Weak solutions to the continuous coagulation equation with multiple fragmentation
aa r X i v : . [ m a t h . A P ] J a n Weak Solutions to the Continuous Coagulation Equation withMultiple Fragmentation
Ankik Kumar Giri , ∗ , Philippe Lauren¸cot , Gerald Warnecke Institute for Applied Mathematics, Montan University Leoben,Franz Josef Straße 18, A-8700 Leoben, Austria Institut de Math´ematiques de Toulouse, CNRS UMR 5219, Universit´e de Toulouse,F–31062 Toulouse Cedex 9, France, Institute for Analysis and Numerics, Otto-von-Guericke University Magdeburg,Universit¨atsplatz 2, D-39106 Magdeburg, Germany
August 18, 2018
Abstract
The existence of weak solutions to the continuous coagulation equation with mul-tiple fragmentation is shown for a class of unbounded coagulation and fragmentation kernels,the fragmentation kernel having possibly a singularity at the origin. This result extends pre-vious ones where either boundedness of the coagulation kernel or no singularity at the originfor the fragmentation kernel were assumed.
Keywords:
Coagulation; Multiple Fragmentation; Unbounded kernels; Existence; Weak com-pactness
The continuous coagulation and multiple fragmentation equation describes the evolution of thenumber density f = f ( x, t ) of particles of volume x ≥ t ≥ ∂f ( x, t ) ∂t = 12 Z x K ( x − y, y ) f ( x − y, t ) f ( y, t ) dy − Z ∞ K ( x, y ) f ( x, t ) f ( y, t ) dy + Z ∞ x b ( x, y ) S ( y ) f ( y, t ) dy − S ( x ) f ( x, t ) , (1)with f ( x,
0) = f ( x ) ≥ . (2)The first two terms on the right-hand side of (1) accounts for the formation and disappearanceof particles as a result of coagulation events and the coagulation kernel K ( x, y ) represents the ∗ Corresponding author. Tel +43 (0)3842-402-1706; Fax +43 (0)3842-402-1702
Email address: [email protected], [email protected] x coalesce with particles of volume y . The remaining twoterms on the right-hand side of (1) describes the variation of the number density resulting fromfragmentation events which might produce more than two daughter particles, and the breakagefunction b ( x, y ) is the probability density function for the formation of particles of volume x from the particles of volume y . Note that it is non-zero only for x < y . The selection function S ( x ) describes the rate at which particles of volume x are selected to fragment. The selectionfunction S and breakage function b are defined in terms of the multiple-fragmentation kernel Γby the identities S ( x ) = Z x yx Γ( x, y ) dy, b ( x, y ) = Γ( y, x ) /S ( y ) . (3)The breakage function is assumed here to have the following properties Z y b ( x, y ) dx = N < ∞ , for all y > , b ( x, y ) = 0 for x > y, (4)and Z y xb ( x, y ) dx = y for all y > . (5)The parameter N represents the number of fragments obtained from the breakage of particlesof volume y and is assumed herein to be finite and independent of y . This is however inessentialfor the forthcoming analysis, see Remark 2.3 below. As for the condition (5), it states that thetotal volume of the fragments resulting from the splitting of a particle of volume y equals y andthus guarantees that the total volume of the system remains conserved during fragmentationevents.The existence of solutions to coagulation-fragmentation equations has already been the subjectof several papers which however are mostly devoted to the case of binary fragmentation, thatis, when the fragmentation kernel Γ satisfies the additional symmetry property Γ( x + y, y ) =Γ( x + y, x ) for all ( x, y ) ∈ ]0 , ∞ [ , see the survey [7] and the references therein. The coagulation-fragmentation equation with multiple fragmentation has received much less attention over theyears though it is already considered in the pioneering work [10], where the existence anduniqueness of solutions to (1)-(2) are established for bounded coagulation and fragmentationkernels K and Γ. A similar result was obtained later on in [9] by a different approach. Theboundedness of Γ was subsequently relaxed in [4] where it is only assumed that S grows atmost linearly, but still for a bounded coagulation kernel. Handling simultaneously unboundedcoagulation and fragmentation kernels turns out to be more delicate and, to our knowledge,is only considered in [5] for coagulation kernels K of the form K ( x, y ) = r ( x ) r ( y ) with nogrowth restriction on r and a moderate growth assumption on Γ (depending on r ) and in[3] for coagulation kernels satisfying K ( x, y ) ≤ φ ( x ) φ ( y ) for some sublinear function φ and amoderate growth assumption on Γ (see also [6] for the existence of solutions for the correspondingdiscrete model). Still, the fragmentation kernel Γ is required to be bounded near the origin in[3, 5] which thus excludes kernels frequently encountered in the literature such as Γ( y, x ) =( α + 2) x α y γ − ( α +1) with α > − γ ∈ R [8].The purpose of this note is to fill (at least partially) this gap and establish the existence ofweak solutions to (1) for simultaneously unbounded coagulation and fragmentation kernels K and Γ, the latter being possibly unbounded for small and large volumes. More precisely, we2ake the following hypotheses on the coagulation kernel K , multiple-fragmentation kernel Γ,and selection rate S . Hypotheses 1.1. (H1) K is a non-negative measurable function on [0 , ∞ [ × [0 , ∞ [ and is sym-metric, i.e. K ( x, y ) = K ( y, x ) for all x, y ∈ ]0 , ∞ [ ,(H2) K ( x, y ) ≤ φ ( x ) φ ( y ) for all x, y ∈ ]0 , ∞ [ where φ ( x ) ≤ k (1 + x ) µ for some ≤ µ < and constant k > .(H3) Γ is a non-negative measurable function on ]0 , ∞ [ × ]0 , ∞ [ such that Γ( x, y ) = 0 if < x < y .Defining S and b by (3), we assume that b satisfies (5) and there are θ ∈ [0 , and two non-negative functions k :]0 , ∞ [ → [0 , ∞ [ and ω :]0 , ∞ [ → [0 , ∞ [ such that, for each R ≥ :(H4) we have Γ( y, x ) ≤ k ( R ) y θ for y > R and x ∈ ]0 , R [ ,(H5) for y ∈ ]0 , R [ and any measurable subset E of ]0 , R [ , we have Z y E ( x )Γ( y, x ) dx ≤ ω ( R, | E | ) , y ∈ ]0 , R [ , where | E | denotes the Lebesgue measure of E , E is the indicator function of E given by E ( x ) := ( if x ∈ E, if x / ∈ E, and we assume in addition that lim δ → ω ( R, δ ) = 0 , (H6) S ∈ L ∞ ]0 , R [ . We next introduce the functional setting which will be used in this paper: define the Banachspace X with norm k · k by X = { f ∈ L (0 , ∞ ) : k f k < ∞} where k f k = Z ∞ (1 + x ) | f ( x ) | dx, together with its positive cone X + = { f ∈ X : f ≥ a.e. } . For further use, we also define the norms k f k x = Z ∞ x | f ( x ) | dx and k f k = Z ∞ | f ( x ) | dx, f ∈ X. The main result of this note is the following existence result:3 heorem 1.2.
Suppose that (H1)–(H6) hold and assume that f ∈ X + . Then (1)-(2) has aweak solution f on ]0 , ∞ [ in the sense of Definition 1.3 below. Furthermore, k f ( t ) k x ≤ k f k x for all t ≥ . Before giving some examples of coagulation and fragmentation kernels satisfying (H1)–(H6), werecall the definition of a weak solution to (1)-(2) [12].
Definition 1.3.
Let T ∈ ]0 , ∞ ] . A solution f of (1)-(2) is a non-negative function f : [0 , T [ → X + such that, for a.e. x ∈ ]0 , ∞ [ and all t ∈ [0 , T [ ,(i) s f ( x, s ) is continuous on [0 , T [ ,(ii) the following integrals are finite Z t Z ∞ K ( x, y ) f ( y, s ) dyds < ∞ and Z t Z ∞ x b ( x, y ) S ( y ) f ( y, s ) dyds < ∞ , (iii) the function f satisfies the following weak formulation of (1)-(2) f ( x, t ) = f ( x ) + Z t (cid:26) Z x K ( x − y, y ) f ( x − y, s ) f ( y, s ) dy − Z ∞ K ( x, y ) f ( x, s ) f ( y, s ) dy + Z ∞ x b ( x, y ) S ( y ) f ( y, s ) dy − S ( x ) f ( x, s ) (cid:27) ds. Coming back to (H1)-(H6), it is clear that coagulation kernels satisfying K ( x, y ) ≤ x µ y ν + x ν y µ for some µ ∈ [0 ,
1[ and ν ∈ [0 ,
1[ which are usually used in the mathematical literature satisfy(H1)-(H2), see also [3] for more complex choices. Let us now turn to fragmentation kernelswhich also fit in the classes considered in Hypotheses 1.1.Clearly, if we assume that Γ ∈ L ∞ (]0 , ∞ [ × ]0 , ∞ [)as in [3, 9], (H4) and (H5) are satisfied with k = k Γ k L ∞ , θ = 0, and ω ( R, δ ) = k Γ k L ∞ δ . Now letus take S ( y ) = y γ and b ( x, y ) = α + 2 y (cid:18) xy (cid:19) α for 0 < x < y, (6)where γ > α ≥
0, see [8, 11]. ThenΓ( y, x ) = ( α + 2) x α y γ − ( α +1) for 0 < x < y. Let us first check (H5). Given
R > y ∈ ]0 , R [, and a measurable subset E of ]0 , R [, we deduce4rom H¨older’s inequality that Z y E ( x )Γ( y, x ) dx = ( α + 2) y γ − ( α +1) Z y E ( x ) x α dx ≤ ( α + 2) y γ − ( α +1) | E | γγ +1 (cid:18) Z y x α ( γ +1) dx (cid:19) γ +1 ≤ ( α + 2) | E | γγ +1 (1 + α ( γ + 1)) − γ +1 y α + γ +1 + γ − ( α +1) ≤ C ( α, γ ) y γ γ +1 | E | γγ +1 ≤ C ( α, γ ) R γ γ +1 | E | γγ +1 . This shows that (H5) is fulfilled with ω ( R, δ ) = C ( α, γ ) R γ γ +1 δ γγ +1 . As for (H4), for 0 < x < R
1[ if γ ∈ ] α +1 , α +2[. Therefore, Theorem 1.2 providesthe existence of weak solutions to (1)-(2) for unbounded coagulation kernels K satisfying (H1)-(H2) and multiple fragmentation kernels Γ given by (6) with α ≥ γ ∈ ]0 , α + 2[. Letus however mention that some fragmentation kernels which are bounded at the origin andconsidered in [3, 5] need not satisfy (H4)-(H5). Remark 1.4.
While the requirement γ < α + 2 restricting the growth of Γ might be only of atechnical nature, the constraint γ > might be more difficult to remove. Indeed, it is well-knownthat there is an instantaneous loss of matter in the fragmentation equation when S ( x ) = x γ and γ < produced by the rapid formation of a large amount of particles with volume zero (dust),a phenomenon refered to as disintegration or shattering [8]. The case γ = 0 thus appears as aborderline case. Let us finally outline the proof of Theorem 1.2. Since the pioneering work [12], it has been real-ized that L -weak compactness techniques are a suitable way to tackle the problem of existencefor coagulation-fragmentation equations with unbounded kernels. This is thus the approachwe use hereafter, the main novelty being the proof of the estimates needed to guarantee theexpected weak compactness in L . These estimates are derived in Section 2.2 on a sequence ofunique global solutions to truncated versions of (1)-(2) constructed in Section 2.1. After estab-lishing weak equicontinuity with respect to time in Section 2.3, we extract a weakly convergentsubsequence in L and finally show that the limit function obtained from the weakly convergentsubsequence is actually a solution to (1)-(2) in Sections 2.4 and 2.5. In order to prove the existence of solutions to (1-2), we take the limit of a sequence of approx-imating equations obtained by replacing the kernel K and selection rate S by their “cut-off”5nalogues K n and S n [12], where K n ( x, y ) := ( K ( x, y ) if x + y < n, x + y ≥ n, S n ( x ) := ( S ( x ) if 0 < x < n, x ≥ n, for n ≥
1. Owing to the boundedness of K n and S n for each n ≥
1, we may argue as in [12,Theorem 3.1] or [13] to show that the approximating equation ∂f n ( x, t ) ∂t = 12 Z x K n ( x − y, y ) f n ( x − y, t ) f n ( y, t ) dy − Z n − x K n ( x, y ) f n ( x, t ) f n ( y, t ) dy + Z nx b ( x, y ) S n ( y ) f n ( y, t ) dy − S n ( x ) f n ( x, t ) , (7)with initial condition f n ( x ) := ( f ( x ) if 0 < x < n, x ≥ n. (8)has a unique non-negative solution f n ∈ C ([0 , ∞ [; L ]0 , n [) such that f n ( t ) ∈ X + for all t ≥ t ∈ [0 , ∞ [, i.e. Z n xf n ( x, t ) dx = Z n xf n ( x ) dx. (9)From now on, we extend f n by zero to ]0 , ∞ [ × [0 , ∞ [, i.e. we set f n ( x, t ) = 0 for x > n and t ≥
0. Observe that we then have the identity S n f n = Sf n .Next, we need to establish suitable estimates in order to apply the Dunford-Pettis Theorem [2, Theorem 4.21.2] and then the equicontinuity of the sequence ( f n ) n ∈ N in time to use the Arzel`a-Ascoli Theorem [1, Appendix A8.5]. This is the aim of the next two sections.
Lemma 2.1.
Assume that (H1)–(H6) hold and fix
T > . Then we have:(i) There is L ( T ) > (depending on T ) such that Z ∞ (1 + x ) f n ( x, t ) dx ≤ L ( T ) for n ≥ and all t ∈ [0 , T ] , (ii) For any ε > there exists R ε > such that for all t ∈ [0 , T ]sup n ≥ (cid:26)Z ∞ R ε f n ( x, t ) dx (cid:27) ≤ ε, (iii) given ε > there exists δ ε > such that, for every measurable set E of ]0 , ∞ [ with | E | ≤ δ ε , n ≥ , and t ∈ [0 , T ] , Z E f n ( x, t ) dx < ε. roof. (i) Let n ≥ t ∈ [0 , T ]. Integrating (7) with respect to x over ]0 ,
1[ and using Fubini’sTheorem, we have ddt Z f n ( x, t ) dx = − Z Z − x K n ( x, y ) f n ( x, t ) f n ( y, t ) dydx − Z Z n − x − x K n ( x, y ) f n ( x, t ) f n ( y, t ) dydx + Z Z nx b ( x, y ) S ( y ) f n ( y, t ) dydx − Z S ( x ) f n ( x, t ) dx. Since K n , f n , and S are non-negative and Γ satisfies (3), we have ddt Z f n ( x, t ) dx ≤ Z Z nx b ( x, y ) S ( y ) f n ( y, t ) dydx = Z Z x Γ( y, x ) f n ( y, t ) dydx + Z Z n Γ( y, x ) f n ( y, t ) dydx, Using Fubini’s Theorem and (H5) (with R = 1 and E =]0 , R = 1) in the second one, we obtain ddt Z f n ( x, t ) dx ≤ Z f n ( y, t ) Z y Γ( y, x ) dxdy + k (1) Z Z n yf n ( y, t ) dydx ≤ ω (1 , Z f n ( x, t ) dx + k (1) k f n ( t ) k x . (10)Recalling that k f n ( t ) k x = k f n (0) k x ≤ k f k for t ≥ ddt Z f n ( x, t ) dx ≤ ω (1 , Z f n ( y, t ) dy + k (1) k f k . Integrating with respect to time, we end up with Z f n ( x, t ) dx ≤ k f k (cid:18) k (1) ω (1 , (cid:19) exp( ω (1 , t ) , t ∈ [0 , T ] . Using (9) again we may estimate Z ∞ (1 + x ) f n ( x, t ) dx = Z f n ( x, t ) dx + Z n f n ( x, t ) dx + Z n xf n ( x, t ) dx ≤ Z f n ( x, t ) dx + Z n xf n ( x, t ) dx + k f k≤ k f k (cid:20)(cid:18) k (1) ω (1 , (cid:19) exp( ω (1 , T ) + 2 (cid:21) =: L ( T ) . (ii) For ε >
0, set R ε := k f k /ε . Then, by (9), for each n ≥ t ∈ [0 , T ] we have Z ∞ R ε f n ( x, t ) dx ≤ R ε Z ∞ R ε xf n ( x, t ) dx ≤ k f k R ε < ε. R >
0. For n ≥ δ ∈ (0 , t ∈ [0 , T ], we define p n ( δ, t ) = sup (cid:26)Z R E ( x ) f n ( x, t ) dx : E ⊂ ]0 , R [ and | E | ≤ δ (cid:27) . Consider a measurable subset E ⊂ ]0 , R [ with | E | ≤ δ . For n ≥ t ∈ [0 , T ], it follows fromthe non-negativity of f n , (3) and (7)-(8) that ddt Z R E ( x ) f n ( x, t ) dx ≤ I n ( t ) + I n ( t ) + I n ( t ) , (11)where I n ( t ) := Z R E ( x ) Z x K n ( x − y, y ) f n ( x − y, t ) f n ( y, t ) dydx,I n ( t ) := Z R E ( x ) Z Rx Γ( y, x ) f n ( y, t ) dydx,I n ( t ) := Z R E ( x ) Z ∞ R Γ( y, x ) f n ( y, t ) dydx. First, applying Fubini’s Theorem to I n ( t ) gives I n ( t ) = Z R f n ( y, t ) Z Ry E ( x ) K n ( y, x − y ) f n ( x − y, t ) dxdy = Z R f n ( y, t ) Z R − y E ( x + y ) K n ( y, x ) f n ( x, t ) dxdy. Setting − y + E := { z > z = − y + x for some x ∈ E } , it follows from (H2) and the aboveidentity that I n ( t ) ≤ k (1 + R ) µ Z R (1 + y ) µ f n ( y, t ) Z R f n ( x, t ) − y + E ∩ ]0 ,R − y [ ( x ) dxdy. Since − y + E ∩ ]0 , R − y [ ⊂ ]0 , R [ and |− y + E ∩ ]0 , R − y [ | ≤ | − y + E | = | E | ≤ δ , we infer fromthe definition of p n ( δ, t ) and Lemma 2.1 (i) that I n ( t ) ≤ k (1 + R ) µ (cid:18)Z R (1 + y ) µ f n ( y, t ) dy (cid:19) p n ( δ, t ) ≤ k L ( T )(1 + R ) µ p n ( δ, t ) . Next, applying Fubini’s Theorem to I n ( t ) and using (H5) and Lemma 2.1 (i) give I n ( t ) = Z R f n ( y, t ) Z y E ( x )Γ( y, x ) dxdy ≤ ω ( R, | E | ) Z R f n ( y, t ) dy ≤ L ( T ) ω ( R, | E | ) . Finally, owing to (H4) and (9), we have I n ( t ) ≤ k ( R ) Z R Z ∞ R E ( x ) y θ f n ( y, t ) dydx ≤ k ( R ) R θ − | E | Z ∞ R yf n ( y, t ) dy ≤ k ( R ) R θ − k f k | E | ≤ k ( R ) R θ − k f k δ. I nj ( t ), 1 ≤ j ≤
3, we infer from (11) that there is C ( R, T ) > ddt Z R E ( x ) f n ( x, t ) dx ≤ C ( R, T ) ( p n ( δ, t ) + ω ( R, δ ) + δ ) . Integrating with respect to time and taking the supremum over all E such that E ⊂ ]0 , R [ with | E | ≤ δ give p n ( δ, t ) ≤ p n ( δ,
0) +
T C ( R, T )[ ω ( R, δ ) + δ ] + C ( R, T ) Z t p n ( δ, s ) ds, t ∈ [0 , T ] . By Gronwall’s inequality (see e.g. [14, p. 310]), we obtain p n ( δ, t ) ≤ [ p n ( δ,
0) +
T C ( R, T )( ω ( R, δ ) + δ )] exp { C ( R, T ) t } , t ∈ [0 , T ] . (12)Now, since f n ( x, ≤ f ( x ) for x >
0, the absolute continuity of the integral guarantees thatsup n { p n ( δ, } → δ → δ → sup n ≥ ,t ∈ [0 ,T ] { p n ( δ, t ) } = 0 . Lemma 2.1 (iii) is then a straightforward consequence of this property and Lemma 2.1 (i).Lemma 2.1 and the
Dunford-Pettis Theorem imply that, for each t ∈ [0 , T ], the sequence offunctions ( f n ( t )) n ≥ lies in a weakly relatively compact set of L ]0 , ∞ [ which does not dependon t ∈ [0 , T ]. Now we proceed to show the time equicontinuity of the sequence ( f n ) n ∈ N . Though the coagula-tion terms can be handled as in [3, 5, 12], we sketch the proof below for the sake of completeness.Let T > ε >
0, and φ ∈ L ∞ ]0 , ∞ [ and consider s, t ∈ [0 , T ] with t ≥ s . Fix R > L ( T ) R < ε , (13)the constant L ( T ) being defined in Lemma 2.1 (i). For each n , by Lemma 2.1 (i), Z ∞ R | f n ( x, t ) − f n ( x, s ) | dx ≤ R Z ∞ R x { f n ( x, t ) + f n ( x, s ) } dx ≤ L ( T ) R . (14)By (7), (13), and (14), we get (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ φ ( x ) { f n ( x, t ) − f n ( x, s ) } dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z R φ ( x ) { f n ( x, t ) − f n ( x, s ) } dx (cid:12)(cid:12)(cid:12)(cid:12) + Z ∞ R | φ ( x ) || f n ( x, t ) − f n ( x, s ) | dx ≤k φ k L ∞ Z ts (cid:20) Z R Z x K n ( x − y, y ) f n ( x − y, τ ) f n ( y, τ ) dydx + Z R Z n − x K n ( x, y ) f n ( x, τ ) f n ( y, τ ) dydx + Z R Z nx b ( x, y ) S ( y ) f n ( y, τ ) dydx + Z R S ( x ) f n ( x, τ ) dx (cid:21) dτ + k φ k L ∞ ε . (15)9y Fubini’s Theorem, (H2), and Lemma 2.1 (i), the first term of the right-hand side of (15) maybe estimated as follows:12 Z R Z x K n ( x − y, y ) f n ( x − y, τ ) f n ( y, τ ) dydx = 12 Z R Z Ry K n ( x − y, y ) f n ( x − y, τ ) f n ( y, τ ) dxdy = 12 Z R Z R − y K n ( x, y ) f n ( x, τ ) f n ( y, τ ) dxdy ≤ k Z R Z R − y (1 + x ) µ (1 + y ) µ f n ( x, τ ) f n ( y, τ ) dydx ≤ k L ( T ) . Similarly, for the second term of the right-hand side of (15), it follows from (H2) that Z R Z n − x K n ( x, y ) f n ( x, τ ) f n ( y, τ ) dydx ≤ k Z R Z n − x (1 + x ) µ (1 + y ) µ f n ( x, τ ) f n ( y, τ ) dydx ≤ k L ( T ) . For the third term of the right-hand side of (15), we use Fubini’s Theorem, (H4), (H5), andLemma 2.1 (i) to obtain Z R Z nx b ( x, y ) S ( y ) f n ( y, τ ) dydx ≤ Z R Z y Γ( y, x ) f n ( y, τ ) dxdy + Z R Z ∞ R Γ( y, x ) f n ( y, τ ) dydx ≤ Z R f n ( y, τ ) Z y ]0 ,R [ ( x )Γ( y, x ) dxdy + k ( R ) Z R Z ∞ R y θ f n ( y, τ ) dydx ≤ ω ( R, R ) Z R f n ( y, τ ) dy + k ( R ) Z R Z ∞ R yf n ( y, τ ) dydx ≤ [ ω ( R, R ) + Rk ( R )] L ( T ) . Finally, the fourth term of the right-hand side of (15) is estimated with the help of (H6) andLemma 2.1 (i) and we get Z R S ( x ) f n ( x, t ) dx ≤ k S k L ∞ ]0 ,R [ L ( T ) . Collecting the above estimates and setting C ( R, T ) = 3 k L ( T ) (cid:8) ω ( R, R ) + Rk ( R ) + k S k L ∞ ]0 ,R [ (cid:9) L ( T )the inequality (15) reduces to (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ φ ( x ) { f n ( x, t ) − f n ( x, s ) } dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( R, T ) k φ k L ∞ ( t − s ) + k φ k L ∞ ε < k φ k L ∞ ε, (16)10henever t − s < δ for some suitably small δ >
0. The estimate (16) implies the time equicon-tinuity of the family { f n ( t ) , t ∈ [0 , T ] } in L ]0 , ∞ [. Thus, according to a refined version of the Arzel`a-Ascoli Theorem , see [12, Theorem 2.1], we conclude that there exist a subsequence ( f n k )and a non-negative function f ∈ L ∞ (]0 , T [; L ]0 , ∞ [) such thatlim n k →∞ sup t ∈ [0 ,T ] (cid:26)(cid:12)(cid:12)(cid:12)(cid:12)Z ∞ { f n k ( x, t ) − f ( x, t ) } φ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:27) = 0 , (17)for all T > φ ∈ L ∞ ]0 , ∞ [. In particular, it follows from the non-negativity of f n and f ,(9), and (17) that, for t ≥ R > Z R xf ( x, t ) dx = lim n k →∞ Z R xf n k ( x, t ) dx ≤ k f k x < ∞ . Letting R → ∞ implies that k f ( t ) k x ≤ k f k x and thus f ( t ) ∈ X + . Now we have to show that the limit function f obtained in (17) is actually a weak solution to(1)-(2). To this end, we shall use weak continuity and convergence properties of some operatorswhich define now: for g ∈ X + , n ≥
1, and x ∈ ]0 , ∞ [, we put Q n ( g )( x ) = 12 Z x K n ( x − y, y ) g ( x − y ) g ( y ) dy, Q n ( g )( x ) = Z n − x K n ( x, y ) g ( x ) g ( y ) dy,Q ( g )( x ) = 12 Z x K ( x − y, y ) g ( x − y ) g ( y ) dy, Q ( g )( x ) = Z ∞ K ( x, y ) g ( x ) g ( y ) dy,Q ( g )( x ) = S ( x ) g ( x ) , Q ( g )( x ) = Z ∞ x b ( x, y ) S ( y ) g ( y ) dy, and Q n = Q n − Q n − Q + Q , Q = Q − Q − Q + Q .We then have the following result: Lemma 2.2.
Let ( g n ) n ∈ N be a bounded sequence in X + , || g n || ≤ L , and g ∈ X + such that g n ⇀ g in L ]0 , ∞ [ as n → ∞ . Then, for each R > and i ∈ { , . . . , } , we have Q ni ( g n ) ⇀ Q i ( g ) in L ]0 , R [ as n → ∞ . (18) Proof.
The proof of (18) for i = 1 , i = 3 is obvious since φS belongs to L ∞ ]0 , R [ by (H6) and (18) follows at once from the weakconvergence of ( g n ) in L ]0 , ∞ [. For i = 4, we consider φ ∈ L ∞ ]0 , R [ and use (3) and Fubini’sTheorem to compute, for r > R , (cid:12)(cid:12)(cid:12)(cid:12)Z R φ ( x ) { Q ( g n )( x ) − Q ( g )( x ) } dx (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z R Z ∞ x φ ( x ) S ( y ) b ( x, y ) { g n ( y ) − g ( y ) } dydx (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z R Z y φ ( x ) S ( y ) b ( x, y ) { g n ( y ) − g ( y ) } dxdy (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ R Z R φ ( x )Γ( y, x ) { g n ( y ) − g ( y ) } dxdy (cid:12)(cid:12)(cid:12)(cid:12) . (cid:12)(cid:12)(cid:12)(cid:12)Z R φ ( x ) { Q ( g n )( x ) − Q ( g )( x ) } dx (cid:12)(cid:12)(cid:12)(cid:12) = J n + J n ( r ) + J n ( r ) , (19)with J n = (cid:12)(cid:12)(cid:12)(cid:12)Z R { g n ( y ) − g ( y ) } Z y φ ( x ) S ( y ) b ( x, y ) dxdy (cid:12)(cid:12)(cid:12)(cid:12) J n ( r ) = (cid:12)(cid:12)(cid:12)(cid:12)Z rR { g n ( y ) − g ( y ) } Z R φ ( x )Γ( y, x ) dxdy (cid:12)(cid:12)(cid:12)(cid:12) J n ( r ) = (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ r { g n ( y ) − g ( y ) } Z R φ ( x )Γ( y, x ) dxdy (cid:12)(cid:12)(cid:12)(cid:12) . We use (H6) and (4) to observe that, for y ∈ ]0 , R [, (cid:12)(cid:12)(cid:12)(cid:12)Z y φ ( x ) S ( y ) b ( x, y ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ k S k L ∞ ]0 ,R [ k φ k L ∞ ]0 ,R [ Z y b ( x, y ) dx ≤ N k S k L ∞ ]0 ,R [ k φ k L ∞ ]0 ,R [ . This shows that the function y R y φ ( x )Γ( y, x ) dx belongs to L ∞ ]0 , R [. Since g n ⇀ g in L ]0 , ∞ [ as n → ∞ , it thus follows that lim n →∞ J n = 0 . (20)We next infer from (H4) that, for y ∈ ]0 , R [, (cid:12)(cid:12)(cid:12)(cid:12)Z R φ ( x )Γ( y, x ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ k ( R ) y θ Z R φ ( x ) dx ≤ Rk ( R ) k φ k L ∞ ]0 ,R [ y θ . (21)On the one hand, (21) guarantees that the function y R R φ ( x )Γ( y, x ) dx belongs to L ∞ ] R, r [and the weak convergence of ( g n ) to g in L ]0 , ∞ [ entails thatlim n →∞ J n ( r ) = 0 for all r > R. (22)On the other hand, we deduce from (21) and the boundedness of ( g n ) and g in X + that (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ r { g n ( y ) − g ( y ) } Z R φ ( x )Γ( y, x ) dxdy (cid:12)(cid:12)(cid:12)(cid:12) ≤ Rk ( R ) k φ k L ∞ ]0 ,R [ Z ∞ r y θ { g n ( y ) + g ( y ) } dy ≤ Rk ( R )( L + k g k ) r − θ k φ k L ∞ ]0 ,R [ which is asymptotically small (as r → ∞ ) uniformly with respect to n . We thus conclude thatlim r →∞ sup n ≥ { J n ( r ) } = 0 . (23)Substituting (20) and (22) into (19), we obtainlim sup n →∞ (cid:12)(cid:12)(cid:12)(cid:12)Z R φ ( x ) { Q ( g n )( x ) − Q ( g )( x ) } dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ sup n ≥ { J n ( r ) } for all r > R . Owing to (23), we may let r → ∞ and conclude that (18) holds true for i = 4thanks to the arbitrariness of φ and the proof of Lemma 2.2 is complete.12 .5 Existence Now we are in a position to prove the main result.
Proof of Theorem 1.2.
Fix
R > T >
0, and consider t ∈ [0 , T ] and φ ∈ L ∞ ]0 , R [. Owing toLemma 2.2, we have for each s ∈ [0 , t ], Z R φ ( x ) { Q n k ( f n k ( s ))( x ) − Q ( f ( s ))( x ) } dx → as n k → ∞ . (24)Arguing as in Section 2.3, it follows from (H2), (H4)–(H6), and Lemma 2.1 (i) that there is C ( R, T ) > n ≥
1, and s ∈ [0 , t ], we have (cid:12)(cid:12)(cid:12)(cid:12)Z R φ ( x ) Q n ( f n ( s ))( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( R, T ) k φ k L ∞ ]0 ,R [ . (25)Since the right-hand side of (25) is in L ]0 , t [, it follows from (24), (25) and the dominatedconvergence theorem that (cid:12)(cid:12)(cid:12)(cid:12)Z t Z R φ ( x ) { Q n k ( f n k ( s ))( x ) − Q ( f ( s ))( x ) } dxds (cid:12)(cid:12)(cid:12)(cid:12) → n k → ∞ . (26)Since φ is arbitrary in L ∞ ]0 , R [, Fubini’s Theorem and (26) give Z t Q n k ( f n k ( s )) ds ⇀ Z t Q ( f ( s )) ds in L ]0 , R [ as n k → ∞ . (27)It is then straightforward to pass to the limit as n k → ∞ in (7)-(8) and conclude that f is asolution to (1)-(2) on [0 , ∞ [ (since T is arbitrary). This completes the proof of Theorem 1.2. Remark 2.3.
It is worth pointing out that the assumption (4) R y b ( x, y ) dx = N is only used toprove (20) and it is clear from that proof that the assumption sup y ∈ ]0 ,R [ Z y b ( x, y ) dx < ∞ for all R > is sufficient. Thus, Theorem 1.2 is actually valid under this weaker assumption. Acknowledgments
A.K. Giri would like to thank International Max-Planck Research School, Magdeburg, Germanyand FWF Austrian Science Fund for their support. Part of this work was done while Ph.Lauren¸cot enjoys the hospitality and support of the Isaac Newton Institute for MathematicalSciences, Cambridge, UK. 13 eferences [1] R.B. Ash.
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