Global solution for the coagulation equation of water droplets in atmosphere between two horizontal planes
aa r X i v : . [ m a t h . A P ] M a y Global solution for the coagulation equationof water droplets in atmosphere betweentwo horizontal planes.
Hanane Belhireche , Steave C. Selvaduray Laboratory of Applied Mathematics and Modeling,University 8 Mai 1945, Guelma, Algeria,[email protected] Dipartimento di Matematica, Universit`a di Torino, Italy,steave [email protected], [email protected]
Abstract.
In this paper we give a global existence and uniqueness theorem for an initial andboundary value problem (IBVP) relative to the coagulation equation of water droplets and weshow the convergence of the global solution to the stationary solution. The coagulation equationis an integro-differential equation that describes the variation of the density σ of water dropletsin the atmosphere. Furthermore, IBVP is considered on a strip limited by two horizontal planesand its boundary condition is such that rain fall from the strip. To obtain this result of globalexistence of the solution σ in the space of bounded continuous functions, through the method ofcharacteristics, we assume bounded continuous and small data, whereas the vector field, besidesbeing bounded continuous, has W , ∞ − regularity in space. Key words: initial and boundary value problem, integro-PDE, method of characteristics,phase transitions, stationary solution, coagulation equation.
MSC:
In [17] has been introduced a model of motion of the air and the phase transition ofwater in the three states in the atmosphere. Since then, many papers have been producedin relation to this model (see for example [15], [3] [1], [18], [19], etc.). In particular, inthe paper [15], under some suitable conditions, has been proved the existence and theuniqueness of the stationary solution for the equation of water droplets, without takinginto account the condensation or evaporation process, assuming a domain with two spatialdimensions and a horizontal constant wind. Moreover in [3] has been shown, under weakassumptions, the global existence and uniqueness of the solution for the same equation,1ithout taking into account the wind, on a domain with one spatial dimension. On theother hand, in the paper [19] has been proved the existence and the uniqueness, in W , ∞ ,of the local solution of an IBVP for the hyperbolic part of the model introduced in [17],on a strip bounded by two horizontal planes with the boundary condition such that rainfall from the strip.Now, a significant question which arises, is that to establish the global existence of thesolution of IVBP studied in [19]. At the moment, this question is very difficult to treat,therefore, in this paper, we have decided to study only one equation of the hyperbolic partof the model seen in [17]. More precisely, we consider the coagulation equation of waterdroplets, including condensation or evaporation process and effects of wind, in atmospherebetween two horizontal planes (or vertical strip), supposing known the initial density andthe density on the upper horizontal plane. For this problem, assuming small data andthat the vertical velocities of droplets are negative, we prove the global existence anduniqueness of the solution and we establish the existence of the stationary solution andthe convergence of the global solution to the stationary solution.Of course, we know that there exists a well-developed mathematical theory of coagu-lation (see for example [9]), but the results obtained in our paper, although they appearelementary, are not present in the literature about the coagulation equation.Afterwards, let us say something about the sections of this paper. In section 2, weintroduce the initial and boundary value problem for the coagulation equation of waterdroplets and its stationary equation associated. In section 3, after having introducedhypotheses about the velocities of water droplets, using the method of characteristics,we transform our IBVP in two sets of infinite Cauchy’s problems for ordinary differentialequations (ODEs) and we give the definitions of generalized solutions for them. Hence,in section 4, we give the two main theorems of this paper. In section 5, we study thelinearized version of the ODEs about the coagulation equation obtained in section 3 andwe establish useful estimates to treat our IBVP. In section 6, using some fixed-pointarguments, we are able to prove the first main theorem about the global existence anduniqueness of the solution for the coagulation equation. In section 7, we prove the secondmain theorem about stationary solution.The authors would like to thank Prof. Hisao Fujita Yashima for having proposed tous the study of this problem. We consider the integro-differential equation which describes the variation of the den-sity σ ( t, x, m ) of water droplets with mass m in the atmosphere (see [17], [2])(2.1) ∂ t σ ( t, x, m ) + ∇ x · ( σ ( t, x, m ) u ( t, x, m )) + ∂ m ( mh gl ( t, x, m ) σ ( t, x, m )) == h gl ( t, x, m ) σ ( t, x, m ) + m Z m β ( m − m ′ , m ′ ) σ ( t, x, m ′ ) σ ( t, x, m − m ′ ) dm ′ + − m Z ∞ β ( m, m ′ ) σ ( t, x, m ) σ ( t, x, m ′ ) dm ′ + g ( m ) (cid:2) N − e N ( σ ) (cid:3) + ( t, x )[ Q ] + ( t, x )+ − g ( m )[ Q ] − ( t, x ) σ ( t, x, m ) ,
2n the domain(2.2) R + × Ω × R + , where Ω = { x ∈ R | ( x , x ) ∈ R , < x < } , t ∈ R + is time, x = ( x , x , x ) ∈ Ωis a point in three-dimensional space and m ∈ R + is the mass of a droplet, whereas [ r ] + and [ r ] − are the positive and the negative part of r ∈ R . In (2.1), u and h gl are givenfunctions defined on R + × Ω × R + , Q is a given function on R + × Ω, g and g are givenfunctions on R + and β is a given function on R + × R + , whereas N and e N ( σ ) are a positiveconstant and a linear functional of σ (that we define below) respectively. Now, we studythe equation (2.1) with the following conditions(2.3) σ (0 , x, m ) = ˜ σ ( x, m ) for x ∈ Ω , m ∈ R + , (2.4) σ ( t, x , x , , m ) = ˜ σ ( t, x , x , m ) for t ∈ R + , ( x , x ) ∈ R , m ∈ R + , where ˜ σ and ˜ σ are given functions defined on Ω × R + and R × R + respectively. Moreover,assuming Γ − = (cid:0) { } × Ω × R + (cid:1) ∪ (cid:0) R + × R × { } × R + (cid:1) , we can rewrite (2.3) and (2.4)as(2.5) σ | Γ − = ˜ σ = (cid:26) ˜ σ on { } × Ω × R + , ˜ σ on R + × R × { } × R + . From a physical point of view, the function u ( t, x, m ) is the velocity of the dropletlocated at x with mass m and has approximately the following expression(2.6) u ( t, x, m ) = v ( t, x ) − gα ( m ) e , e = (0 , , T , ( t, x, m ) ∈ R + × Ω × R + , where v ( t, x ) is the velocity of air, g is the gravitational acceleration and α ( m ) is deter-mined by air friction on droplet with mass m . However, in this paper, we do not assumethe expression (2.6) for u . On the other hand, the function h gl ( t, x, m ) is the amount of H O that turns from gas to liquid condensing on a droplet with mass m and Q = Q ( t, x )is the difference between the vapor density π ( t, x ) and the density π vs ( T ) of a saturatedvapor relative to the liquid state at temperature T ; more precisely we have(2.7) Q ( t, x ) = π ( t, x ) − π vs ( T ( t, x )) , ( t, x ) ∈ R + × Ω;therefore a well approximation for h gl is given by the relation(2.8) h gl ( t, x, m ) = η ( m ) Q ( t, x ) , ( t, x, m ) ∈ R + × Ω × R + , where the coefficient η ( m ) ≥ R + .Moreover β ( m , m ) is the encounter probability between a droplet with mass m andanother with masse m , whereas g ( m )[ N − e N ( σ )] + is the coefficient of appearance fordroplets with mass m and g ( m ) is that of disappearance for droplets with mass m ; N and e N ( σ ) are the number of aerosol compared to the unit of air volume and the numberof aerosol already present in droplets respectively. (For details of physical meaning ofthese functions, see [17], [2]). 3o determine the distribution for σ , we introduce two numbers m a and m A (with0 < m a < m A < ∞ ) and we consider that droplets are absent apart from an interval[ m a , m A ], then we have(2.9) σ ( m ) = 0 for m ∈ [0 , m a [ ∪ ] m A , ∞ [ , where m a , m A correspond respectively to the lower mass and the upper mass of droplets(see [16]).Now we suppose that β is a continuous function defined on R + × R + such that(2.10) β ( m , m ) ≥ , β ( m , m ) = β ( m , m ) ∀ ( m , m ) ∈ R + × R + , (2.11) max (cid:2) sup We make the following assumptions on the velocity u and Q (3.1) u ∈ C b ( R + × Ω × R + ; R ) ∩ L loc (cid:0) R + ; W , ∞ (Ω × R + ; R ) (cid:1) ,u ∈ L x (cid:0) , W , ∞ ( t loc ,x ,x ,m ) ( R + × R × R + ) (cid:1) , ∇ · u ∈ C b ( R + × Ω × R + ) , where C b ( X ; R ) is the space of continuous and bounded functions on a generic metricspace X ,(3.2) L loc (cid:0) R + ; W , ∞ (Ω × R + ; R ) (cid:1) = \ T > L (cid:0) (0 , T ); W , ∞ (Ω × R + ; R ) (cid:1) , and(3.3) L x (cid:0) , W , ∞ ( t loc ,x ,x ,m ) ( R + × R × R + ) (cid:1) = \ T > L (cid:0) (0 , W , ∞ ((0 , T ) × R × R + ) (cid:1) ;moreover there exists a strictly positive constant A such that(3.4) u ( t, x, m ) ≤ − A , ∀ ( t, x, m ) ∈ R + × Ω × R + and we suppose that(3.5) Q ∈ L loc (cid:0) R + ; W , ∞ (Ω) (cid:1) . Therefore the condition (3.4) determines the fall of rain from the strip.Afterwards, for data ˜ σ and ˜ σ , we suppose that(3.6) ˜ σ ∈ C b (Ω × R + ) , ˜ σ ≥ , (3.7) ˜ σ ∈ C b ( R + × R × R + ) , ˜ σ ≥ , (3.8) ˜ σ ( x, m ) = 0 , ˜ σ ( t, x , x , m ) = 0 if ( t, x ) ∈ R + × Ω , m [ m a , m A ] . Furthermore, we assume that(3.9) IK (cid:0) − e − J (cid:1) < J, (3.10) k ˜ σ k L ∞ (Γ − ) < IJ e − J (cid:0) J + KI (1 − e − J ) (cid:1) , where(3.11) I = k ˜ σ k L ∞ (Γ − ) + (1 /A ) k g k L ∞ ( R + ) N k Q k L ∞ ( R + × Ω) , (3.12) J = (1 /A ) (cid:0) k∇ · u k L ∞ ( R + × Ω × R + ) + k ∂ m ( mh gl ) k L ∞ ( R + × Ω × R + ) ++ k g k L ∞ ( R + ) k Q k L ∞ ( R + × Ω) + k g k L ∞ ( R + ) k n k L ( R + ) k Q k L ∞ ( R + × Ω) (cid:1) , K = (1 /A ) (cid:0) sup m ∈ R + (cid:12)(cid:12) m + ∞ Z β ( m, m ′ ) dm ′ (cid:12)(cid:12) + sup m ∈ R + (cid:12)(cid:12) m m Z β ( m − m ′ , m ′ ) dm ′ (cid:12)(cid:12)(cid:1) . Now, for the stationary speed u ∗ we make the following assumptions(3.14) u ∗ ∈ W , ∞ (Ω × R + ; R ) , ∇ · u ∗ ∈ C b (Ω × R + ) , (3.15) u ∗ ( x, m ) ≤ − A ∀ ( x, m ) ∈ Ω × R + . Hence, for ˜ σ ∗ we assume that(3.16) ˜ σ ∗ ∈ C b ( R × R + ) , ˜ σ ∗ ≥ , (3.17) ˜ σ ∗ ( x , x , m ) = 0 if ( x , x ) ∈ R , m [ m a , m A ] , (3.18) k ˜ σ ∗ k L ∞ ( R × R + ) < I ∗ J ∗ e − J ∗ (cid:0) J ∗ + KI ∗ (1 − e − J ∗ ) (cid:1) , where(3.19) I ∗ = k ˜ σ ∗ k L ∞ ( R × R + ) + (1 /A ) k g k L ∞ ( R + ) N k Q ∗ k L ∞ (Ω) , (3.20) J ∗ = (1 /A ) (cid:0) k∇ · u ∗ k L ∞ (Ω × R + ) + k ∂ m ( mh ∗ gl ) k L ∞ (Ω × R + ) ++ k g k L ∞ ( R + ) k Q ∗ k L ∞ (Ω) + k g k L ∞ ( R + ) k n k L ( R + ) k Q ∗ k L ∞ (Ω) (cid:1) , (3.21) I ∗ K (cid:0) − e − J ∗ (cid:1) < J ∗ . Now, after introducing key assumptions about the velocities and data, we are readyto give the definitions of generalized solutions for the problems (2.1) (or (2.19)), (2.5) and(2.16) (or (2.20)), (2.17).First of all, we consider the following Cauchy’s problem related to the flow X associatedto the vector field e U (3.22) dds t ( s ) = 1 , dds X ( s ) = e U ( t ( s ) , X ( s )) , ( t (0) , X (0)) = (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ) ∈ Γ − , where X ( s ) = (cid:0) X ( s ) , X ( s ) , X ( s ) , M ( s ) (cid:1) . We observe that the first equation of (3.22)gives(3.23) t ( s ) = ˜ t + s. On the other hand, thanks to the assumptions on u and h gl (see also the hypothesis(3.5) about Q ), we deduce there exists one and only one solution X (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ; · ) on7 − ˜ t, + ∞ (cid:2) for Cauchy’s problem (3.22). Therefore, we formally can transform the problem(2.1), (2.5) in the following form(3.24) dds σ (˜ t + s, X (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ; s )) = − σ (˜ t + s, X (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ; s )) ×× (cid:2)e g (˜ t + s, X (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ; s )) + f [ σ ](˜ t + s, X (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ; s )) (cid:3) ++Φ[ σ ](˜ t + s, X (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ; s )) + h [ σ ](˜ t + s, X (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ; s )) , (3.25) σ (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ) = ˜ σ (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ) ∀ (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ) ∈ Γ − , that is equivalent, in Caratheodory’s theory, to the integral equation(3.26) σ (˜ t + s, X (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ; s )) = ˜ σ (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ) − s Z n σ (˜ t + r, X (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ; r )) ×× (cid:2)e g (˜ t + r, X (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ; r )) + f [ σ ](˜ t + r, X (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ; r )) (cid:3) ++Φ[ σ ](˜ t + r, X (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ; r )) + h [ σ ](˜ t + r, X (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ; r )) o dr. Hence, we give the following definition Definition 3.1. A continuous solution σ for the integral equation (3.26) is called ageneralized solution for the problem (2.1) and (2.5) . Now, in a similar way, we can transform the stationary problem (2.16)-(2.17) in thefollowing form(3.27) dds σ ∞ ( X ∗ ( ˜ x , ˜ x , ˜ x , ˜ m ; s )) = − σ ∞ ( X ∗ ( ˜ x , ˜ x , ˜ x , ˜ m ; s )) ×× (cid:2)e g ∗ ( X ∗ ( ˜ x , ˜ x , ˜ x , ˜ m ; s )) + f [ σ ∞ ]( X ∗ ( ˜ x , ˜ x , ˜ x , ˜ m ; s )) (cid:3) ++Φ[ σ ∞ ]( X ∗ ( ˜ x , ˜ x , ˜ x , ˜ m ; s )) + h ∗ [ σ ∞ ]( X ∗ ( ˜ x , ˜ x , ˜ x , ˜ m ; s )) , (3.28) σ ∞ ( ˜ x , ˜ x , , ˜ m ) = ˜ σ ∗ ( ˜ x , ˜ x , ˜ m ) ∀ ( ˜ x , ˜ x , ˜ m ) ∈ R × R + , where the flow X ∗ ( ˜ x , ˜ x , ˜ x , ˜ m ; s ) is the solution of the following problem(3.29) ( dds X ∗ ( s ) = e U ∗ ( X ∗ ( s )) ,X (0) = ( ˜ x , ˜ x , , ˜ m ) , with X ∗ ( s ) = (cid:0) X ∗ ( s ) , X ∗ ( s ) , X ∗ ( s ) , M ∗ ( s ) (cid:1) .Of course, the problem (3.27)-(3.28) is equivalent, in Caratheodory’s theory, to thefollowing integral equation(3.30) σ ∞ ( X ∗ ( ˜ x , ˜ x , ˜ x , ˜ m ; s )) = ˜ σ ∗ ( ˜ x , ˜ x , ˜ m ) − s Z n σ ∞ ( X ∗ ( ˜ x , ˜ x , ˜ x , ˜ m ; r )) × (cid:2)e g ∗ ( X ∗ ( ˜ x , ˜ x , ˜ x , ˜ m ; r )) + f [ σ ∞ ]( X ∗ ( ˜ x , ˜ x , ˜ x , ˜ m ; r )) (cid:3) ++Φ[ σ ∞ ]( X ∗ ( ˜ x , ˜ x , ˜ x , ˜ m ; r )) + h ∗ [ σ ∞ ]( X ∗ ( ˜ x , ˜ x , ˜ x , ˜ m ; r )) o dr. Afterwards, we give the following definition of a generalized solution for the stationaryequation Definition 3.2. A continuous solution σ ∞ for the integral equation (3.30) is called ageneralized solution for the problem (2.16) and (2.17) . Now, we are ready to give the first important result of this paper. Theorem 4.1. We assume all hypotheses stated above on all functions involved inthe problem (2.1) and (2.5) . Then there exists one and only one generalized solution σ for the problem (2.1) , (2.5) , such that (4.1) σ ∈ C b ( R + × Ω × R + ) , σ ≥ , supp σ ⊆ R + × Ω × [0 , m B ] , where (4.2) m B = m A exp (cid:0) (1 /A ) k h gl k ∞ (cid:1) . Furthermore the solution σ verifies the inequality (4.3) k σ k L ∞ ( R + × Ω × R + ) ≤ IJ e − J J + KI (1 − e − J ) . ( for I , J , K , see (3.11) - (3.13) ) . Finally, we can give the last main theorem of this paper. Theorem 4.2. We assume all hypotheses stated above on all functions involved inthe problem (2.16) and (2.17) . Then there exists one and only one generalized solutionfor the problem (2.16) and (2.17) , such that (4.4) σ ∞ ∈ C b (Ω × R + ) , σ ∞ ≥ , supp σ ∞ ⊆ Ω × [0 , m B ] , where (4.5) m B = m A exp (cid:0) (1 /A ) (cid:13)(cid:13) h ∗ gl (cid:13)(cid:13) ∞ (cid:1) . Furthermore the solution σ ∞ verifies the inequality (4.6) k σ ∞ k L ∞ (Ω × R + ) ≤ I ∗ J ∗ e − J ∗ J ∗ + KI ∗ (1 − e − J ∗ ) . ( for I ∗ , J ∗ , see (3.19) - (3.20) ) .Finally, if we assume (4.7) g ( m ) = 0 , η ( m ) = 0 if m [ m a , m B ] , σ → L ∞ (Ω × R + ) ˜ σ ∗ , Q ( t, · ) → L ∞ (Ω) Q ∗ ( · ) , (4.9) u ( t, · ) → L ∞ (Ω × R + ; R ) u ∗ ( · ) , ∇ x · u ( t, · ) → L ∞ (Ω × R + ) ∇ x · u ∗ ( · ) , for t → ∞ , then (4.10) σ ( t, · ) → L ∞ (Ω × R + ) σ ∞ ( · ) , for t → ∞ , where σ is defined in Theorem 4.1. First of all, we observe that from the second equation of Cauchy’s problem (3.22)follows(5.1) dmdx = mh gl u ≤ m k h gl k L ∞ ( R + × Ω × R + ) A . If, we define m B such that(5.2) m B Z m A dmm = Z k h gl k L ∞ ( R + × Ω × R + ) A dx , then we deduce (4.2). After defining m B , we consider the following cone K + of C b ( R + × Ω × R + )(5.3) K + = { λ | λ ∈ C b ( R + × Ω × R + ) , λ ≥ , supp λ ⊆ R + × Ω × [0 , m B ] } . In this section we study the linear differential equation(5.4) dds σ (˜ t + s, X (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ; s )) = − σ (˜ t + s, X (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ; s )) ×× he g (˜ t + s, X (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ; s )) + f [ σ ](˜ t + s, X (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ; s )) i ++Φ[ σ ](˜ t + s, X (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ; s )) + h [ σ ](˜ t + s, X (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ; s )) , where σ ∈ K + . A first result about this ODE is the following Lemma 5.1. The equation (5.4) with the condition (3.25) has one and only onesolution σ (˜ t + · , X (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ; · )) on the interval [0 , s ] , where (5.5) X (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ; s ) = 0 and s satisfies the inequality (5.6) 0 < s ≤ A . oreover we have (5.7) σ (˜ t + s, X (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ; s )) ≥ ∀ s ∈ [0 , s ] , (5.8) σ (˜ t + s, X (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ; s )) = 0 if M (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ; s ) > m B . Proof. As X (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ; · ) is well defined and the equation (5.4) is linear, theexistence and the uniqueness of the solution for the problem (5.4), (3.25) on [0 , s ] followfrom the classical theory. To determine (5.6), it is sufficient to observe that(5.9) dds X (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ; s ) = u (˜ t + s, X (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ; s )) ≤ − A . Therefore, the solution σ (˜ t + · , X (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ; · )) of the problem (5.4), (3.25) can be sorepresented on the interval [0 , s ] by the expression(5.10) σ (˜ t + s, X (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ; s )) = ˜ σ (˜ t, X (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ) ×× exp n Z s (cid:0)e g + f [ σ ] (cid:1)(cid:0) ˜ t + s ′ , X (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ; s ′ ) (cid:1) ds ′ o ++ Z s h(cid:0) Φ[ σ ] + h [ σ ] (cid:1) (˜ t + s ′ , X (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ; s ′ )) ×× exp (cid:16) Z ss ′ e g (˜ t + s ′′ , X (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ; s ′′ )) ds ′′ (cid:17)i ds ′ . Consequently (5.7) can be directly obtained from (5.10).Now to prove (5.8) it is sufficient to show thata) σ (˜ t + s, X (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ; s )) = 0 if ˜ m > m A ;b) M (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ; s ) ≤ m B if ˜ m ≤ m A .The relation a) can be deduced from the consideration that M (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ; · ) is nondecreasing function and that the conditions (3.8), (2.12), (2.13) are satisfied. To obtainb), we observe that(5.11) dds M (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ; s ) ≤ k h gl k L ∞ ( R + × Ω × R + ) M (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ; s ) , therefore we have(5.12) M (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ; s ) ≤ ˜ m exp (cid:0) s k h gl k L ∞ ( R + × Ω × R + ) (cid:1) ≤ m B . (cid:3) Now we are ready to give the following important result Lemma 5.2. Let us assume σ ∈ K + and σ the solution of the problem (5.4) , (3.25) .Then σ is a function on R + × Ω × R + that belongs to K + . Proof. As the relations (5.7) and (5.8) are already established, we just have to showthat σ is a continuous function on R + × Ω × R + . If we denote by τ − ( t, x , x , x , m ) the11inimum time of existence for X ( t, x , x , x , m ; · ) (see [19]), then we have the followingrepresentation formula for σ (5.13) σ ( t, x , x , x , m ) = ˜ σ (cid:0) τ − ( t, x , x , x , m ) , X ( t, x , x , x , m ; τ − ( t, x , x , x , m )) (cid:1) ++ t Z τ − ( t,x ,x ,x ,m ) h σ (cid:0) − e g − f [ σ ] (cid:1) + Φ [ σ ] + h [ σ ] i(cid:0) s, X ( t, x , x , x , m ; s ) (cid:1) ds. Now, from the hypotheses (3.1)-(3.5) follow that, thanks to Lemma 4.2 in [19], τ − ∈ W , ∞ ( t loc ,x ,x ,x ,m ) ( R + × Ω × R + ) = T T > W , ∞ ((0 , T ) × Ω × R + ) and X ◦ τ − ∈ W , ∞ ( t loc ,x ,x ,x ,m )) ( R + × Ω × R + ) ; therefore, remembering that the given functions appearingin (5.13) are continuous, we deduce that σ is continuous. (cid:3) Lemma 5.3. Let δ be a number such that < δ ≤ . We suppose that there exists asolution σ on R + × Ω δ × R + , where Ω δ = R × (1 − δ, , for the equation (3.24) with thefollowing condition (5.14) σ (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ) = ˜ σ (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ) ∀ (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ) ∈ Γ δ − , where Γ δ − = (cid:2) { } × R × (1 − δ, × R + (cid:3) ∪ (cid:2) R + × R × { } × R + (cid:3) . Under these hypotheses,we have (5.15) k σ k L ∞ ( R + × Ω δ × R + ) ≤ IJ e − J J + KI (1 − e − J ) . Proof. From the equations relative to the flow associated to the vector field ˜ U (see(3.22)), we have(5.16) ddx σ (cid:0) s, X (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ; s ) (cid:1) = 1 u dds σ (cid:0) s, X (˜ t, ˜ x , ˜ x , ˜ x , ˜ m ; s ) (cid:1) . Therefore, performing integration on (3.24) suitably transformed and after some calcula-tions, we obtain(5.17) k σ ( · , x , · ) k L ∞ ( R + × R × R + ) ≤ k ˜ σ k L ∞ (Γ δ − ) ++ Z x (1 /A ) k σ ( · , z, · ) k L ∞ ( R + × R × R + ) (cid:0) k e g k L ∞ ( R + × Ω δ × R + ) + k f [ σ ] ( · , z, · ) k L ∞ ( R + × R × R + ) (cid:1) dz ++ Z x (1 /A ) (cid:2) k Φ [ σ ] ( · , z, · ) k L ∞ ( R + × R × R + ) + k h [ σ ] ( · , z, · ) k L ∞ ( R + × R × R + ) (cid:3) dz ≤≤ I + Z x (cid:0) J k σ ( · , z, · ) k L ∞ ( R + × R × R + ) + K k σ ( · , z, · ) k L ∞ ( R + × R × R + ) (cid:1) dz. Finally, taking into account the condition (3.9) and using the techniques of comparisonlemma we deduce (5.15). (cid:3) Remark 5.4. We observe that the estimate (5.15) does not depending on δ .12 Proof of the Theorem 4.1. We start to prove that there exists 0 < δ < δ withthe condition (5.14) has one and only one solution. For this purpose, we consider thefollowing operator Λ δ : K δ → C b ( R + × Ω δ × R + ), where the domain of K δ is so defined(6.1) K δ = n σ ∈ C b ( R + × Ω δ × R + ) | σ ≥ , supp σ ⊆ R + × Ω δ × [0 , m B ] , k σ k L ∞ ( R + × Ω δ × R + ) ≤ IJ e − J J + KI (1 − e − J ) o and Λ δ is the operator that to each σ ∈ K δ associates the solution σ to the linear equation(5.4) with the condition (5.14).Now, we determine an upper bound for δ such that Λ δ ( K δ ) ⊆ K δ . Using for theequation (5.4) an analogous method to that seen for (3.24) in lemma 5.3, we deduce(6.2) k σ ( · , x , · ) k L ∞ ( R + × R × R + ) ≤ k ˜ σ k L ∞ (Γ δ − ) + Z x F (cid:0) k σ ( · , z, · ) k L ∞ ( R + × R × R + ) (cid:1) dz, where x ∈ ]1 − δ, 1[ and F ( · ) is(6.3) F (cid:0) k σ ( · , z, · ) k L ∞ ( R + × R × R + ) (cid:1) = (1 /A ) h k σ ( · , z, · ) k L ∞ ( R + × R × R + ) ×× (cid:0) k e g k L ∞ ( R + × Ω δ × R + ) + k f [ σ ] ( · , z, · ) k L ∞ ( R + × R × R + ) (cid:1) ++ k Φ [ σ ] k L ∞ ( R + × Ω δ × R + ) + k h [ σ ] k L ∞ ( R + × Ω δ × R + ) i ≤ C (1 + k σ ( · , z, · ) k L ∞ ( R + × R × R + ) ) , where C is a constant not depending on δ and σ . Therefore, using Gronwall’s lemma wehave(6.4) k σ k L ∞ ( R + × Ω δ × R + ) ≤ exp( Cδ ) (cid:0) k ˜ σ k L ∞ (Γ δ − ) + Cδ (cid:1) . Hence, using (3.10), we prove that there exists 0 < δ < k σ k L ∞ ( R + × Ω δ × R + ) ≤ IJ e − J J + KI (cid:0) − e − J (cid:1) , for every 0 < δ < δ . Therefore, thanks to Lemma 5.2 and (6.5) we deduce that Λ δ ( K δ ) ⊆ K δ if 0 < δ < δ .We proceed to study Λ δ and in particular we want to establish an upper bound for δ such that this map is a contraction. For this purpose, we consider σ , σ ∈ K δ and wedefine σ j = Λ δ ( σ j ) with j = 1 , 2. Representing the solutions σ , σ in integral form, wededuce, after some transformations, the inequality(6.6) k σ ( · , x , · ) − σ ( · , x , · ) k L ∞ ( R + × R × R + ) ≤ A h(cid:0) k e g k L ∞ ( R + × Ω δ × R + ) + k f [ σ ] k L ∞ ( R + × Ω δ × R + ) (cid:1) ×× Z x k σ ( · , z, · ) − σ ( · , z, · ) k L ∞ ( R + × R × R + ) dz ++ δ (cid:0) k σ k L ∞ ( R + × Ω δ × R + ) k f [ σ ] − f [ σ ] k L ∞ ( R + × Ω δ × R + ) ++ k Φ [ σ ] − Φ [ σ ] k L ∞ ( R + × Ω δ × R + ) + k h [ σ ] − h [ σ ] k L ∞ ( R + × Ω δ × R + ) (cid:1)i ≤≤ Cδ k σ − σ k L ∞ ( R + × Ω δ × R + ) + C Z x k σ ( · , x , · ) − σ ( · , z, · ) k L ∞ ( R + × R × R + ) dz, where C is a constant not depending on δ , σ and σ . Hence, a direct application ofGronwall’s lemma gives(6.7) k σ − σ k L ∞ ( R + × Ω δ × R + ) ≤ Cδ exp( Cδ ) k σ − σ k L ∞ ( R + × Ω δ × R + ) . Therefore there exists 0 < δ < δ such that Λ δ is a contraction for every 0 < δ < δ .Consequently, the map Λ δ admits one and only one fixed point if 0 < δ < δ .We have thus proved that the problem (3.24), (5.14) has one and only one solutionon Ω δ if we assume 0 < δ < δ . Now, taking into account the estimate (5.15) and usinga simple absurd reasoning about the maximal width of the strip on which the solution isdefined, we arrive to show that the solution is defined on the whole strip R + × Ω × R + .Therefore, the main result has been proved. (cid:3) The proof of the existence and the uniqueness of generalized solution σ ∞ together withthe properties (4.4)-(4.6) can be showed in analogous way to that seen for the generalizedsolution σ in Theorem 4.1. Therefore, we only prove the last part of this theorem, i.e. weshow the convergence σ ( t, · ) → σ ∞ , in L ∞ (Ω × R + ), for t → ∞ .Let δ be a number such that ˜ t > A . Hence, we have σ (˜ t, ˜ x , ˜ x , , ˜ m ) = ˜ σ (˜ t, ˜ x , ˜ x , ˜ m )in the integral equation (3.26). Therefore, we can deduce(7.1) σ (˜ t + s, X (˜ t, ˜ x, ˜ m, s )) = ˜ σ (˜ t, ˜ x , ˜ x , ˜ m )+ − Z s (cid:16)(cid:0) f [ σ ](˜ t + s ′ , X (˜ t, ˜ x, ˜ m, s ′ )) + ˜ g (˜ t + s ′ , X (˜ t, ˜ x, ˜ m, s ′ )) (cid:1) σ (˜ t + s ′ , X (˜ t, ˜ x, ˜ m, s ′ ))++ (cid:0) Φ[ σ ] + h [ σ ] (cid:1) (˜ t + s ′ , X (˜ t, ˜ x, ˜ m, s ′ )) (cid:17) ds ′ , if ˜ t > A . On the other hand, assuming X (0) = X ∗ (0) and recalling the definitions of the char-acteristics X (˜ t, ˜ x, ˜ m, s ) and X ∗ (˜ x, ˜ m, s ) given in (3.22) and (3.29), we obtain the followingestimate(7.2) | X (˜ t, ˜ x, ˜ m, s ) − X ∗ (˜ x, ˜ m, s ) | ≤ Z s (cid:16)(cid:12)(cid:12) ˜ U (˜ t + s ′ , X (˜ t, ˜ x, ˜ m, s ′ )) − ˜ U ∗ ( X (˜ t, ˜ x, ˜ m, s ′ )) (cid:12)(cid:12) + (cid:12)(cid:12) ˜ U ∗ ( X (˜ t, ˜ x, ˜ m, s ′ )) − ˜ U ∗ ( X ∗ (˜ x, ˜ m, s ′ )) (cid:12)(cid:12)(cid:17) ds ′ . Remembering (2.18), (3.14), we have that there exists a constant C > | X (˜ t, ˜ x, ˜ m, s ) − X ∗ (˜ x, ˜ m, s ) | ≤ Z s (cid:16) k u (˜ t + s ′ , · ) − u ∗ ( · ) k L ∞ (Ω × R + ; R ) ++ C k Q (˜ t + s ′ , · ) − Q ∗ ( · ) k L ∞ (Ω) + C (cid:12)(cid:12) X (˜ t, ˜ x, ˜ m, s ′ ) − X ∗ (˜ x, ˜ m, s ′ ) (cid:12)(cid:12)(cid:17) ds ′ . Now, applying the comparison lemma to (7.3) and using (4.9), we obtain the followinguseful convergence(7.4) | X (˜ t, ˜ x, ˜ m, s ) − X ∗ (˜ x, ˜ m, s ) | → ˜ t →∞ . Afterwards, a direct application of (7.4) into (7.1) gives(7.5) σ (˜ t + s, X ∗ (˜ x, ˜ m, s )) = ˜ σ (˜ t, ˜ x , ˜ x , ˜ m )+ − Z s (cid:16)(cid:0) f [ σ ](˜ t + s ′ , X ∗ (˜ x, ˜ m, s ′ )) + ˜ g (˜ t + s ′ , X ∗ (˜ x, ˜ m, s ′ )) (cid:1) σ (˜ t + s ′ , X ∗ (˜ x, ˜ m, s ′ ))++ (cid:0) Φ[ σ ] + h [ σ ] (cid:1) (˜ t + s ′ , X ∗ (˜ x, ˜ m, s ′ )) (cid:17) ds ′ , if ˜ t > A . Hence, making the difference between the equation (7.5) and the equation (3.30), aftersimple algebraic manipulations, we obtain | σ (˜ t + s, X ∗ (˜ x, ˜ m, s )) − σ ∞ ( X ∗ (˜ x, ˜ m, s )) | ≤ k ˜ σ (˜ t, · ) − ˜ σ ∗ ( · ) k L ∞ (Ω × R + ) ++ Z s h(cid:16) k∇ · (cid:0) u (˜ t + s ′ , · ) − u ∗ ( · ) (cid:1) k L ∞ (Ω × R + ) + C ′ k Q (˜ t + s ′ , · ) − Q ∗ ( · ) k L ∞ (Ω) ++ C ′ k σ (˜ t + s ′ , · ) − σ ∞ ( · ) k L ∞ (Ω × R + ) (cid:17) k σ (˜ t + s ′ , · ) k L ∞ (Ω × R + ) + N k Q (˜ t + s ′ , · ) − Q ∗ ( · ) k L ∞ (Ω) ++ (cid:16) k∇ · u ∗ ( · ) k L ∞ (Ω) + C ′ (cid:0) k Q ∗ ( · ) k L ∞ (Ω × R + ) + k σ ∞ ( · ) k L ∞ (Ω × R + ) + k σ (˜ t + s ′ , · ) k L ∞ (Ω × R + ) (cid:1)(cid:17) ××k σ (˜ t + s ′ , · ) − σ ∞ ( · ) k L ∞ (Ω × R + ) i ds ′ , where C ′ is a positive constant. Now, remembering (4.8) and applying the comparisonlemma, we immediately prove the convergence (4.10). 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