:13)(cid:13) L ( [0 ,t ] × R + ; R ) ≤ e ( G + M ) t t (cid:20) e G t (cid:13)(cid:13) u ′ o (cid:13)(cid:13) L ( R + ; R ) + t F ∞ (cid:21) (cid:13)(cid:13) m ′ − m ′′ (cid:13)(cid:13) L ∞ ([0 ,t ] × R + ; R ) + e Mt (cid:13)(cid:13) u ′ o − u ′′ o (cid:13)(cid:13) L ( R + ; R ) + e (2 G + M ) t (cid:13)(cid:13) f ′ − f ′′ (cid:13)(cid:13) L ( [0 ,t ] × R + ; R )concluding the proof of (4.27). (3.1) Proof of Theorem 3.1.
Fix t ∗ >
0, with t ∗ ∈ I , and let J = [0 , t ∗ ]. Define the constants K > k u o k L ( R + ; R n ) + B K ∞ > max (cid:16) nB L ˇ g (cid:17) k u o k L ∞ ( R ; R n ) + B ∞ ˇ g (cid:16) G ∞ ˇ g (cid:17) (cid:18) B ∞ ˇ g + (cid:16) nB L ˇ g (cid:17) (cid:16) k u o k L ∞ ( R + ; R n ) + TV( u o ; R + ) (cid:17)(cid:19) (4.29)and the complete metric space ( X n , d X n ) where X = u ∈ C (cid:0) J ; L ( R + ; R ) (cid:1) : k u k C ( J ; L ( R + ; R )) ≤ K k u k C ( J ; L ∞ ( R + ; R )) ≤ K ∞ sup t ∈ J TV( u ( t, · ); R + ) ≤ K ∞ d X n ( u ′ , u ′′ ) = max i ∈{ ,...,n } (cid:13)(cid:13) u ′′ i − u ′ i (cid:13)(cid:13) C ( J ; L ( R + ; R )) . (4.30)Define the map T : X n → X n , such that, for w = ( w , · · · , w n ) ∈ X n , T ( w ) = u , where u = ( u , · · · , u n ) solves ∂ t u i + ∂ x ( g i ( t, x ) u i ) + m i ( t, x ) u i = f i ( t, x ) u i (0 , x ) = u oi ( x ) u i ( t, b i ( t ) i = 1 , . . . , n (4.31)where m i ( t, x ) = − (cid:0) α i [ w ( t )]( x ) (cid:1) i − (cid:0) γ i ( t, x ) (cid:1) i f i ( t, x ) = P j = i (cid:16)(cid:0) α i [ w ( t )]( x ) (cid:1) j + (cid:0) γ i ( t, x ) (cid:1) j (cid:17) w j ( t, x ) b i ( t ) = β i (cid:0) t, u ( t, ¯ x − ) , . . . , u n ( t, ¯ x n − ) (cid:1) . (4.32)Remark that in the last line above an essential role is going to be played by the assumption ∂ u j β i ( t, u ) = 0 for all j ≥ i . 17 he map T is well defined. (i.e. T ( w ) ∈ X n for every w ∈ X n ). Aiming at the use ofLemma 4.5, we verify that the assumptions (g) , (m) , (f ) , and (b) therein hold. (g) holds. It is immediate by (IBVP.1) . (m) holds. The continuity of x → m ( t, x ) follows from (IBVP.2) and (IBVP.3) . Ob-serve that the map t → α i [ w ( t )]( x ) is continuous, indeed: (cid:13)(cid:13) α i [ w ( t )]( x ) − α i [ w ( t )]( x ) (cid:13)(cid:13) ≤ (cid:13)(cid:13) α i [ w ( t )] − α i [ w ( t )] (cid:13)(cid:13) C ( R + ; R n ) ≤ (cid:13)(cid:13) α i [ w ( t ) − w ( t )] (cid:13)(cid:13) C ( R + ; R n ) [By linearity] ≤ A L (cid:13)(cid:13) w ( t ) − w ( t ) (cid:13)(cid:13) L ( R + ; R ) [By (3.4)]and the fact that w ∈ C ( J ; L ( R + ; R )) allows to conclude.Fix ( t , x ) , ( t , x ) in J × R + and compute, for i = 1 , . . . , n , (cid:12)(cid:12) m i ( t, x ) − m i ( t, x ) (cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:0) α i [ w ( t )] (cid:1) i ( x ) − (cid:0) α i [ w ( t )] (cid:1) i ( x ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:0) γ i ( t, x ) (cid:1) i − (cid:0) γ i ( t, x ) (cid:1) i (cid:12)(cid:12)(cid:12) [By (4.32)] ≤ (cid:13)(cid:13) α i [ w ( t )]( x ) − α i [ w ( t )]( x ) (cid:13)(cid:13) + (cid:13)(cid:13) γ i ( t, x ) − γ i ( t, x ) (cid:13)(cid:13) ≤ A | x − x | + C L | x − x | . [By (3.6) and (3.7)Moreover, k m i k L ∞ ( J × R + ; R ) ≤ ess sup t ∈ J (cid:13)(cid:13)(cid:13)(cid:0) α i [ w ( t )] (cid:1) i (cid:13)(cid:13)(cid:13) C ( R + ; R ) + k γ i k L ∞ ( J × R + ; R n ) [By (4.32)] ≤ A L k w k C ( J ; L ( R + ; R n )) + C ∞ [By (3.4) and (3.8)] ≤ A L K + C ∞ [By (4.30)]so that m i is a Caratheodory function. Finally,sup t ∈ J TV( m i ( t, · ); R + ) ≤ sup t ∈ J TV( α i [ w ( t )]; R + ) + sup t ∈ J TV( γ i ( t, · ); R + ) [By (4.32)] ≤ A L k w k C ( J ; L ( R + ; R )) + C ∞ [By (3.4) and (3.8)] ≤ A L K + C ∞ [By (4.30)]completing the proof of (m) with M = A L K + C ∞ . (4.33) (f ) holds. Compute the terms in the right hand side of (cid:13)(cid:13) f i ( t ) − f i ( t ) (cid:13)(cid:13) L ( R + ; R ) ≤ X j = i (cid:13)(cid:13)(cid:13)(cid:0) α i [ w ( t )] (cid:1) j w j ( t ) − (cid:0) α i [ w ( t )] (cid:1) j w j ( t ) (cid:13)(cid:13)(cid:13) L ( R + ; R ) + X j = i (cid:13)(cid:13)(cid:13)(cid:0) γ i ( t ) (cid:1) j w j ( t ) − (cid:0) γ i ( t ) (cid:1) j w j ( t ) (cid:13)(cid:13)(cid:13) L ( R + ; R ) (cid:13)(cid:13)(cid:13)(cid:0) α i [ w ( t )] (cid:1) j w j ( t ) − (cid:0) α i [ w ( t )] (cid:1) j w j ( t ) (cid:13)(cid:13)(cid:13) L ( R + ; R ) [Use the linearity of α i ] ≤ (cid:13)(cid:13) α i [ w ( t )] (cid:13)(cid:13) C ( R + ; R n ) (cid:13)(cid:13) w ( t ) − w ( t ) (cid:13)(cid:13) L ( R + ; R n ) [Use (3.4)]+ (cid:13)(cid:13) α i [ w ( t ) − w ( t )] (cid:13)(cid:13) C ( R + ; R n ) (cid:13)(cid:13) w ( t ) (cid:13)(cid:13) L ( R + ; R n ) [Use (3.4)] ≤ A L (cid:13)(cid:13) w ( t ) (cid:13)(cid:13) L ( R + ; R ) (cid:13)(cid:13) w ( t ) − w ( t ) (cid:13)(cid:13) L ( R + ; R n ) [Use (4.30)]+ A L (cid:13)(cid:13) w ( t ) − w ( t ) (cid:13)(cid:13) L ( R + ; R n ) (cid:13)(cid:13) w ( t ) (cid:13)(cid:13) L ( R + ; R ) [Use (4.30)] ≤ A L K (cid:13)(cid:13) w ( t ) − w ( t ) (cid:13)(cid:13) L ( R + ; R n ) . and similarly (cid:13)(cid:13)(cid:13)(cid:0) γ i ( t ) (cid:1) j w j ( t ) − (cid:0) γ i ( t ) (cid:1) j w j ( t ) (cid:13)(cid:13)(cid:13) L ( R + ; R ) ≤ (cid:13)(cid:13) γ i ( t ) (cid:13)(cid:13) L ∞ ( R + ; R n ) (cid:13)(cid:13) w ( t ) − w ( t ) (cid:13)(cid:13) L ( R + ; R n ) [Use (3.8)]+ (cid:13)(cid:13) w ( t ) (cid:13)(cid:13) L ∞ ( R + ; R n ) (cid:13)(cid:13) γ i ( t ) − γ i ( t ) (cid:13)(cid:13) L ( R + ; R n ) [Use (4.30)] ≤ C ∞ (cid:13)(cid:13) w ( t ) − w ( t ) (cid:13)(cid:13) L ( R + ; R n ) + K ∞ (cid:13)(cid:13) γ i ( t ) − γ i ( t ) (cid:13)(cid:13) L ( R + ; R n ) which show that f i ∈ C ( J ; L ( R + ; R )), by (IBVP.3) and (4.30).We prove now the L and L ∞ bounds on f : k f i k L ∞ ( J ; L ( R + ; R )) ≤ X j = i (cid:18)(cid:13)(cid:13)(cid:13)(cid:0) α i [ w ( t )] (cid:1) j w j ( t ) (cid:13)(cid:13)(cid:13) L ( R + ; R ) + (cid:13)(cid:13)(cid:13)(cid:0) γ i ( t ) (cid:1) j w j ( t ) (cid:13)(cid:13)(cid:13) L ( R + ; R ) (cid:19) [By (4.32)] ≤ ( A L K + C ∞ K ) n [By (3.4), (3.8) and (4.30)]proving the L ∞ bound on f i with F = ( A L K + C ∞ K ) n . (4.34)The L ∞ bound is proved similarly: k f i k L ∞ ( J × R + ; R )) ≤ X j = i (cid:18)(cid:13)(cid:13)(cid:13)(cid:0) α i [ w ] (cid:1) j w j (cid:13)(cid:13)(cid:13) L ∞ ( J × R + ; R ) + (cid:13)(cid:13) ( γ i ) j w j (cid:13)(cid:13) L ∞ ( J × R + ; R ) (cid:19) [By (4.32)] ≤ ( A L K K ∞ + C ∞ K ∞ ) n [By (3.4), (3.8) and (4.30)]Moreover,TV( f i ( t, · ); R + ) ≤ X j = i TV (cid:16)(cid:0) α i [ w ( t )]( · ) (cid:1) j w j ( t, · ) (cid:17) + X j = i TV (cid:16)(cid:0) γ i ( t, · ) (cid:1) j w j ( t, · ) (cid:17) [By (4.32) ≤ X j = i TV (cid:16)(cid:0) α i [ w ( t )]( · ) (cid:1) j ; R + (cid:17) (cid:13)(cid:13) w j ( t ) (cid:13)(cid:13) L ∞ ( R + ; R ) [Use (3.4) and (4.30)]19 X j = i (cid:13)(cid:13)(cid:13)(cid:0) α i [ w ( t )]( · ) (cid:1) j (cid:13)(cid:13)(cid:13) L ∞ ( R + ; R ) TV (cid:16) w j ( t, · ); R + (cid:17) [Use (3.4) and (4.30)]+ X j = i TV (cid:16) ( γ i ( t )) j ; R + (cid:17) (cid:13)(cid:13) w j (cid:13)(cid:13) L ∞ ( J × R + ; R ) [Use (3.8) and (4.30)]+ X j = i (cid:13)(cid:13) ( γ j ( t ) i ) (cid:13)(cid:13) L ∞ ( R + ; R ) TV (cid:16) w j ( t ); R + (cid:17) [Use (3.8) and (4.30)] ≤ n A L K ∞ K + 2 n C ∞ K ∞ completing the proof that (f ) holds with F ∞ = 2 nK ∞ ( C ∞ + A L K ) . (4.35) (b) holds for i = 1 . Note that, in this case, β ( t, u ) is independent of u , so that us-ing (3.12), TV( b ; J ) = TV( β ( · ); J ) ≤ B ∞ . T w is Lipschitz continuous in time with respect to the L norm. Simply ap-ply (SP.7) , observing that (f ) , (g) , (m) and (b) were proved above exhibiting bounds thathold uniformly on X n , once the norm of the initial datum u o and the constants in (IBVP.1) – (IBVP.5) are fixed. (cid:0) T ( w ) (cid:1) ∈ X . By Lemma 4.5, u is well defined as solution to the Initial Boundary ValueProblem (4.31) with i = 1. By (SP.2) , (IBVP.4) and (3.10), we have that k u k C ( J ; L ( R + ; R )) ≤ (cid:20)(cid:13)(cid:13) u o (cid:13)(cid:13) L ( R + ; R ) + 1ˇ g k β k L ( J ; R ) + ( A K + C ∞ K ) n t ∗ (cid:21) e ( A L K + C ∞ ) t ∗ ≤ (cid:20)(cid:13)(cid:13) u o (cid:13)(cid:13) L ( R + ; R ) + B ˇ g + ( A K + C ∞ K ) n t ∗ (cid:21) e ( A L K + C ∞ ) t ∗ ≤ K provided t ∗ is small, since by (4.29) K > (cid:13)(cid:13) u o (cid:13)(cid:13) L ( R + ; R ) + B ˇ g . To bound the L ∞ norm, Use (SP.2) and (3.11) to obtain (cid:13)(cid:13) u ( τ ) (cid:13)(cid:13) L ∞ ([0 ,t ∗ ] × R + ; R ) ≤ (cid:18)(cid:13)(cid:13) u o (cid:13)(cid:13) L ∞ ( R + ; R ) + B ∞ ˇ g +2 n K ∞ ( C ∞ + A L K ) t ∗ (cid:19) e ( G + A L K + C ∞ ) t ∗ < K ∞ provided t ∗ is small, since K ∞ > (cid:13)(cid:13) u o (cid:13)(cid:13) L ∞ ( R + ; R ) + B ∞ ˇ g . Using (SP.6) , we have u ∈ C (cid:0) J ; L ( R + ; R ) (cid:1) . By (SP.3) and (IBVP.4) , for t ∈ J , we haveTV( u ( t ); R + ) ≤ H ( t ) F ∞ t + k β k L ∞ ([0 ,t ]; R ) + TV( β ; R + )ˇ g + (cid:13)(cid:13) u o (cid:13)(cid:13) L ∞ ( R + ; R ) + TV( u o ; R + ) ! ≤ H ( t ∗ ) (cid:18) n K ∞ ( C ∞ + A L K ) t ∗ + 2ˇ g B ∞ + (cid:13)(cid:13) u o (cid:13)(cid:13) L ∞ ( R + ; R ) + TV( u o ; R + ) (cid:19) ≤ K ∞ provided t ∗ is small, since by (4.29) K ∞ > (cid:18) G ∞ ˇ g (cid:19) (cid:18) B ∞ ˇ g + k u o k L ∞ ( R + ; R n ) + TV( u o ; R + ) (cid:19) , u = ( T w ) ∈ X . Remark for later use that, by (SP.4) , u ( · , ¯ x ) ∈ BV ( J ; R ) . (4.36) (b) holds for i > . Fix now an index i >
1, assume that u , . . . , u i − ∈ X and considerthe Initial Boundary Value Problem (4.31)–(4.32). By (IBVP.4) , the function β i depends onlyon t and on u , · · · , u i − . Moreover, by (IBVP.4) and (4.36), which hold for every u j with j < i , the map t b i ( t ) = β i (cid:0) t, u ( t, ¯ x ) , · · · , u i − ( t, ¯ x i − ) (cid:1) is of bounded variation and hence satisfies (b) . (cid:0) T ( w ) (cid:1) i ∈ X for i > . Lemma 4.5 implies that there exists a solution u i to (4.31).By (SP.2) and (3.9) (3.10), we have that (cid:13)(cid:13) u i ( t ) (cid:13)(cid:13) C ( J ; L ( R + ; R )) ≤ h(cid:13)(cid:13) u oi (cid:13)(cid:13) L ( R + ; R ) + k b i k L ( J ; R ) + F t ∗ i e Mt ∗ ≤ (cid:13)(cid:13) u oi (cid:13)(cid:13) L ( R + ; R ) + B + B L i − X j =1 Z t ∗ (cid:12)(cid:12) u j ( s, ¯ x j ) (cid:12)(cid:12) d s + n ( A L K + C ∞ K ) t ∗ e ( A L K + C ∞ ) t ∗ ≤ h(cid:13)(cid:13) u oi (cid:13)(cid:13) L ( R + ; R ) + B + ( i − B L K ∞ t ∗ + n ( A L K + C ∞ K ) t ∗ i e ( A L K + C ∞ ) t ∗ ≤ K provided t ∗ is small, since by (4.29) K > (cid:13)(cid:13) u oi (cid:13)(cid:13) L ( R + ; R ) + B . Using (SP.6) , u i ∈ C (cid:0) J ; L ( R + ; R ) (cid:1) . To bound the L ∞ norm, Use (SP.2) , (4.9), (3.9),and (3.11) to obtain k u i k L ∞ ( J × R + ; R ) ≤ (cid:13)(cid:13) u oi (cid:13)(cid:13) L ∞ ( R + ; R ) + k b i k L ∞ ( J ; R ) ˇ g + F ∞ t ∗ ! e ( G + M ) t ∗ ≤ (cid:13)(cid:13) u oi (cid:13)(cid:13) L ∞ ( R + ; R ) + B ∞ ˇ g + B L ˇ g i − X j =1 (cid:12)(cid:12) u j ( t, ¯ x j ) (cid:12)(cid:12) + 2 n K ∞ ( C ∞ + A L K ) t ∗ e ( G + A L K + C ∞ ) t ∗ ≤ (cid:18) nB L ˇ g e ( G + A L K + C ∞ ) t ∗ (cid:19) k u o k L ∞ ( R + ; R n ) + B ∞ ˇ g + 2 nB L K ∞ ˇ g ( C ∞ + A L K ) t ∗ e ( G + A L K + C ∞ ) t ∗ + 2 n K ∞ ( C ∞ + A L K ) t ∗ ! × e ( G + A L K + C ∞ ) t ∗ < K ∞ provided t ∗ is small, since by (4.29) K ∞ > (cid:18) nB L ˇ g (cid:19) k u o k L ∞ ( R ; R n ) + B ∞ ˇ g . We pass now to estimate the total variation. By (SP.3) and (IBVP.4) , for t ∈ J , we haveTV( u i ( t ); R + ) 21 H ( t ) F ∞ t + k b i k L ∞ ([0 ,t ]; R ) + TV( b i ; J )ˇ g + (cid:13)(cid:13) u oi (cid:13)(cid:13) L ∞ ( R + ; R ) + TV( u oi ; R + ) ! ≤ H ( t ∗ ) nK ∞ ( C ∞ + A L K ) t ∗ + 1ˇ g B ∞ + B L i − X j =1 (cid:16)(cid:13)(cid:13) u j ( · , ¯ x j ) (cid:13)(cid:13) L ∞ ( J ; R ) + TV( u j ( · , ¯ x j ); J ) (cid:17) + (cid:13)(cid:13) u oi (cid:13)(cid:13) L ∞ ( R + ; R ) + TV( u oi ; R + ) [By (3.9), (3.11) and (3.12)] ≤ H ( t ∗ ) (cid:18) n K ∞ ( C ∞ + A L K ) t ∗ + 2 B ∞ ˇ g [Use (4.9), (4.12), (f ) , (4.14)]+ n B L ˇ g (cid:16) k u o k L ∞ ( R + ; R n ) + 2 n K ∞ ( C ∞ + A L K ) t ∗ (cid:17) e ( G + A L K + C ∞ ) t ∗ + n B L ˇ g h TV( u o ; R + ) + 2( G + A L K + C ∞ ) k u o k L ∞ ( R + ; R n ) t ∗ i e ( G + A L K + C ∞ ) t ∗ + 8 n B L ˇ g [1 + ( G + A L K + C ∞ ) t ∗ ] K ∞ ( A L K + C ∞ ) t ∗ e ( G + A L K + C ∞ ) t ∗ + k u o k L ∞ ( R + ; R n ) + TV( u o ; R + ) (cid:17) < K ∞ provided t ∗ is small, since by (4.29) K ∞ > (cid:18) G ∞ ˇ g (cid:19) (cid:18) B ∞ ˇ g + nB L ˇ g (cid:16) k u o k L ∞ ( R + ; R n ) + TV( u o ; R + ) (cid:17)(cid:19) . This concludes the proof of the well posedness of T . The map T is a contraction. Fix w, ¯ w ∈ X . For later use, we prepare some estimates.By (IBVP.2) and (4.30), for every i ∈ { , · · · , n } , we deduce that (cid:13)(cid:13) ( α i [ w ]) i − ( α i [ ¯ w ]) i (cid:13)(cid:13) L ∞ ( J × R + ; R ) ≤ A L d X n ( w, ¯ w ) . (4.37)Moreover, by (IBVP.2) and (4.30), for every i ∈ { , · · · , n } and j ∈ { , · · · , n } \ { i } , we obtain (cid:13)(cid:13) ( α i [ w ]) j w j − ( α i [ ¯ w ]) j ¯ w j (cid:13)(cid:13) L ( J × R + ; R ) ≤ (cid:13)(cid:13) ( α i [ w ]) j w j − ( α i [ ¯ w ]) j w j (cid:13)(cid:13) L ( J × R + ; R ) + (cid:13)(cid:13) ( α i [ ¯ w ]) j ( w j − ¯ w j ) (cid:13)(cid:13) L ( J × R + ; R ) ≤ (cid:13)(cid:13) ( α i [ w − ¯ w ]) j (cid:13)(cid:13) L ∞ ( J × R + ; R ) d X n ( w, t ∗ + (cid:13)(cid:13) ( α i [ ¯ w ]) j (cid:13)(cid:13) L ∞ ( J × R + ; R ) t ∗ d X n ( w, ¯ w ) ≤ A L K t ∗ d X n ( w, ¯ w ) . (4.38)Finally by (IBVP.3) , for every i ∈ { , · · · , n } and j ∈ { , · · · , n } \ { i } , we also have (cid:13)(cid:13) ( γ i ) j w j − ( γ i ) j ¯ w j (cid:13)(cid:13) L ( J × R + ; R ) ≤ C ∞ t ∗ d X n ( w, ¯ w ) . (4.39)For i = 1 and t ∈ J , Lemma 4.7 implies that (cid:13)(cid:13) ( T w ) ( t, · ) − ( T ¯ w ) ( t, · ) (cid:13)(cid:13) L ( R + ; R ) [Use (4.24) and (4.32)]22 e G + A L K + C ∞ ) t ∗ n X j =2 (cid:13)(cid:13) ( α [ w ]) j w j − ( α [ ¯ w ]) j ¯ w j (cid:13)(cid:13) L ( J × R + ; R ) + 2 e G + A L K + C ∞ ) t ∗ n X j =2 (cid:13)(cid:13) ( γ ) j w j − ( γ ) j ¯ w j (cid:13)(cid:13) L ( J × R + ; R ) + e (2 G + A L K + C ∞ ) t ∗ h(cid:13)(cid:13) u o (cid:13)(cid:13) L ( R + ; R ) + 2 nK ( A L K + C ∞ ) t ∗ + B i × t ∗ (cid:13)(cid:13) ( α [ w ]) − ( α [ ¯ w ]) (cid:13)(cid:13) L ∞ ( J × R + ; R ) . Therefore, using (4.37), (4.38), and (4.39), we get that (cid:13)(cid:13) ( T w ) ( t, · ) − ( T ¯ w ) ( t, · ) (cid:13)(cid:13) L ( R + ; R ) ≤ n e G + A L K + C ∞ ) t ∗ A L K t ∗ d X n ( w, ¯ w ) + 2 n e G + A L K + C ∞ ) t ∗ C ∞ t ∗ d X n ( w, ¯ w )+ e (2 G + A L K + C ∞ ) t ∗ h(cid:13)(cid:13) u o (cid:13)(cid:13) L ( R + ; R ) + 2 nK ( A L K + C ∞ ) t ∗ + B i A L d X n ( w, ¯ w )and so, choosing t ∗ sufficiently small, we obtain that (cid:13)(cid:13) ( T w ) − ( T ¯ w ) (cid:13)(cid:13) C ( J ; L ( R + ; R )) ≤ n d X n ( w, ¯ w ) . (4.40)For i > t ∈ J , Lemma 4.7 implies that (cid:13)(cid:13) ( T w ) i ( t, · ) − ( T ¯ w ) i ( t, · ) (cid:13)(cid:13) L ( R + ; R ) ≤ e G + A L K + C ∞ ) t ∗ n X j =1 ,j = i (cid:13)(cid:13) ( α i [ w ]) j w j − ( α i [ ¯ w ]) j ¯ w j (cid:13)(cid:13) L ( J × R + ; R ) + 2 e G + A L K + C ∞ ) t ∗ n X j =2 ,j = i (cid:13)(cid:13) ( γ i ) j w j − ( γ i ) j ¯ w j (cid:13)(cid:13) L ( J × R + ; R ) + e G + A L K + C ∞ ) t ∗ (cid:13)(cid:13) b i − ¯ b i (cid:13)(cid:13) L ( J ; R ) + e (2 G + A L K + C ∞ ) t ∗ h(cid:13)(cid:13) u o (cid:13)(cid:13) L ( R + ; R ) + 2 nK ( A L K + C ∞ ) t ∗ + (cid:13)(cid:13) ¯ b i (cid:13)(cid:13) L ( J ; R ) i × t ∗ (cid:13)(cid:13) ( α i [ w ]) i − ( α i [ ¯ w ]) i (cid:13)(cid:13) L ∞ ( J × R + ; R ) where b i ( t ) = β i (cid:0) t, u ( t, ¯ x ) , · · · , u i − ( t, ¯ x i − ) (cid:1) and ¯ b i ( t ) = β i (cid:0) t, ¯ u ( t, ¯ x ) , · · · , ¯ u i − ( t, ¯ x i − ) (cid:1) (4.41)are the boundary terms respectively for w and ¯ w . We thus have that (cid:13)(cid:13) b i − ¯ b i (cid:13)(cid:13) L ( J ; R ) ≤ B L i − X j =1 (cid:13)(cid:13) u j ( · , ¯ x j ) − ¯ u j ( · , ¯ x j ) (cid:13)(cid:13) L ( J ; R ) [By (3.9), (4.41)] ≤ B L i − X j =1 e ( G + A L K + C ∞ ) t ∗ t ∗ " e G t ∗ (cid:13)(cid:13)(cid:13) u oj (cid:13)(cid:13)(cid:13) L ( R + ; R )+ t ∗ nK ∞ ( A L K + C ∞ ) [By (4.27), (4.32)]23 (cid:13)(cid:13)(cid:13)(cid:0) α j [ w ] (cid:1) j − (cid:0) α j [ ¯ w ] (cid:1) j (cid:13)(cid:13)(cid:13) L ∞ ( J × R + ; R ) + e (2 G + A L K + C ∞ ) t ∗ n X h =1 ,h = j (cid:13)(cid:13)(cid:13)(cid:0) α j [ w ] (cid:1) h w h − (cid:0) α j [ ¯ w ] (cid:1) h ¯ w h (cid:13)(cid:13)(cid:13) L ( J × R + ; R ) + e (2 G + A L K + C ∞ ) t ∗ n X h =1 ,h = j (cid:13)(cid:13) ( γ j ) h w h − ( γ j ) h ¯ w h (cid:13)(cid:13) L ( J × R + ; R ) ≤ B L ne ( G + A L K + C ∞ ) t ∗ t ∗ h e G t ∗ k u o k L ( R + ; R n ) + t ∗ nK ∞ ( A L K + C ∞ ) i × A L d X n ( w, ¯ w ) [By (4.37)]+ e (2 G + A L K + C ∞ ) t ∗ nA L K t ∗ d X n ( w, ¯ w ) [By (4.38)]+ e (2 G + A L K + C ∞ ) t ∗ nC ∞ t ∗ d X n ( w, ¯ w ) . [By (4.39)]Moreover (cid:13)(cid:13) ¯ b i (cid:13)(cid:13) L ( J ; R ) ≤ B L i − X j =1 (cid:13)(cid:13) u j ( · , ¯ x j ) (cid:13)(cid:13) L ( J ; R ) + B [By 3.9, 3.10 and (4.41)] ≤ nB L e (2 G + A L K + C ∞ ) t ∗ t ∗ k u o k L ∞ ( R + ; R n ) [By 4.9, (4.12), and (g) ]+ 2 n B L e ( G + A L K + C ∞ ) t ∗ K ∞ ( A L K + C ∞ ) t ∗ . [By 4.9, (4.12), and (f ) ]Finally, using again (4.37), (4.38), and (4.39), we obtain (cid:13)(cid:13) ( T w ) i ( t, · ) − ( T ¯ w ) i ( t, · ) (cid:13)(cid:13) L ( R + ; R ) ≤ n A L K t ∗ d X n ( w, ¯ w ) e G + A L K + C ∞ ) t ∗ + 2 n C ∞ t ∗ d X n ( w, ¯ w ) e G + A L K + C ∞ ) t ∗ + n A L B L h e G t ∗ k u o k L ( R + ; R n ) + t ∗ nK ∞ ( A L K + C ∞ ) i t ∗ d X n ( w, ¯ w ) e G + A L K + C ∞ ) t ∗ + 2 n A L K t ∗ d X n ( w, ¯ w ) e G + A L K + C ∞ ) t ∗ + 2 n C ∞ t ∗ d X n ( w, ¯ w ) e G + A L K + C ∞ ) t ∗ + A L h(cid:13)(cid:13) u o (cid:13)(cid:13) L ( R + ; R ) + 2 nK ( A L K + C ∞ ) t ∗ + (cid:13)(cid:13) ¯ b i (cid:13)(cid:13) L ( J ; R ) i t ∗ d X n ( w, ¯ w ) e (2 G + A L K + C ∞ ) t ∗ . Choosing t ∗ sufficiently small, we obtain (cid:13)(cid:13) ( T w ) i − ( T ¯ w ) i (cid:13)(cid:13) C ( J ; L ( R + ; R )) ≤ n d X n ( w, ¯ w ). To-gether with (4.40), this implies that d X n ( T w, T ¯ w ) ≤ d X n ( w, ¯ w ), hence T is a contraction. Existence, Uniqueness and Lipschitz Continuity in Time.
On the basis of Defini-tion 4.3, a map u is a solution to (3.1) on [0 , t ∗ ] if and only if it is a fixed point of T , whencewe have existence and uniqueness of the solution on the time interval [0 , t ∗ ]. The Lipschitzcontinuity of u ∗ in time follows from (SP.6) . Dependence on the Boundary and Initial Data.
Call u ′ , respectively u ′′ , the solutioncorresponding to the boundary datum β ′ , respectively β ′′ , and to the initial datum u ′ o , respec-tively u ′′ o . In the following, the constants K and K ∞ satisfy (4.29) for both u ′ o and u ′′ o . Fix24 ∈ { , · · · , n } and t ∈ [0 , t ∗ ]. Estimate (4.25) implies that (cid:13)(cid:13) u ′ i ( t ) − u ′′ i ( t ) (cid:13)(cid:13) L ( R + ; R ) ≤ e Mt (cid:13)(cid:13)(cid:13) u ′ o,i ( t ) − u ′′ o,i ( t ) (cid:13)(cid:13)(cid:13) L ( R + ; R )+ 2 e G + M ) t Z t Z R + n X j =1 ,j = i (cid:12)(cid:12)(cid:12)(cid:0) α i [ u ′ ( s )]( x ) (cid:1) j u ′ j ( s, x ) − (cid:0) α i [ u ′′ ( s )]( x ) (cid:1) j u ′′ j ( s, x ) (cid:12)(cid:12)(cid:12) d s d x + 2 e G + M ) t Z t Z R + n X j =1 ,j = i (cid:12)(cid:12)(cid:12)(cid:0) γ i ( s, x ) (cid:1) j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u ′ j ( s, x ) − u ′′ j ( s, x ) (cid:12)(cid:12)(cid:12) d s d x + e G + M ) t Z t (cid:12)(cid:12)(cid:12) β ′ i (cid:0) s, · · · , u ′ i − ( s, ¯ x i − − ) (cid:1) − β ′′ i (cid:0) s, · · · , u ′′ i − ( s, ¯ x i − − ) (cid:1)(cid:12)(cid:12)(cid:12) d s + e G + M ) t "(cid:13)(cid:13)(cid:13) u ′′ o,i (cid:13)(cid:13)(cid:13) L ∞ ( R + ; R ) + 2 tF ∞ + 1ˇ g (cid:13)(cid:13)(cid:13) β ′′ i (cid:0) · , · · · , u ′′ i − ( · , ¯ x i − − ) (cid:1)(cid:13)(cid:13)(cid:13) L ∞ ( [0 ,t ]; R ) ×× Z t (cid:13)(cid:13)(cid:13)(cid:0) α i [ u ′ ( s )] (cid:1) i − (cid:0) α i [ u ′′ ( s )] (cid:1) i (cid:13)(cid:13)(cid:13) L ( R + ; R ) d s . (4.42)We need to estimate every term in the right hand side of (4.42). Preliminary, using (IBVP.4) ,(3.9) and (4.28), we deduce, for t ∈ [0 , t ∗ ], that Z t (cid:12)(cid:12)(cid:12) β ′ i (cid:0) s, · · · , u ′ i − ( s, ¯ x i − − ) (cid:1) − β ′′ i (cid:0) s, · · · , u ′′ i − ( s, ¯ x i − − ) (cid:1)(cid:12)(cid:12)(cid:12) d s ≤ B L i − X j =1 (cid:13)(cid:13)(cid:13) u ′ j (cid:0) · , ¯ x j − (cid:1) − u ′′ j (cid:0) · , ¯ x j − (cid:1)(cid:13)(cid:13)(cid:13) L ( (0 ,t ); R ) + (cid:13)(cid:13) β ′ i − β ′′ i (cid:13)(cid:13) L ∞ ([0 ,t ] × [0 ,K ∞ ] i − ; R ) t ≤ B L e ( G + M ) t G ∞ i − X j =1 " e G t (cid:13)(cid:13)(cid:13) u ′ o,j (cid:13)(cid:13)(cid:13) L ∞ ( R + ; R ) + tF ∞ ×× Z t Z + ∞ (cid:12)(cid:12)(cid:12)(cid:0) α j [ u ′ ( s )]( x ) (cid:1) j − (cid:0) α j [ u ′′ ( s )]( x ) (cid:1) j (cid:12)(cid:12)(cid:12) d x d s + B L e Mt (cid:13)(cid:13) u ′ o − u ′′ o (cid:13)(cid:13) L ( R + ; R n ) + B L e (2 G + M ) t ×× i X j =1 n X h =1 ,h = j Z t Z + ∞ (cid:12)(cid:12)(cid:12)(cid:0) α j [ u ′ ( s )]( x ) (cid:1) h u ′ h ( s, x ) − (cid:0) α j [ u ′′ ( s )]( x ) (cid:1) h u ′′ h ( s, x ) (cid:12)(cid:12)(cid:12) d x d s + B L e (2 G + M ) t i X j =1 n X h =1 ,h = j Z t Z + ∞ (cid:12)(cid:12)(cid:12)(cid:0) γ j ( s, x ) (cid:1) h (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u ′ h ( s, x ) − u ′′ h ( s, x ) (cid:12)(cid:12) d x d s + (cid:13)(cid:13) β ′ i − β ′′ i (cid:13)(cid:13) L ∞ ([0 ,t ] × [0 ,K ∞ ] i − ; R ) t. (4.43)25or j, h ∈ { , · · · , n } , j = h , and t ∈ [0 , t ∗ ], using (IBVP.2) , (3.4), (3.5), and (4.29), we have Z t Z R + (cid:12)(cid:12)(cid:12)(cid:0) α j [ u ′ ( s )]( x ) (cid:1) h u ′ h ( s, x ) − (cid:0) α j [ u ′′ ( s )]( x ) (cid:1) h u ′′ h ( s, x ) (cid:12)(cid:12)(cid:12) d x d s ≤ Z t Z R + (cid:12)(cid:12)(cid:12)(cid:0) α j [ u ′ ( s )]( x ) (cid:1) h (cid:0) u ′ h ( s, x ) − u ′′ h ( s, x ) (cid:1)(cid:12)(cid:12)(cid:12) d x d s + Z t Z R + (cid:12)(cid:12)(cid:12)(cid:0) α j [ u ′ ( s ) − u ′′ ( s )]( x ) (cid:1) h u ′′ h ( s, x ) (cid:12)(cid:12)(cid:12) d x d s ≤ A L K Z t (cid:13)(cid:13) u ′ h ( s ) − u ′′ h ( s ) (cid:13)(cid:13) L ( R + ; R ) d s + A K ∞ Z t (cid:13)(cid:13) u ′ ( s ) − u ′′ ( s ) (cid:13)(cid:13) L ( R + ; R n ) d s . (4.44)Moreover, for j, h ∈ { , · · · , n } , j = h , and t ∈ [0 , t ∗ ], using (IBVP.3) , we get Z t Z R + (cid:12)(cid:12)(cid:12)(cid:0) γ j ( s, x ) (cid:1) h (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u ′ h ( s, x ) − u ′′ h ( s, x ) (cid:12)(cid:12) d x d s ≤ C ∞ Z t (cid:13)(cid:13) u ′ h ( s ) − u ′′ h ( s ) (cid:13)(cid:13) L ( R + ; R ) d s . (4.45)Finally, for j ∈ { , · · · , n } and t ∈ [0 , t ∗ ], using (IBVP.2) and (3.5), we have that Z t (cid:13)(cid:13)(cid:13)(cid:0) α j [ u ′ ( s )] (cid:1) j − (cid:0) α j [ u ′′ ( s )] (cid:1) j (cid:13)(cid:13)(cid:13) L ( R + ; R ) d s = Z t Z + ∞ (cid:12)(cid:12)(cid:12)(cid:0) α j [ u ′ ( s ) − u ′′ ( s )] (cid:1) j ( x ) (cid:12)(cid:12)(cid:12) d x d s ≤ A Z t (cid:13)(cid:13) u ′ ( s ) − u ′′ ( s ) (cid:13)(cid:13) L ( R + ; R n ) d s . (4.46)Inserting (4.43), (4.44), (4.45), and (4.46) into (4.42) we obtain that, for t ∈ [0 , t ∗ ], (cid:13)(cid:13) u ′ ( t ) − u ′′ ( t ) (cid:13)(cid:13) L ( R + ; R n ) ≤ H ( t ) Z t (cid:13)(cid:13) u ′ ( s ) − u ′′ ( s ) (cid:13)(cid:13) L ( R + ; R n ) d s + H ( t ) (cid:13)(cid:13) u ′ o ( t ) − u ′′ o ( t ) (cid:13)(cid:13) L ( R + ; R n )+ e G + M ) t (cid:13)(cid:13) β ′ − β ′′ (cid:13)(cid:13) L ∞ ([0 ,t ] × [0 ,K ∞ ] n ; R n ) t, where H ( t ) = ne G + M ) t [2 A L K + 2 nA K ∞ + 2 C ∞ ]+ n e G + M ) t B L A G ∞ (cid:20) e G t (cid:13)(cid:13) u ′ o (cid:13)(cid:13) L ∞ ( R + ; R n ) + tF ∞ (cid:21) + n e (4 G +3 M ) t B L [ A L K + nA K ∞ + C ∞ ]+ n e G + M ) t A (cid:20)(cid:13)(cid:13) u ′′ o (cid:13)(cid:13) L ∞ ( R + ; R n ) + 2 ntF ∞ + n ˇ g ( B ∞ + nB L K ∞ ) (cid:21) ; H ( t ) = ne Mt (cid:16) B L e G + M ) t (cid:17) . An application of Gronwall Lemma yields (3.14). (cid:3) roof of Corollary 3.3. We proceed with the same notation used in the proof of Theorem 3.1, u being the solution to (3.1) on J .The positivity of each u i directly follows from (4.9)–(4.10).Assume, by contradiction, that there exists a maximal time of existence ¯ t for the solution u to (3.1). A direct consequence of (NEG) and (EQ) is that ∂ t n X i =1 u i + ∂ x g ( t, x ) n X i =1 u i = n X i =1 (cid:0) α i [ u ( t )] + γ i ( t, x ) (cid:1) · u ≤ . Therefore, (SP.2) and (SP.3) ensure that (cid:13)(cid:13)P ni =1 u i ( t ) (cid:13)(cid:13) L ( R + ; R ) , (cid:13)(cid:13)P ni =1 u i ( t ) (cid:13)(cid:13) L ∞ ( R + ; R ) andTV (cid:0)P ni =1 u i ( t ); R + (cid:1) are uniformly bounded on [0 , ¯ t [. The uniform continuity of u in timeensures that u can be defined also at time ¯ t . A further application of Theorem 3.1 allows touniquely extend u beyond time ¯ t , yielding the contradiction.The functional Lipschitz continuous dependence of u ∗ on the initial datum u o and on theboundary inflow β now follows from (SP.8) and (SP.10) , possibly iterating the estimate (4.27)to comply with the condition γ ( t ) < ¯ x therein. (cid:3) (1.1) – (1.2) and (1.1) – (1.5) Proof of Theorem 2.1.
The present proof consists in showing that with the present assump-tions, Theorem 3.1 and Corollary 3.3 can be applied to (1.1)–(1.2)–(2.1). To this aim, set fornotational simplicity ¯ a = 0, ¯ a N +1 = + ∞ and define u j ( t, a ) = S ( t, a + ¯ a j ) u j ( t, a ) = I ( t, a + ¯ a j ) u j ( t, a ) = R ( t, a + ¯ a j ) for j = 0 , , . . . , Nt ∈ I a ∈ R + . (4.47)Set g i ( t, a ) = 1 for i = 1 , . . . , n and n = 3 N + 3. Define for j = 0 , , . . . , N and i = 1 , . . . , n (cid:2) α j [ u ]( a ) (cid:3) i = − N +1 X ℓ =1 Z ¯ a ℓ ¯ a ℓ − λ ( a + ¯ a j , a ′ ) u ℓ ( a ′ − ¯ a ℓ − ) d a ′ i = 1 + 3 j (cid:2) α j [ u ]( a ) (cid:3) i = N +1 X ℓ =1 Z ¯ a ℓ ¯ a ℓ − λ ( a + ¯ a j , a ′ ) u ℓ ( a ′ − ¯ a ℓ − ) d a ′ i = 1 + 3 j (cid:2) α j [ u ]( a ) (cid:3) i = 0 (4.48) (cid:2) γ j ( t, a ) (cid:3) i = ( − d S ( t, a + ¯ a j ) i = 1 + 3 j (cid:2) γ j ( t, a ) (cid:3) i = ( − d I ( t, a + ¯ a j ) − r I ( t, a + ¯ a j ) i = 2 + 3 j (cid:2) γ j ( t, a ) (cid:3) i = − d R ( t, a + ¯ a j ) i = 3 + 3 jr I ( t, a + ¯ a j ) i = 2 + 3 j x i = ¯ a j − ¯ a j − for all i ∈ { , · · · , n } and for all j = { , · · · , N + 1 } such that i − j − ∈ { , , } . Moreover β ( t, u ) = S b ( t ) β ( t, u ) = I b ( t ) β ( t, u ) = R b ( t ) β j +1 ( t, u ) = (cid:0) − η j ( t ) (cid:1) u j − β j +2 ( t, u ) = u j − β j +3 ( t, u ) = η j ( t ) u j − + u j (4.50)We now verify that the assumptions required in Theorem 3.1 on the functions above hold. (IBVP.1) holds. It is immediate, since g i ( t, a ) = 1 for all ( t, a ) ∈ I × R + . (IBVP.2) holds. On the basis of (4.48), we have: (cid:13)(cid:13) α i [ w ] (cid:13)(cid:13) L ∞ ( R + ; R n ) ≤ Λ ∞ k u k L ( R + ; R n ) [By (2.2), (3.4) holds with A L = Λ ∞ ]TV( α i [ w ]; ( R + ) n ) ≤ N +1 X ℓ =1 Z ¯ a ℓ ¯ a ℓ − TV( λ ( · , a ′ ); R + ) (cid:12)(cid:12) w ℓ ( a ′ − ¯ a ℓ − ) (cid:12)(cid:12) d a ′ [By 4.5] ≤ Λ ∞ k u k L ( R + ; R n ) [By (2.2), (3.4) holds with A L = Λ ∞ ] (cid:13)(cid:13) α i [ u ]( a ) − α i [ u ]( a ) (cid:13)(cid:13) ≤ N +1 X ℓ =1 Z ¯ a ℓ ¯ a ℓ − (cid:12)(cid:12) λ ( a +¯ a j , a ′ ) − λ ( a +¯ a j , a ′ ) (cid:12)(cid:12) | u ℓ | ( a ′ − ¯ a ℓ − ) d a ′ [By ( λ ) ] ≤ Λ l N +1 X ℓ =1 Z ¯ a ℓ ¯ a ℓ − | u ℓ | ( a ′ − ¯ a ℓ − ) d a ′ | a − a | [By (2.3)] ≤ N Λ L k u k L ( R + ; R n ) | a − a | proving (3.6) in (IBVP.2) with A = N Λ L k u k L ( R + ; R n ) . (IBVP.3) holds. Refer to (4.49). The Lipschitz continuity of γ directly follows from (2.5),proving (3.7). The other conditions are immediate consequences of (2.4), (2.6) and (2.4). (IBVP.4) holds. Refer to (4.50). Condition (3.9) is immediate, thanks to the bounded-ness of η . (3.10), (3.11) and (3.12) follow from (2.7). (IBVP.5) holds. It is an immediate consequence of (2.7), using (4.47) at t = 0. (POS) holds. It immediately follows from (IB) . (NEG) holds. Long but straightforward computations based on (1.1), (4.48) and (4.49)show that (NEG) holds. (EQ) holds.
It is a direct consequence of (1.1).
Dependence on η . Theorem 3.1 and Corollary 3.3 ensure the Lipschitz continuous depen-dence of the solution (
S, I, R ) in L on the boundary datum through its L norm. In viewof (4.50), this yields the continuous dependence of the solution ( S, I, R ) in L on η in L ∞ . (cid:3) t and a . However, for completeness, we provide an independent proof. Proof of Theorem 2.2.
We now show that Theorem 3.1 can be iteratively applied to prob-lem (1.1)–(2.1)–(1.5). To this aim, define n = 3 and u ( t, a ) = S ( t, a ) , u ( t, a ) = I ( t, a ) , u ( t, a ) = R ( t, a ) . (cid:2) α [ u ]( a ) (cid:3) i = ( − R R + λ ( a, a ′ ) u ( a ′ ) d a ′ i = 10 i = 2 , (cid:2) γ ( t, a ) (cid:3) i = ( − d S ( t, a ) i = 10 i = 2 , (cid:2) α [ u ]( a ) (cid:3) i = (R R + λ ( a, a ′ ) u ( a ′ ) d a ′ i = 10 i = 2 , (cid:2) γ ( t, a ) (cid:3) i = ( − d I ( t, a ) − r I ( t, a ) i = 20 i = 1 , (cid:2) α ( a ) (cid:3) i = 0 (cid:2) γ ( t, a ) (cid:3) i = i = 1 r I ( t, a ) i = 2 − d R ( t, a ) i = 3We now iteratively apply Theorem 3.1 on the time interval [¯ t k , ¯ t k +1 ] assigning, on the basisof (2.1), the initial and boundary data k = 0 t ∈ [0 , ¯ t ] a ∈ R + u o ( a ) = S o ( a ) u o ( a ) = I o ( a ) u o ( a ) = R o ( a ) β ( t, u ) = S b ( t ) β ( t, u ) = I b ( t ) β ( t, u ) = R b ( t ) k ≥ t ∈ [¯ t k , ¯ t k +1 ] a ∈ R + u o (¯ t k , a ) = (cid:0) − ν k ( a ) (cid:1) u (¯ t k − , a ) u o (¯ t k , a ) = u (¯ t k − , a ) u o (¯ t k , a ) = u (¯ t k − , a ) + ν k ( a ) u (¯ t k − , a ) β ( t, u ) = S b ( t ) β ( t, u ) = I b ( t ) β ( t, u ) = R b ( t ) (4.51)We now verify that the assumptions required in Theorem 3.1 on the functions above hold. (IBVP.1) holds. It is immediate, since g i ( t, a ) = 1 for all ( t, a ) ∈ I × R + . (IBVP.2) holds. (cid:13)(cid:13) α i [ w ] (cid:13)(cid:13) L ∞ ( R + ; R n ) ≤ Λ ∞ k u k L ( R + ; R n ) [By (2.2), (3.4) holds with A L = Λ ∞ ]TV( α i [ w ]; ( R + ) n ) ≤ Z R + TV( λ ( · , a ′ ); R + ) (cid:12)(cid:12) w j ( a ′ ) (cid:12)(cid:12) d a ′ [By 4.5] ≤ Λ ∞ k u k L ( R + ; R n ) [By (2.2), (3.4) holds with A L = Λ ∞ ] (cid:13)(cid:13) α i [ u ]( a ) − α i [ u ]( a ) (cid:13)(cid:13) ≤ Z R + (cid:12)(cid:12) λ ( a , a ′ ) − λ ( a , a ′ ) (cid:12)(cid:12) | u | ( a ′ ) d a ′ ≤ Λ l Z R + (cid:12)(cid:12) u ( a ′ ) (cid:12)(cid:12) d a ′ | a − a | [By (2.3)] ≤ Λ L k u k L ( R + ; R n ) | a − a | . [(3.6) holds with A = Λ L k u k L ( R + ; R n ) ] (IBVP.3) holds. The Lipschitz continuity of γ directly follows from (2.5), proving (3.7).The other conditions are immediate consequences of (2.4), (2.6) and (2.4). (IBVP.4) holds. The definitions (4.51) and (2.7) directly imply (3.10), (3.11) and (3.12).29
IBVP.5) holds.
It directly follows from (4.51). (POS) holds.
It immediately follows from (IB) . (NEG) holds. Long but straightforward computations based on (1.1), (4.48) and (4.49)show that (NEG) holds. (EQ) holds.
It is a direct consequence of (1.1).
Dependence on ν . Repeat the same argument used in the final part of the proof ofTheorem 2.1, replacing the boundary datum with the initial datum. (cid:3)
Acknowledgments:
Part of this work was supported by the PRIN 2015 project
HyperbolicSystems of Conservation Laws and Fluid Dynamics: Analysis and Applications and by theGNAMPA 2018 project
Conservation Laws: Hyperbolic Games, Vehicular Traffic and Fluiddynamics . The
IBM Power Systems Academic Initiative substantially contributed to the nu-merical integrations.
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