3.1. Mathematical Setting and Analysis
We note that the operators on the left-hand side of (12) have the property that one of the variables is a parameter and, for each value of this parameter, the operator has a certain desirable property (like being the generator of a semigroup) with respect to the other variable. Thus we need to work with parameter-dependent operators that can be “glued” together in such a way that the resulting operator inherits the properties of the individual ones. Let us provide a framework for such a technique called the method of semigroups with a parameter [2].
Let Λ=ℝ3×ℕ and consider the space 𝒳:=Lp(V,X) where 1≤p<∞, (V,dm) is a measure space and X a Banach space. Let us suppose that we are given a family of operators {(Av,D(Av))}v∈V in X and define the operator (𝔸,D(𝔸)) acting in 𝒳 according to the following formulae:
(14)𝒟(𝔸):={g∈𝒳;g(v)∈D(Av) for almost every: =v∈V, 𝔸g∈𝒳;g(v)∈D(Av)},
and, for g∈𝒟(𝔸),
(15)(𝔸g)(v):=Avg(v)
for every v∈𝒳1. We have the following proposition.
Proposition 2 (see [2, Proposition 3.28]).
If for almost any v∈V the operator Av is m-dissipative in X, and the function v→R(λ,Av)g(v) is measurable for any λ>0 and g∈𝒳, then the operator 𝔸 is an m-dissipative operator in 𝒳. If (Gv(t))t≥0 and (𝒢(t))t≥0 are the semigroups generated by Av and 𝔸, respectively, then for almost every v∈V, t≥0, and g∈𝒳 we have
(16)[𝒢(t)g](v):=Gv(t)g(v).
Using the above ideas we introduce operators that cover the case of the present applications. Here the n variable is the parameter and x is the main variable. We set
(17)Xx:=L1(ℝ3,dx):={ψ:=∥ψ∥=∫ℝ3|ψ(x,n)|dx<∞}
and define in Xx the operators (𝒜n,D(𝒜n)) as
(18)𝒜np(t,x,n)=an(x)p(t,x,n),D(𝒜n):={pn∈Xx,𝒜npn∈Xx}, n∈ℕ.
Using the above proposition, we can take 𝔸=A, 𝒳=𝒳1=L1(ℕ,Xx)=L1(Λ,dμdς)=L1(ℝ3×ℕ,dμdς), where ℕ is equipped with the counting measure dς and dμ=dx is the Lebesgue measure in ℝ3. In the notation of the proposition, (ℕ,dς)=(V,dm), Xx=X, and Av=𝒜n; therefore (𝒜n,D(𝒜n))n∈ℕ is a family of operators in Xx, and using (15), we have
(19)(Ap)n:=𝒜npn.
Theorem 3.
There is an extension K of A+B that generates a positive semigroup of contractions (SK(t))t≥0 on 𝒳1. Moreover, for each po =(pon(x))n∈ℕ∈D(K) there is a measurable representation p of SK(t)po which is absolutely continuous with respect to t≥0 for almost any (x,n) and such that (10) is satisfied almost everywhere.
Proof.
To prove the first part of the theorem, let’s show that for each n∈ℕ, 𝒜n generates a positive semigroup of contractions. This assertion follows from (1) and the fact that the operator 𝒜n is a multiplication operator on Xx induced by the measurable function a; then, it is closed and densely defined [12]. Also for any λ>0, it is obvious that λI-𝒜n is bijective and the resolvent R(λ,𝒜n) of 𝒜n satisfies the estimate
(20)∥R(λ,𝒜n)ψ∥≤1λ∥ψ∥
for any ψ∈Xx. Furthermore for any positive λ, the operator R(λ,𝒜n) is nonnegative. Therefore (𝒜n,D(𝒜n)) generates a positive semigroup of contractions. Thus, using relation (16) of Proposition 2, we claim that (A,D(A)) also generates a positive semigroup of contractions.
It is clear that (B,D(B)) is positive. Furthermore for any p∈D(A), by the calculations in the Lemma 1, we have ∥Ag∥1=∥Bg∥1. Then
(21)∫Λ(-Ap+Bp)dμ dς =-∫ℝ3∑m=1∞mam(y)|g(y,m)|dy +∫ℝ3(∑n=1∞n∑m=n+1∞∫ℝ3am(y)bn,m(y)h(x,n,m,y) ×|g(y,m)|dy∑n=1∞n∑m=n+1∞∫ℝ3)dx =0.
Thus the assumptions of Kato’s Theorem in L1 (see [2, Corollary 5.17]) are satisfied. Therefore there is an extension K of A+B generating a substochastic semigroup (GK(t))t≥0. Let Kn be the nth of K according to (19) and Proposition 2; then from (16), it suffices to prove the assertions of the theorem for each Kn, n∈ℕ. For any pon∈D(Kn), the function t→GKn(t)pon is a C1-function in the norm of Xx and satisfies the equation
(22)ddtGKn(t)pon=KnGKn(t)pon,
where the equality holds for any t>0 in the sense of equality in Xx. The initial condition is satisfied in the following sense:
(23)limt→0+GKn(t)pon=pon,
where the convergence is in the Xx-norm.
In order to prove the second part of this theorem we make use of the theory of extensions and the theory of L spaces [2]. Let Θ be the set of finite almost everywhere measurable functions defined on ℝ3. We recall that Θ is a lattice with respect to the usual relation (≤almost everywhere), Xx⊂Θ, and Xx is a sublattice of Θ. We denote by (Xx)+ and Θ+ the positive cones of Xx and Θ, respectively. For each n∈ℕ we introduce the operator Dn defined such that for any nondecreasing sequence (ψk)k∈ℕ in (Xx)+ with supk∈ℕψk=ψ∈Θ+,
(24)Dnψ:=supk∈ℕℬnψk,
where ℬn is given by 𝔅p=(ℬn)n=1∞ defined in (11). Since ℬn is an integral operator with positive kernel, Lebesgue’s monotone convergence theorem yields that Dn=ℬn. Thus, [2, Theorem 6.20] yields Kn⊂𝒜n+ℬn. Hence GKn(t)pon satisfies
(25)[ddtGKn(t)pon](x,n)=[𝒜nGKn(t)pon](x,n) +[ℬnGKn(t)pon](x,n),
for each fixed t>0, where the right-hand side does not depend (in the sense of equality almost everywhere) on what representation of the solution GKn(t)pon is considered. Making use of the fact that Xx is an L space, from [13, Theorem 3.4.2], we have that since the function GKn(t)pon is strongly differentiable, there is a representation p(t,x,n)=pn of GKn(t)pon that is absolutely continuous with respect to t≥0 for almost any (x,n)∈ℝ+×ℝ3, and that satisfies (∂/∂t)p(t,x,n)=[(d/dt)GKn(t)pon](x,n) for any t≥0 and almost any (x,n). Hence, taking this representation, we obtain that
(26)∂p∂t(t,x,n) =-an(x)pn +∑m=n+1∞∫ℝ3am(y)bn,m(y)h(x,n,m,y)pm(y)dy
holds almost everywhere on ℝ+×ℝ3. Moreover, the continuity of pn with respect to t for almost every (x,n) shows that limt→0+pn=p¯(x,n) exists almost everywhere. From (23) we see that there is a sequence (tk)k∈ℕ converging to 0 such that limk→∞p(tk,x,n)=pon(x,n), for almost every (x,n). Here we can use the same representation as above because we are dealing with a (countable) sequence. Indeed, changing the representation on a set of measure zero for each n and further taking the union of all these sets still produce a set of measure zero. Thus pon=p¯n almost everywhere.
In general for each n∈ℕ, the function GKn(t)pon is not differentiable if pon∈Xx∖D(Kn). Therefore it cannot be a classical solution of the Cauchy problem (22) and (23). However it is a mild solution, that is, it is a continuous function such that ∫0tpn(τ)dτ∈D(Kn) for any t≥0, satisfying the integrated version of (22) and (23):
(27)pn(t)=pon+Kn∫0tpn(τ)dτ.
Corollary 4.
If pon∈Xx∖D(Kn), then pn=[GKn(t)pon](x,n) satisfies the equation
(28)p(t,x,n)=pon(x,n)-an(x)∫0tp(τ,x,n)dτ +∑m=n+1∞∫ℝ3am(y)bn,m(y)h(x,n,m,y)pm(y) + ∑m=n+1∞∫ℝ3×(∫0tp(τ,y,n)dτ)dy.
Proof.
Because pn is continuous in the norm of Xx=L1(ℝ3,dx), we can use the fact that Xx is of type L (see [2, Theorem 2.39]) to claim that aan(x)∫0tp(τ,x,n)dτ is defined for almost any (x,n) and any t, and hence we can write
(29)[(𝒜n+ℬn)∫0tp(τ)dτ](x,n) =-an(x)∫0tp(τ,x,n)dτ +∑m=n+1∞∫ℝ3am(y)bn,m(y)h(x,n,m,y)pm(y) +∑m=n+1∞∫ℝ3×(∫0tp(τ,y,n)dτ)dy.
Thus, combining the result used in the previous theorem that Kn⊂𝒜n+ℬn with (27) we obtain (28).
Next we provide a fairly general condition for honesty of (GKn(t))t≥0.