On Two-Parameter Regularized Semigroups and the Cauchy Problem

Abstract

Suppose that 𝑋 is a Banach space and 𝐶 is an injective operator in 𝐵(𝑋), the space of all bounded linear operators on 𝑋. In this note, a two-parameter 𝐶-semigroup (regularized semigroup) of operators is introduced, and some of its properties are discussed. As an application we show that the existence and uniqueness of solution of the 2-abstract Cauchy problem (𝜕/(𝜕𝑡_𝑖))𝑢(𝑡₁,𝑡₂)=𝐻_𝑖𝑢(𝑡₁,𝑡₂),𝑖=1,2, 𝑡_𝑖>0, 𝑢(0,0)=𝑥, 𝑥∈𝐶(𝐷(𝐻₁)∩𝐷(𝐻₂)) is closely related to the two-parameter 𝐶-semigroups of operators.

1. Introduction and Preliminaries

Suppose that 𝑋 is a Banach space and 𝐴 is a linear operator in 𝑋 with domain 𝐷(𝐴) and range 𝑅(𝐴). For a given 𝑥∈𝐷(𝐴), the abstract Cauchy problem for 𝐴 with the initial value 𝑥 consists of finding a solution 𝑢(𝑡) to the initial value problem ACP(𝐴;𝑥)𝑑𝑢(𝑡)𝑑𝑡=𝐴𝑢(𝑡),𝑡∈ℝ₊,𝑢(0)=𝑥, (1.1) where by a solution we mean a function 𝑢∶ℝ₊→𝑋, which is continuous for 𝑡≥0, continuously differentiable for 𝑡>0, 𝑢(𝑡)∈𝐷(𝐴) for 𝑡∈ℝ₊, and 𝐴𝐶𝑃(𝐴;𝑥) is satisfied.

If 𝐶∈𝐵(𝑋), the space of all bounded linear operators on X, is injective, then a one-parameter 𝐶-semigroup (regularized semigroup) of operators is a family {𝑇(𝑡)}_𝑡∈ℝ+⊂𝐵(𝑋) for which 𝑇(0)=𝐶, 𝑇(𝑠+𝑡)𝐶=𝑇(𝑠)𝑇(𝑡), and for each 𝑥∈𝑋, the mapping 𝑡↦𝑇(𝑡)𝑥 is continuous. An operator 𝐴∶𝐷(𝐴)→𝑋 with 𝐷(𝐴)={𝑥∈𝑋∶lim_𝑡→0𝑇(𝑡)𝑥−𝐶𝑥𝑡existsintherangeof𝐶}, (1.2) and where, for 𝑥∈𝐷(𝐴), 𝐴𝑥∶=𝐶⁻¹lim_𝑡→0((𝑇(𝑡)𝑥−𝐶𝑥)/𝑡) is called the infinitesimal generator of 𝑇(𝑡).

Regularized semigroups and their connection with the 𝐴𝐶𝑃(𝐴;𝑥) have been studied in [1–6] and some other papers. Also the concept of local 𝐶-semigroups and their relation with the 𝐴𝐶𝑃(𝐴;𝑥) have been considered in [7–10].

In Section 2, we introduce the concept of two-parameter regularized semigroups of operators and their generator. Some basic properties of two-parameter regularized semigroups and their relation with the generators are studied in this section.

In Section 3, two-parameter abstract Cauchy problems are considered. It is proved that the existence and uniqueness of its solutions is closely related with two-parameter regularized semigroups of operators.

2. Two-Parameter Regularized Semigroups

In this section we introduce two-parameter regularized semigroup and its generator on Banach spaces. Then some properties of two-parameter regularized semigroups are studied.

Definition 2.1. Suppose that 𝑋 is a Banach space and 𝐶∈𝐵(𝑋) is an injective operator. A family {𝑊(𝑠,𝑡)}_{𝑠,𝑡∈ℝ+}⊂𝐵(𝑋) is called a two-parameter regularized semigroup (or two parameter 𝐶-semigroup) if(i)𝑊(0,0)=𝐶,(ii)𝑊(𝑠+𝑠^′,𝑡+𝑡^′)𝐶=𝑊(𝑠,𝑡)𝑊(𝑠^′,𝑡^′), for all 𝑠,𝑠^′,𝑡,𝑡^′∈ℝ₊,(iii)lim_{(𝑠′,𝑡′)→(𝑠,𝑡)}𝑊(𝑠^′,𝑡^′)𝑥=𝑊(𝑠,𝑡)𝑥, for all 𝑥∈𝑋.It is called exponentially bounded if ‖𝑊(𝑠,𝑡)‖≤𝑀𝑒^{(𝑠+𝑡)𝜔}, for some 𝑀,𝜔>0.

Suppose that {𝑊(𝑠,𝑡)}_{𝑠,𝑡∈ℝ+} is a two-parameter 𝐶-semigroup. Put 𝑢(𝑠)∶=𝑊(𝑠,0) and 𝑣(𝑡)∶=𝑊(0,𝑡), then it is easy to see that these families are two commuting one-parameter 𝐶-semigroups such that 𝑊(𝑠,𝑡)𝐶=𝑢(𝑠)𝑣(𝑡). Also 𝑢(𝑠) and 𝑣(𝑡) commute with 𝐶. If 𝐻₁ and 𝐻₂ are their generators, respectively, then we will think of (𝐻₁,𝐻₂) as the generator of 𝑊(𝑠,𝑡).

From the one-parameter case (see [8]), one can prove that 𝑅(𝐶)⊆𝐷(𝐻₁)∩𝐷(𝐻₂), and 𝐶⁻¹𝐻_𝑖𝐶=𝐻_𝑖, 𝑖=1,2.

Also if {𝑈(𝑠)}_𝑠∈ℝ+ and {𝑉(𝑡)}_𝑡∈ℝ+ are two commuting one-parameter 𝐶-semigroups, then one can see that 𝑊(𝑠,𝑡)∶=𝑈(𝑠)𝑉(𝑡) is a two-parameter 𝐶²-semigroup of operators.

The following is an example of a two-parameter 𝐶-semigroup which is not exponentially bounded.

Example 2.2. Let 𝑋=𝐿²(ℂ), and [𝑊(𝑠,𝑡)𝑓](𝑧)∶=𝑒^{−|𝑧|2+(𝑠+𝑡)𝑧}𝑓(𝑧), (𝐶𝑓)(𝑧)∶=𝑒^−|𝑧|2𝑓(𝑧), then 𝑊(𝑠,𝑡) is a two-parameter 𝐶-semigroup which is not exponentially bounded.

In the following theorem we can see some elementary properties of a two-parameter 𝐶-semigroup.

Theorem 2.3. Suppose that 𝑊(𝑠,𝑡) is a two-parameter 𝐶-semigroup with the infinitesimal generator (𝐻₁,𝐻₂). Then, one has the following.(i)For each 𝑥∈𝑋 and for every 𝑠,𝑡≥0, ∫^𝑡₀∫^𝑠₀𝑊(𝜇,𝜈)𝑥𝑑𝜇𝑑𝜈, is in 𝐷(𝐻₁)∩𝐷(𝐻₂). Also lim_{(ℎ,𝑘)→(0,0)}1ℎ𝑘_𝑡^𝑡+ℎ_𝑠^𝑠+𝑘𝑊(𝜇,𝜈)𝑥𝑑𝜇𝑑𝜈=𝑊(𝑠,𝑡)𝑥. (2.1)(ii)For each 𝑥∈𝑋, and for every 𝑠,𝑡∈ℝ₊, ∫^𝑠₀𝑊(𝜇,𝑡)𝑥𝑑𝜇∈𝐷(𝐻₁) and ∫^𝑡₀𝑊(𝑠,𝜈)𝑥𝑑𝜈∈𝐷(𝐻₂); furthermore 𝐻₁^𝑠₀𝐻𝑊(𝜇,𝑡)𝑥𝑑𝜇=𝑊(𝑠,𝑡)𝑥−𝑊(0,𝑡)𝑥,_2𝑡0𝑊(𝑠,𝜈)𝑥𝑑𝜈=𝑊(𝑠,𝑡)𝑥−𝑊(𝑠,0)𝑥. (2.2)(iii)𝑅(𝐶)⊆𝐷(𝐻₁)∩𝐷(𝐻₂) and 𝐻₁ and 𝐻₂ are closed.(iv)For any 𝑥∈𝐷(𝐻₁)∩𝐷(𝐻₂), and each 𝑠,𝑡>0, 𝑢(𝑠)𝑥 and 𝑣(𝑡)𝑥 are in 𝐷(𝐻₁)∩𝐷(𝐻₂). Also for this 𝑥, and 𝑖=1,2, 𝜕𝜕𝑡_𝑖𝑊(𝑡₁,𝑡₂)𝑥=𝐻_𝑖𝑊(𝑡₁,𝑡₂)𝑥=𝑊(𝑡₁,𝑡₂)𝐻_𝑖𝑥. (2.3)(v)For any 𝑎,𝑏>0, 𝑇(𝑡)∶=𝑊(𝑡𝑎,𝑡𝑏) is a one-parameter 𝐶-semigroup whose generator is an extension of 𝑎𝐻₁+𝑏𝐻₂.

Proof. To prove (i), suppose 𝑥∈𝑋. First we note that for any 𝜈≥0, lim_ℎ→01ℎ_𝑡^𝑡+ℎ𝑊(𝜇,𝜈)𝐶𝑥𝑑𝜇=𝑊(0,𝜈)lim_ℎ→01ℎ_𝑡^𝑡+ℎ𝑊(𝜇,0)𝑥𝑑𝜇=𝑊(0,𝜈)𝑊(𝑡,0)𝑥=𝑊(𝑡,𝜈)𝐶𝑥. (2.4) Thus 1ℎ(𝑊(ℎ,0)^𝑠₀^𝑡₀𝑊(𝜇,𝜈)𝑥𝑑𝜇𝑑𝜈−𝐶^𝑠₀^𝑡₀=1𝑊(𝜇,𝜈)𝑥𝑑𝜇𝑑𝜈)ℎ𝐶(^𝑠_0ℎ^𝑡+ℎ𝑊(𝜇,𝜈)𝑥𝑑𝜇𝑑𝜈−^𝑠₀^𝑡₀=𝑊(𝜇,𝜈)𝑥𝑑𝜇𝑑𝜈)_𝑠0(1ℎ[_𝑡^𝑡+ℎ𝑊(𝜇,𝜈)𝐶𝑥𝑑𝜇−^ℎ₀𝑊(𝜇,𝜈)𝐶𝑥𝑑𝜇])𝑑𝜈, (2.5) which tends to 𝐶∫^𝑠₀(𝑊(𝑡,𝜈)−𝑊(0,𝜈))𝑥𝑑𝜈 as ℎ→0. This implies that ∫^𝑠₀∫^𝑡₀𝑊(𝜇,𝜈)𝑥𝑑𝜇𝑑𝜈 is in 𝐷(𝐻₁) and 𝐻₁^𝑠₀^𝑡₀𝑊(𝜇,𝜈)𝑥𝑑𝜇𝑑𝜈=^𝑠₀(𝑊(𝑡,𝜈)−𝑊(0,𝜈))𝑥𝑑𝜈. (2.6) A similar argument implies that it is in 𝐷(𝐻₂) and 𝐻₂^𝑠₀^𝑡₀𝑊(𝜇,𝜈)𝑥𝑑𝜇𝑑𝜈=^𝑡₀(𝑊(𝜇,𝑠)−𝑊(𝜇,0))𝑥𝑑𝜈. (2.7) For the second part, from the continuity of 𝐶 we have 𝐶lim_{(ℎ,𝑘)→(0,0)}1ℎ𝑘_𝑡^𝑡+ℎ_𝑠^𝑠+𝑘𝑊(𝜇,𝜈)𝑥𝑑𝜇𝑑𝜈=lim_{(ℎ,𝑘)→(0,0)}1ℎ𝑘_𝑡^𝑡+ℎ_𝑠^𝑠+𝑘𝑊(𝜇,𝜈)𝐶𝑥𝑑𝜇𝑑𝜈=lim_{(ℎ,𝑘)→(0,0)}1ℎ_𝑡^𝑡+ℎ1𝑊(0,𝜈)𝑘_𝑠^𝑠+𝑘𝑊(𝜇,0)𝑥𝑑𝜇𝑑𝜈=lim_ℎ→01ℎ_𝑡^𝑡+ℎ𝑊(0,𝜈)(lim_𝑘→01𝑘_𝑠^𝑠+𝑘𝑊(𝜇,0)𝑥𝑑𝜇)𝑑𝜈=𝑊(0,𝑡)𝑊(𝑠,0)𝑥=𝑊(𝑠,𝑡)𝐶𝑥. (2.8) Now the fact that 𝐶 is injective completes the proof of this part.
The proof of (ii) has a process similar to the first part of (i).
To prove (iii), we first note that 𝐻₁ and 𝐻₂ are closed as a trivial consequence of the one-parameter case (see [2]). For any 𝑥∈𝑋 we saw that 1ℎ^ℎ₀^ℎ₀𝑊(𝜇,𝜈)𝑥𝑑𝜇𝑑𝜈∈𝐷(𝐻₁)∩𝐷(𝐻₂), (2.9) which tends to 𝑊(0,0)𝑥=𝐶𝑥∈𝑅(𝐶), as ℎ→0. This implies that 𝑅(𝐶)⊆𝐷(𝐻₁)∩𝐷(𝐻₂).
To prove (iv), we let 𝑥∈𝐷(𝐻₁)∩𝐷(𝐻₂). If 𝑢(𝑠)=𝑊(𝑠,0) and 𝑣(𝑡)=𝑊(𝑠,𝑡), there is 𝑦∈𝑋 such that lim_𝑠→0𝑢(𝑠)𝑥−𝐶𝑥𝑠=𝐶𝑦. (2.10) Hence lim_𝑠→0𝑢(𝑠)𝑣(𝑡)𝑥−𝐶𝑣(𝑡)𝑥𝑠=𝑣(𝑡)𝐶𝑦=𝐶𝑣(𝑡)𝑦, (2.11) which is in the 𝑅(𝐶), and this implies that 𝑣(𝑡)𝑥 is in 𝐷(𝐻₁), similarly it is in 𝐷(𝐻₂).
Now from [2, Theorem  2.4(b)], for 𝑥∈𝐷(𝐻₁)∩𝐷(𝐻₂), from the fact that 𝑣(𝑡)𝑥 is in 𝐷(𝐻₁), 𝜕𝑑𝜕𝑠𝑊(𝑠,𝑡)𝐶𝑥=𝑑𝑠(𝑢(𝑠)(𝑣(𝑡)𝑥))=𝐻₁𝑢(𝑠)(𝑣(𝑡)𝑥)=𝐻₁𝑊(𝑠,𝑡)𝐶𝑥=𝐶𝐻₁𝑊(𝑠,𝑡)𝑥. (2.12) On the other hand from the part (ii) and closedness of 𝐻₁, ^𝑠₀𝐻₁𝑊(𝜇,𝑡)𝑥𝑑𝜇=𝐻₁^𝑠₀𝑊(𝜇,𝑡)𝑥𝑑𝜇=𝑊(𝑠,𝑡)𝑥−𝑊(0,𝑡)𝑥, (2.13) which implies that (𝜕/𝜕𝑠)𝑊(𝑠,𝑡)𝑥 exists. Hence from the continuity of 𝐶𝐶𝜕𝜕𝜕𝑠𝑊(𝑠,𝑡)𝑥=𝜕𝑠𝑊(𝑠,𝑡)𝐶𝑥=𝐶𝐻₁𝑊(𝑠,𝑡)𝑥. (2.14) But 𝐶 is injective so 𝜕𝜕𝑠𝑊(𝑠,𝑡)𝑥=𝐻₁𝑊(𝑠,𝑡)𝑥=𝑊(𝑠,𝑡)𝐻₁𝑥. (2.15) The second one is similar.
To prove (v), first we note that 𝑇(𝑡) is a one-parameter 𝐶-semigroup. Now if 𝑥∈𝐷(𝑎𝐻₁+𝑏𝐻₂)=𝐷(𝐻₁)∩𝐷(𝐻₂), 𝐶lim_𝑡→0+𝑇(𝑡)𝑥−𝐶𝑥𝑡=lim_𝑡→0+𝑊(𝑡𝑎,0)𝑊(0,𝑡𝑏)𝑥−𝑊(𝑡𝑎,0)𝐶𝑥+𝑊(𝑡𝑎,0)𝐶𝑥−𝐶²𝑥𝑡=𝑏lim_𝑡→0+𝑊(𝑡𝑎,0)𝑊(0,𝑡𝑏)𝑥−𝐶𝑥𝑏𝑡+𝑎lim_𝑡→0+𝑊(𝑎𝑡,0)𝐶𝑥−𝐶²𝑥𝑡=𝑏𝐶²𝐻₂𝑥+𝑎𝐻₁𝐶²𝑥. (2.16) Now the fact that 𝐶 is injective implies that 𝐶⁻¹lim_𝑡→0+𝑇(𝑡)𝑥−𝐶𝑥𝑡=𝑎𝐻₁𝑥+𝑏𝐻₂𝑥. (2.17)

For an exponentially bounded one-parameter 𝐶-semigroup 𝑇(𝑡) with the generator 𝐴, from [1] the existence of 𝐿_𝜆(𝐴)𝑥=∫^∞₀𝑒^−𝜆𝑡𝑇(𝑡)𝑥𝑑𝑡 is guaranteed for sufficiently large 𝜆∈ℝ. Now we have the following lemma for one-parameter 𝐶-semigroups of operators which is similar to the Yosida-approximation theorem for strongly continuous semigroups. This will be applied in our study of two-parameter regularized semigroups.

Lemma 2.4. Let {𝑇(𝑡)}_𝑡∈ℝ+ be a one-parameter 𝐶-semigroup satisfying the condition ‖𝑇(𝑡)‖≤𝑀𝑒^𝜔𝑡, for some 𝜔>0 and 𝑀>0, with the generator 𝐴. If for 𝜆>𝜔, 𝐴_𝜆∶=𝜆𝐴𝐿_𝜆(𝐴), then one has the following.(i)For any 𝑥∈𝑋, ‖𝐿_𝜆(𝐴)𝑥‖≤(𝑀/(𝜆−𝜔))‖𝑥‖, 𝐴_𝜆=𝜆²𝐿_𝜆(𝐴)−𝜆𝐶, and so 𝐴_𝜆 is bounded. Also 𝑆(𝑡)∶=𝐶𝑒^𝑡𝐴𝜆 is a one-parameter 𝐶-semigroup which is exponentially bounded.(ii)For any 𝑥∈𝐷(𝐴), lim_𝜆→∞𝜆𝐿_𝜆(𝐴)𝑥=𝐶𝑥 and for all 𝑥∈𝐷(𝐴), lim_𝜆→∞𝐴_𝜆𝑥=𝐶𝐴𝑥. Also if 𝑅(𝐶) is dense in 𝑋, then the first equality holds on 𝑋.(iii)For any 𝑥∈𝐷(𝐴), 𝑇(𝑡)𝑥=lim_𝜆→∞𝐶𝑒^𝑡𝐴𝜆𝑥.

Proof. The first inequality of (i) is trivial. From [2, Lemma  2.8], we know that for any 𝑥∈𝑋, (𝜆−𝐴)𝐿_𝜆(𝐴)𝑥=𝐶𝑥; thus, −𝜆(𝜆−𝐴)𝐿_𝜆(𝐴)𝑥=−𝜆𝐶𝑥. (2.18) This implies our desired equality.
For the second part, first we show that 𝐶𝐴_𝜆=𝐴_𝜆𝐶. For this we note that 𝐶𝐿_𝜆(𝐴)=𝐶^∞₀𝑒^−𝜆𝑡=𝑇(𝑡)𝑥𝑑𝑥_∞0𝐶𝑒^−𝜆𝑡=𝑇(𝑡)𝑥𝑑𝑥_∞0𝑒^−𝜆𝑡𝑇(𝑡)𝐶𝑥𝑑𝑥=𝐿_𝜆(𝐴)𝐶𝑥. (2.19) This and the first part imply that 𝐶𝐴_𝜆=𝐴_𝜆𝐶. Now we prove the 𝐶-semigroup properties of 𝑆(𝑡). Trivially 𝑆(0)=𝐶. Also from the last equality, 𝑆(𝑠+𝑡)𝐶=𝐶𝑒^{(𝑠+𝑡)𝐴𝜆}𝐶=𝐶𝑒^𝑠𝐴𝜆𝐶𝑒^𝑡𝐴𝜆=𝑆(𝑠)𝑆(𝑡). (2.20) The fact that 𝐴_𝜆, 𝜆>𝜔, is a bounded operator trivially implies that 𝑆(⋅) is exponentially bounded. Now the continuity of the mapping 𝑡↦𝑆(𝑡)𝑥 at zero implies the strongly continuity of 𝑆(𝑡).
To prove (ii), for 𝑥∈𝐷(𝐴), from (i) and the fact that 𝐴 is closed, we have ‖𝜆𝐿_𝜆(𝐴)𝑥−𝐶𝑥‖=‖𝐴𝐿_𝜆(𝐴)𝑥‖=‖𝐿_𝜆(𝐴)𝐴𝑥‖≤‖𝐿_𝜆≤𝑀(𝐴)‖‖𝐴𝑥‖(𝜆−𝜔)‖𝐴𝑥‖→0as𝜆→∞. (2.21) The continuity of 𝐶 and 𝐿_𝜆(𝐴) implies that for any 𝑥∈𝐷(𝐴), lim_𝜆→∞𝜆𝐿_𝜆(𝐴)𝑥=𝐶𝑥.
Now for 𝑥∈𝐷(𝐴), lim_𝜆→∞𝐴_𝜆𝑥=lim_𝜆→∞𝜆𝐿_𝜆(𝐴)𝐴𝑥=𝐶𝐴𝑥=𝐴𝐶𝑥. (2.22) For the last part of (ii), if 𝐶 has a dense range, then by [8, Lemma  1.1.3], 𝑅(𝐶)⊆𝐷(𝐴), and so 𝑋=𝑅(𝐶)⊆𝐷(𝐴)⊆𝑋, which means that 𝐷(𝐴)=𝑋.
To prove (iii), for any 𝑥∈𝐷(𝐴), we have ‖𝐶𝑒^𝑡𝐴𝜆𝑥−𝐶𝑒^𝑡𝐴𝜇𝑥‖=‖¹₀𝑑𝑑𝑠(𝐶𝑒^{𝑡𝑠𝐴𝜆}𝑒^{𝑡(1−𝑠𝐴𝜇)}≤𝑥)‖₁₀𝑡‖𝐶𝑒^{𝑡𝑠𝐴𝜆}𝑒^{𝑡(1−𝑠𝐴𝜇)}(𝐴_𝜆𝑥−𝐴_𝜇𝑥)‖𝑑𝑠≤𝑡‖𝐶‖‖𝐴_𝜆𝑥−𝐴_𝜇𝑥‖≤𝑡‖𝐶‖(‖𝐴_𝜆𝑥−𝐴𝐶𝑥‖+‖𝐴𝐶𝑥−𝐴_𝜇𝑥‖). (2.23) This and the previous part prove the existence of lim_𝜆→∞𝐶𝑒^𝑡𝐴𝜆𝑥.

Using this theorem we may find the following approximation theorem for two-parameter regularized semigroups.

Corollary 2.5. Suppose that (𝐻,𝐾) is the infinitesimal generator of an exponentially bounded two-parameter 𝐶-semigroup 𝑊(𝑠,𝑡), then for each 𝑥∈𝐷(𝐻)∩𝐷(𝐾), 𝑊(𝑠,𝑡)𝑥=𝐶lim_𝜆→∞𝑒^{𝑠𝐻𝜆+𝑡𝐾𝜆}𝑥. (2.24)

For exponentially bounded 𝐶-semigroup 𝑊(𝑠,𝑡) satisfying ‖𝑊(𝑠,𝑡)‖≤𝑀𝑒^{(𝑠+𝑡)𝜔}, with the infinitesimal generator (𝐻,𝐾), define 𝐿_𝜆1(𝐻)𝑥∶=∫^∞₀𝑒^−𝜆1𝑠𝑊(𝑠,0)𝑥𝑑𝑠 and 𝐿_𝜆2(𝐾)𝑥∶=∫^∞₀𝑒^−𝜆2𝑡𝑊(0,𝑡)𝑥𝑑𝑡, where Re(𝜆_𝑖)>𝜔. From the previous Lemma 𝐿_𝜆1(𝐻) and 𝐿_𝜆2(𝐾) are bounded operators.

Theorem 2.6. (i) Let (𝐻,𝐾) be the generator of an exponentially bounded two-parameter 𝐶-semigroup, then for large enough 𝜆₁, 𝜆₂𝐿_𝜆1(𝐻)𝐿_𝜆2(𝐾)=𝐿_𝜆2(𝐾)𝐿_𝜆1(𝐻). (2.25)(ii)Let (𝐻,𝐾) be the generator of an exponentially bounded two-parameter 𝐶-semigroup, then 𝐷(𝐻)∩𝐷(𝐻𝐾)⊆𝐷(𝐾𝐻), and for 𝑥∈𝐷(𝐻)∩𝐷(𝐻𝐾), 𝐻𝐾𝑥=𝐾𝐻𝑥. (2.26)(iii)Suppose that 𝐻 and 𝐾 are the generators of two exponentially bounded one-parameter 𝐶-semigroups {𝑢(𝑠)}_𝑠∈ℝ+ and {𝑣(𝑡)}_𝑡∈ℝ+, respectively. If their resolvents commute and 𝑅(𝐶) is dense in 𝑋, then 𝑊(𝑠,𝑡)∶=𝑢(𝑠)𝑣(𝑡) is a two-parameter 𝐶²-semigroup.

Proof. The proof of (i) follows trivially from the properties of two-parameter 𝐶-semigroups.
To prove (ii), we let 𝑥∈𝐷(𝐻)∩𝐷(𝐻𝐾); from the strongly continuity of 𝑊(𝑠,𝑡) and the fact that 𝐾 is closed, we have 𝐶²𝐻𝐾𝑥=𝐶lim_𝑠→0𝑊(𝑠,0)𝐾𝑥−𝐶𝐾𝑥𝑠=lim_𝑠→01𝑠(𝑊(𝑠,0)(lim_𝑡→0𝑊(0,𝑡)𝑥−𝐶𝑥𝑡)−lim_𝑡→0𝑊(0,𝑡)𝑥−𝐶𝑥𝑡)=lim_𝑠→0lim_𝑡→01𝑠𝑡(𝑊(𝑠,0)𝑊(0,𝑡)𝑥−𝑊(𝑠,0)𝐶𝑥−𝑊(0,𝑡)𝑥+𝐶𝑥)=lim_𝑠→0lim_𝑡→01𝑠𝑡(𝑊(0,𝑡)𝑊(𝑠,0)𝑥−𝑊(𝑠,0)𝐶𝑥−𝑊(0,𝑡)𝑥+𝐶𝑥)=lim_𝑠→0lim_𝑡→01𝑡(𝑊(0,𝑡)(𝑊(𝑠,0)𝑥−𝐶𝑥𝑠)−𝑊(𝑠,0)𝑥−𝐶𝑥𝑠)=𝐶lim_𝑠→0𝐾(𝑊(𝑠,0)𝑥−𝐶𝑥𝑠)=𝐶²𝐾𝐻𝑥. (2.27) However, 𝐶 is injective, and this completes the proof of (i).
To prove (iii), from our hypothesis, for sufficiently large 𝜆, 𝜆^′, we know that 𝐿_𝜆(𝐻)𝐿_𝜆′(𝐾)=𝐿_𝜆′(𝐾)𝐿_𝜆(𝐻). (2.28) By Lemma 2.4, 𝐻_𝜆=𝜆²𝐿_𝜆(𝐻)−𝜆𝐶 and 𝐾_𝜆′=𝜆^′2𝐿_𝜆′(𝐻)−𝜆^′𝐶, thus 𝐻_𝜆𝐾_𝜆′=𝐾_𝜆′𝐻_𝜆. From (iii) of Lemma 2.4, for each 𝑥∈𝐷(𝐻)∩𝐷(𝐾), 𝑢(𝑠)𝑥=lim_𝜆→∞𝐶𝑒^𝑠𝐻𝜆𝑥,𝑣(𝑡)=lim_{𝜆′→∞}𝐶𝑒^{𝑡𝐾𝜆′}𝑥. (2.29) So 𝑢(𝑠)𝑣(𝑡)𝑥=𝐶lim_𝜆→∞𝑒^𝑠𝐻𝜆𝑣(𝑡)𝑥=𝐶²lim_𝜆→∞𝑒^𝑠𝐻𝜆(lim_{𝜆′→∞}𝑒^{𝑡𝐾𝜆′}𝑥),(𝑒^𝑠𝐻𝜆iscontinuous)=𝐶²lim_𝜆→∞lim_{𝜆′→∞}𝑒^𝑠𝐻𝜆𝑒^{𝑡𝐾𝜆′}𝑥=𝐶²lim_𝜆→∞lim_{𝜆′→∞}𝑒^{𝑡𝐾𝜆′}𝑒^𝑠𝐻𝜆𝑥=𝐶lim_𝜆→∞𝑣(𝑡)𝑒^𝑠𝐻𝜆𝑥=𝑣(𝑡)𝑢(𝑠)𝑥. (2.30) Now the continuity of 𝑢(𝑠) and 𝑣(𝑡) and the fact that 𝐷(𝐻)∩𝐷(𝐾)=𝑅(𝐶)=𝑋 imply that for each 𝑥∈𝑋, 𝑢(𝑠)𝑣(𝑡)𝑥=𝑣(𝑡)𝑢(𝑠)𝑥. Thus 𝑊(𝑠,𝑡)𝑊(𝑠^′,𝑡^′)=𝑢(𝑠)𝑣(𝑡)𝑢(𝑠^′)𝑣(𝑡^′)=𝑢(𝑠)𝑢(𝑠^′)𝑣(𝑡)𝑣(𝑡^′)=𝐶𝑢(𝑠+𝑠^′)𝐶𝑣(𝑡+𝑡^′)=𝑊(𝑠+𝑠^′,𝑡+𝑡^′)𝐶². (2.31) On the other hand 𝑊(0,0)=𝐶², which completes the proof.

If 𝐻 and 𝐾 are two closed operators on 𝑋, then 𝑋₁∶=𝐷(𝐻)∩𝐷(𝐾) with ‖𝑥‖₁=‖𝑥‖+‖𝐻𝑥‖+‖𝐾𝑥‖, 𝑥∈𝑋₁, is a Banach space.

Proposition 2.7. Suppose that 𝐶∈𝐵(𝑋) is injective and {𝑊(𝑠,𝑡)} is a two-parameter 𝐶-semigroup with the generator (𝐻,𝐾). Then 𝑊₁(𝑠,𝑡)∶=𝑊(𝑠,𝑡)|_𝑋1 defines a two-parameter 𝐶₁-semigroup, with the generator (𝐻₁,𝐾₁), where 𝐶₁=𝐶|_𝑋1, and 𝐻₁, 𝐾₁ are the part of 𝐻 and 𝐾 on 𝑋₁, respectively.

Proof. The 𝐶₁-semigroup properties of 𝑊₁(𝑠,𝑡) are obvious. Let (𝐴,𝐵) be the generator of 𝑊₁(𝑠,𝑡); we show that 𝐴=𝐻₁ and 𝐵=𝐻₂. First we note that 𝐷(𝐻₁)={𝑥∈𝑋₁∶𝐻𝑥∈𝑋₁}={𝑥∈𝐷(𝐻)∩𝐷(𝐾)∶𝑥∈𝐷(𝐻²)∩𝐷(𝐾𝐻)}=𝐷(𝐾)∩𝐷(𝐻²)∩𝐷(𝐾𝐻). (2.32) Let 𝑥∈𝐷(𝐻₁). So we have 𝑊₁(𝑠,0)𝑥−𝐶₁𝑥𝑡=𝑊(𝑠,0)𝑥−𝐶𝑥𝑡→𝐶𝐻𝑥=𝐶₁𝐻₁𝐻𝑊𝑥,₁(𝑠,0)𝑥−𝐶₁𝑥𝑡=𝑊(𝑠,0)𝐻𝑥−𝐶𝐻𝑥𝑡→𝐶𝐻²𝑥=𝐻𝐶₁𝐻₁𝐾𝑊𝑥,₁(𝑠,0)𝑥−𝐶₁𝑥𝑡=𝑊(𝑠,0)𝐾𝑥−𝐶𝐾𝑥𝑡→𝐶𝐻𝐾𝑥=𝐾𝐶𝐻𝑥=𝐾𝐶₁𝐻₁𝑥. (2.33) These show that (𝑊₁(𝑠,0)𝑥−𝐶₁𝑥)/𝑡→𝐶₁𝐻₁𝑥 in ‖⋅‖₁, that is, 𝑥∈𝐷(𝐴) and 𝐴𝑥=𝐻₁𝑥. Hence 𝐻₁⊆𝐴. Conversely, if 𝑥∈𝐷(𝐴)⊆𝑋₁, then ‖⋅‖₁−lim_𝑡→0𝑊(𝑠,0)𝑥−𝐶𝑥𝑡=‖⋅‖₁−lim_𝑡→0𝑊₁(𝑠,0)𝑥−𝐶₁𝑥𝑡=𝐶₁𝐴𝑥=𝐶𝐴𝑥, (2.34) so 𝐻𝑥=𝐴𝑥∈𝑋₁. Hence 𝑥∈𝐷(𝐾)∩𝐷(𝐻²)∩𝐷(𝐾𝐻)=𝐷(𝐻₁) and 𝐻₁𝑥=𝐻𝑥=𝐴𝑥.
A similar argument shows that 𝐾₁=𝐵, which completes the proof.

3. Two-Parameter Abstract Cauchy Problems

Suppose that 𝐻_𝑖∶𝐷(𝐻_𝑖)⊆𝑋→𝑋, 𝑖=1,2, is linear operator. Consider the following two-parameter Cauchy problem: 2-𝐴𝐶𝑃(𝐻₁,𝐻₂𝜕;𝑥)𝜕𝑡_𝑖𝑢(𝑡₁,𝑡₂)=𝐻_𝑖𝑢(𝑡₁,𝑡₂),𝑡_𝑖>0,𝑖=1,2,𝑢(0,0)=𝑥,𝑥∈𝐶(𝐷(𝐻₁)∩𝐷(𝐻₂)). (3.1) We mean by a solution a continuous Banach-valued function 𝑢(⋅,⋅)∶[0,∞)×[0,∞)→𝑋 which has continuous partial derivative and satisfies 2-𝐴𝐶𝑃(𝐻₁,𝐻₂;𝑥).

In this section first we prove that if (𝐻₁,𝐻₂) is the infinitesimal generator of a two-parameter 𝐶-semigroup of operators, then 2-𝐴𝐶𝑃(𝐻₁,𝐻₂;𝑥) has a unique solution for any 𝑥∈𝐶(𝐷(𝐻₁)∩𝐷(𝐻₂)). Next it is proved that under some condition on 𝐶, existence and uniqueness of solutions of 2-𝐴𝐶𝑃(𝐻₁,𝐻₂;𝐶𝑥), for every 𝑥∈𝐷(𝐻₁)∩𝐷(𝐻₂), imply that this unique solution is induced by a two-parameter regularized semigroup.

Theorem 3.1. Suppose that an extension of (𝐻₁,𝐻₂) is the generator of a two-parameter 𝐶-semigroup 𝑊(𝑠,𝑡), then 2-𝐴𝐶𝑃(𝐻₁,𝐻₂;𝑥) has the unique solution 𝑢(𝑠,𝑡;𝑥)∶=𝑊(𝑠,𝑡)𝐶⁻¹𝑥, for all 𝑥∈𝐶(𝐷(𝐻₁)∩𝐷(𝐻₂)).

Proof. The fact that 𝑢(𝑠,𝑡;𝑥)∶=𝑊(𝑠,𝑡)𝐶⁻¹𝑥 is a solution of 2-𝐴𝐶𝑃(𝐻₁,𝐻₂;𝑥) is obvious from Theorem 2.3. It is enough to show that 2-𝐴𝐶𝑃(𝐻₁,𝐻₂;𝑥) has the unique solution 𝑢(𝑠,𝑡)=0, for the initial value 𝑥=0. From one-parameter case (see [2]), we know that the systems 𝑑𝑢(𝑡)𝑑𝑡=𝐻₁𝑢(𝑡),𝑡∈ℝ₊,𝑢(0)=0,𝑑𝑣(𝑡)𝑑𝑡=𝐻₂𝑣(𝑡),𝑡∈ℝ₊,𝑣(0)=0 (3.2)(3.3) have the unique solution zero. Now if 𝑢(𝑠,𝑡;0) is a solution of 2-𝐴𝐶𝑃(𝐻₁,𝐻₂;0), then 𝑢₁(𝑠)∶=𝑊(𝑠,0)𝐶⁻¹𝑢(0,𝑡;0),𝑢₂(𝑠)∶=𝑢(𝑠,𝑡;0) (3.4) are two solutions of (3.2), for the initial value 𝑢(0,𝑡;0), since 𝑑𝑢𝑑𝑠¹𝑑(𝑠)=𝑑𝑠𝑊(𝑠,0)𝐶⁻¹𝑢(0,𝑡;0)=𝐻₁𝑊(𝑠,0)𝐶⁻¹𝑢(0,𝑡;0)=𝐻₁𝑢₁𝑑(𝑠),𝑢𝑑𝑠²𝜕(𝑠)=𝜕𝑠𝑢(𝑠,𝑡;0)=𝐻₁𝑢(𝑠,𝑡;0)=𝐻₁𝑢₂(𝑠). (3.5) The uniqueness of solution in one-parameter case implies that 𝑢₁(𝑠)=𝑢₂(𝑠). So 𝑊(𝑠,0)𝐶⁻¹𝑢(0,𝑡;0)=𝑢(𝑠,𝑡;0). (3.6) Also 𝑣₁(𝑡)∶=𝑊(0,𝑡)𝐶⁻¹𝑢(𝑠,0;0) and 𝑣₂(𝑡)∶=𝑢(𝑠,𝑡;0) are two solutions of (3.3) for the initial value 𝑢(𝑠,0;0). From the uniqueness of solution in (3.3), 𝑊(0,𝑡)𝐶⁻¹𝑢(𝑠,0;0)=𝑢(𝑠,𝑡;0), for all 𝑠,𝑡≥0. Thus 𝑢(𝑠,𝑡;0)=𝑊(𝑠,0)𝐶⁻¹𝑢(0,𝑡;0)=𝑊(𝑠,0)𝐶⁻¹𝑊(0,𝑡)𝑢(0,0;0)=0. (3.7)

The uniqueness of solution 2-𝐴𝐶𝑃(𝐻,𝐾;𝐶𝑥), for all 𝑥∈𝐷(𝐻)∩𝐷(𝐾), also leads us to a two-parameter 𝐶-semigroup. This will be shown in the following theorem.

In this theorem 𝑋₁ and 𝐶₁ have their meaning in Proposition 2.7.

Theorem 3.2. Suppose that 𝐶∈𝐵(𝑋) is injective and 𝐻,𝐾 are two closed operators satisfying 𝐶𝑥∈𝑋₁,𝐾𝐶𝑥=𝐶𝐾𝑥,𝐻𝐶𝑥=𝐶𝐻𝑥,∀𝑥∈𝑋₁. (3.8) If, for each 𝑥∈𝑋₁, the Cauchy problem 2-𝐴𝐶𝑃(𝐻,𝐾;𝐶𝑥) has a unique solution 𝑢(⋅,⋅;𝐶𝑥), then there exists a two-parameter 𝐶₁-semigroup 𝑊₁(⋅,⋅) on 𝑋₁ such that 𝑢(⋅,⋅;𝐶𝑥)=𝑊₁(⋅,⋅)𝑥. Moreover, the infinitesimal generator of 𝑊₁(⋅,⋅) is a restriction of (𝐻₁,𝐾₁), where 𝐻₁ and 𝐾₁ are the part of 𝐻 and 𝐾 on 𝑋₁, respectively.

Proof. Suppose that, for any 𝑥∈𝑋₁, 2-𝐴𝐶𝑃(𝐻,𝐾;𝐶𝑥) has a unique solution 𝑢(⋅,⋅;𝐶𝑥)∈𝐶¹([0,∞)×[0,∞),𝑋). For 𝑥∈𝑋₁ and 0<𝑠,𝑡<∞, define 𝑊₁(𝑠,𝑡)𝑥∶=𝑢(𝑠,𝑡;𝐶𝑥).
From the uniqueness of solution 𝑊₁(𝑠,𝑡) is a well-defined and linear operator on 𝑋₁ and 𝑊₁(0,0)𝑥=𝑢(0,0;𝑥)=𝐶𝑥. (3.9) By uniqueness of solutions one can see that 𝑊₁(𝑠+𝑠^′,𝑡+𝑡^′)𝐶₁=𝑊₁(𝑠,𝑡)𝑊₁(𝑠^′,𝑡^′). (3.10)
We are going to show that 𝑊₁(𝑠,𝑡) is a bounded operator on (𝑋₁,‖⋅‖₁). Let 0<𝑠,𝑡<∞. Define the mapping 𝜙_𝑠,𝑡∶𝑋₁→𝐶([0,𝑠]×[0,𝑡],𝑋₁) by 𝜙_𝑠,𝑡𝑥=𝑊₁(⋅,⋅)𝑥=𝑢(⋅,⋅;𝐶𝑥). Obviously 𝜙_𝑠,𝑡 is linear. We claim that this mapping is closed. Suppose that 𝑥_𝑛∈𝑋₁, 𝑥_𝑛→𝑥 and 𝑢(⋅,⋅;𝐶𝑥_𝑛)=𝜙_𝑠,𝑡(𝑥_𝑛)→𝑦 in 𝐶([0,𝑠]×[0,𝑡],𝑋₁) with its usual supremum norm. From the Cauchy problem we know that 𝑢(𝜇,𝜈;𝐶𝑥_𝑛)=𝐶𝑥_𝑛+^𝜇₀𝐻𝑢(𝜂,𝜈;𝐶𝑥_𝑛)𝑑𝜂,𝑢(𝜇,𝜈;𝐶𝑥_𝑛)=𝐶𝑥_𝑛+^𝜈₀𝐾𝑢(𝜇,𝜂;𝐶𝑥_𝑛)𝑑𝜂. (3.11) Letting 𝑛→∞, we obtain 𝑦(𝜇,𝜈)=𝐶𝑥+^𝜇₀𝐻𝑦(𝜂,𝜈)𝑑𝜂,𝑦(𝜇,𝜈)=𝐶𝑥+^𝜈₀𝐾𝑦(𝜇,𝜂)𝑑𝜂 (3.12) for any (𝜇,𝜈)∈[0,𝑠]×[0,𝑡]. Now define 𝑦 on [0,∞)×[0,∞) by 𝑊𝑦(𝜇,𝜈)=𝐶𝑦(𝜇,𝜈),0≤𝜇≤𝑠,0≤𝜈≤𝑡,₁𝑊(0,𝜈−𝑡)𝑦(𝜇,𝑡),0≤𝜇≤𝑠,𝑡<𝜈<∞,₁𝑊(𝜇−𝑠,0)𝑦(𝑠,𝜈),𝑠<𝜇<∞,0≤𝜈≤𝑡,₁(𝜇−𝑠,𝜈−𝑡)𝑦(𝑠,𝑡),𝑠<𝜇<∞,𝑡<𝜈<∞. (3.13) One can see that 𝑦 is a solution of 2-𝐴𝐶𝑃(𝐻,𝐾;𝐶²𝑥). Indeed from (3.12) 𝑦(0,0)=𝐶𝑦(0,0)=𝐶²𝑥. (3.14) Also (3.12) and the fact that 𝐶 commutes with 𝐻 and 𝐾 imply that 𝜕𝜕𝜇𝑦(𝜇,𝜈)=𝐻𝑦(𝜇,𝜈),0≤𝜇≤𝑠,0≤𝜈≤𝑡,𝐻𝑊₁(0,𝜈−𝑡)𝑦(𝜇,𝑡),0≤𝜇≤𝑠,𝑡<𝜈<∞,𝐻𝑊₁(𝜇−𝑠,0)𝑦(𝑠,𝜈),0<𝜇<∞,0≤𝜈≤𝑡,𝐻𝑊₁(𝜇−𝑠,𝜈−𝑡)𝑦(𝑠,𝑡),0<𝜇<∞,0<𝜈<∞,=𝐻𝑦(𝜇,𝜈). (3.15) Similarly 𝜕𝜕𝜈𝑦(𝜇,𝜈)=𝐾𝑦(𝜇,𝜈). (3.16) Uniqueness of the solution implies that 𝑦(⋅,⋅)=𝑢(⋅,⋅;𝐶𝑥²)=𝑊₁(⋅,⋅)𝐶𝑥=𝐶𝑊₁(⋅,⋅)𝑥. (3.17) In particular for 0≤𝜇≤𝑠 and 0≤𝜈≤𝑠, 𝐶𝑦(𝜇,𝜈)=𝑦(𝜇,𝜈)=𝐶𝑊₁(𝜇,𝜈)𝑥=𝐶𝜙_𝑠,𝑡(𝑥)(𝜇,𝜈). (3.18) The fact that 𝐶 is injective implies that 𝑦=𝜙_𝑠,𝑡(𝑥), which shows that 𝜙_𝑠,𝑡 is closed operator.
By the Closed Graph Theorem 𝜙_𝑠,𝑡 is a continuous operator from Banach space 𝑋₁ into the Banach space 𝐶([0,𝑠]×[0,𝑡],𝑋₁). So if 𝑥_𝑛→𝑥 in 𝑋₁, then 𝜙_𝑠,𝑡(𝑥_𝑛)→𝜙_𝑠,𝑡(𝑥) in 𝐶([0,𝑠]×[0,𝑡],𝑋₁); thus for each (𝜇,𝜈)∈[0,𝑠]×[0,𝑡], 𝑊₁(𝑠,𝑡)𝑥_𝑛=𝜙_𝑠,𝑡(𝑥_𝑛)(𝜇,𝜈)→𝜙_𝑠,𝑡(𝑥)(𝜇,𝜈)=𝑊₁(𝜇,𝜈)𝑥. (3.19) But 𝑠 and 𝑡 were arbitrary; hence 𝑊₁(𝜇,𝜈) is continuous for any 𝜇,𝜈∈[0,∞). Also for every 𝑥∈𝑋₁, 𝑊₁(⋅,⋅)𝑥=𝜙_𝑠,𝑡(𝑥) is continuous on [0,𝑠]×[0,𝑡]; that is, 𝑊₁(⋅,⋅) is strongly continuous family of operators.
Now let (𝐴,𝐵) be its infinitesimal generator and 𝑥∈𝐷(𝐴), then ‖⋅‖₁−lim_𝑠→0𝑊₁(𝑠,0)𝑥−𝐶₁𝑥𝑠=𝐶₁𝐴𝑥, (3.20) which implies that lim_𝑠→0((𝑊₁(𝑠,0)𝑥−𝐶𝑥)/𝑠)=𝐶𝐴𝑥, but 𝐷(𝐴)⊆𝐷(𝐻)lim_𝑠→0𝑊₁(𝑠,0)𝑥−𝐶𝑥𝑠=lim_𝑠→0𝑢(𝑠,0;𝐶𝑥)−𝐶𝑥𝑠=𝜕𝜕𝑠𝑢(0,0;𝐶𝑥)=𝐻𝐶𝑥=𝐶𝐻𝑥. (3.21) Hence 𝐶𝐻𝑥=𝐶𝐴𝑥. The injectivity of 𝐶 implies that 𝐻𝑥=𝐴𝑥∈𝑋₁=𝐷(𝐻)∩𝐷(𝐾). Thus 𝑥∈𝐷(𝐾)∩𝐷(𝐻²)∩𝐷(𝐾𝐻)=𝐷(𝐻₁) and 𝐻₁𝑥=𝐴𝑥. This shows that 𝐴 is a restriction of 𝐻₁. Similarly one can see that 𝐵 is a restriction of 𝐾₁, which completes the proof.

We conclude this section with a simple example as an application of our discussion. Consider the following sequence of initial value problems:

𝜕𝑢𝜕𝑠^𝑛(𝑠,𝑡)=𝑛𝑢_𝑛𝜕(𝑠,𝑡),𝑢𝜕𝑡^𝑛(𝑠,𝑡)=𝑛²𝑢_𝑛𝑢(𝑠,𝑡),𝑛∈ℕ,_𝑛(0,0)=𝑒^−𝑛2𝑞_𝑛. (3.22) Suppose that 𝑋=𝑐₀, the space of all complex sequences in ℂ which vanish at infinity. Now define linear operators 𝐻 and 𝐾 in 𝑋 and operator 𝐶 on 𝑋 as follows: 𝐻(𝑥_𝑛)_𝑛∈ℕ=(𝑛𝑥_𝑛)_𝑛∈ℕ,𝐾(𝑥_𝑛)_𝑛∈ℕ=(𝑛²𝑥_𝑛)_𝑛∈ℕ,𝐶(𝑥_𝑛)_𝑛∈ℕ=(𝑒^−𝑛2𝑥_𝑛)_𝑛∈ℕ. (3.23) Using these operators the initial value problem (3.22) can be rewrite as follows: 𝜕𝜕𝜕𝑠𝑢(𝑠,𝑡)=𝐻𝑢(𝑠,𝑡),𝜕𝑡𝑢(𝑠,𝑡)=𝐾𝑢(𝑠,𝑡),𝑢(0,0)=𝐶𝑞, (3.24) where 𝑢(𝑠,𝑡)=(𝑢_𝑛(𝑠,𝑡))_𝑛∈ℕ and 𝑞=(𝑞_𝑛)_𝑛∈ℕ. One can easily see that (𝐻,𝐾) is the generator of the following two-parameter 𝐶-semigroup: 𝑊(𝑠,𝑡)(𝑥_𝑛)_𝑛∈ℕ=(𝑒^{𝑛2(𝑡−1)+𝑠𝑛}𝑥_𝑛)_𝑛∈ℕ (3.25) on 𝑋. Hence for every 𝑞=(𝑞_𝑛)_𝑛∈ℕ∈𝐷(𝐻)∩𝐷(𝐾), by Theorem 3.1, the abstract Cauchy problem (3.24) has the unique solution

𝑢(𝑠,𝑡)=𝑊(𝑠,𝑡)𝑞=(𝑒^{𝑛2(𝑡−1)+𝑠𝑛}𝑞_𝑛)_𝑛∈ℕ. (3.26) This implies that for each 𝑛∈ℕ, 𝑢_𝑛(𝑠,𝑡)=𝑒^{𝑛2(𝑡−1)+𝑡𝑛}𝑞_𝑛 is a solution of (3.22).

Acknowledgment

The author is grateful to the referees for their very useful suggestions which helped him to improve the presentation considerably.