Schauder-Tychonoff Fixed-Point Theorem in Theory of Superconductivity

Abstract

We study the existence of mild solutions to the time-dependent Ginzburg-Landau ((TDGL), for short) equations on an unbounded interval. The rapidity of the growth of those solutions is characterized. We investigate the local and global attractivity of solutions of TDGL equations and we describe their asymptotic behaviour. The TDGL equations model the state of a superconducting sample in a magnetic field near critical temperature. This paper is based on the theory of Banach space, Fréchet space, and Sobolew space.

1. Introduction

The objective of the paper is to investigate the existence and asymptotic behaviour of mild solutions on an unbounded interval of time-dependent Ginzburg-Landau equations (TDGL, for short) in superconductivity.

In the Ginzburg-Landau theory of phase transitions [1], the state of a superconducting material near the critical temperature is described by a complex-valued order parameter 𝜓, a real vector-valued vector potential 𝐀, and, when the system changes with time, a real-valued scalar potential 𝜙. The latter is a diagnostic variable; 𝜓 and 𝐀 are prognostic variables, whose evolution is governed by a system of coupled differential equations: 𝜂𝜕𝑖𝜕𝑡+𝑖𝜅𝜙𝜓=−𝜅∇+𝐀²𝜓+1−𝜓²𝜓,𝜕𝐀𝜕𝑡+∇𝜙=−∇×∇×𝐀+𝐉_𝑠+∇×𝐇. (1)(2) The supercurrent density 𝐉_𝑠 is a nonlinear function of 𝜓 and 𝐀, 𝐉_𝑠≡𝐉_𝑠=1𝜓,𝐀𝜓2𝑖𝜅^∗∇𝜓−𝜓∇𝜓^∗−𝜓²𝐀=−Re𝜓^∗𝑖𝜅∇+𝐀𝜓. (3) The system of (1)–(3) must be satisfied everywhere in Ω, the region occupied by the superconducting material, and at all times 𝑡>0. The boundary conditions associated with the differential equations have the form 𝑖𝐧⋅𝜅𝑖∇+𝐀𝜓+𝜅()𝛾𝜓=0,𝐧×∇×𝐀−𝐇=𝟎 (4)(5) on 𝜕Ω, where 𝜕Ω is the boundary of Ω and 𝐧  is the local outer unit normal to 𝜕Ω. They must be satisfied at all times 𝑡>0.

We prove that the systems of (1)–(5) can be reduced to a semilinear equation; to use the appropriate theorem, we investigate the local and global attractivity of solutions of equations in question and describe their asymptotic behaviour.

In this paper, we consider the existence and asymptotic behaviour of mild solutions on an unbounded interval of the semilinear evolution equation of the following form: 𝑑𝑢((𝑡𝑑𝑡+𝒜𝑢=ℱ𝑡,𝑢)),𝑡∈ℝ₊=[),𝑢(0)0,+∞=𝑢₀∈𝔛, (6)(7) where the operator 𝒜∶𝐷(𝒜)⊂𝔛→𝔛 generates a 𝐶₀-semigroup {𝑒^{−𝒜(𝑡−𝑠)}}_𝑡≥0 and 𝔛 is a real Banach space.

Recently, a lot of papers have appeared that deal with the same or similar equations on a bounded interval (see [2–21]). However, only in a few papers, problem (6)-(7) was considered on an unbounded interval [10, 22]. Additionally, in assumptions concerning the semigroup {𝑒^{−𝒜(𝑡−𝑠)}}_𝑡≥0 or the function ℱ(𝑡,𝑢), rather restrictive conditions have been imposed which frequently require the compactness of ℱ(𝑡,𝑢) or {𝑒^{−𝒜(𝑡−𝑠)}}_𝑡≥0 or equicontinuity of semigroup {𝑒^{−𝒜(𝑡−𝑠)}}_𝑡≥0 [2–11, 13–22]. It is worthwhile mentioning that only a few papers have discussed asymptotic behaviour of solutions, mostly without the formulation of existence theorems [10, 23, 24].

In this paper, we present conditions guaranteeing the existence of mild solutions on an unbounded interval of problem (6)-(7). We dispense with assumptions on the compactness of ℱ(𝑡,𝑢) or {𝑒^{−𝒜(𝑡−𝑠)}}_𝑡≥0.

Moreover, we formulate theorems about asymptotic properties and both local and global attractivity of solutions of problem (6)-(7). The existence theorems concerning that problem will be proved with the help of the technique of a family of measures of noncompactness in the Fréchet space 𝐶(ℝ₊,𝔛) and Schauder-Tychonoff fixed-point principle.

The approach applied here was introduced and developed in [20, 25–35], for instance.

The paper is organized as follows. In Section 2, there are given notation and auxiliary facts that are needed further on. In Section 3, we formulate and prove a theorem on the existence of mild solution of (6) with condition (7). Moreover, the rate of the growth of those solutions is characterized.

Section 4 contains a theorem on local and global attractivity of solutions of problem (6)-(7). In Section 5, we give a theorem describing the asymptotic behaviour of solutions of (6)-(7).

Finally, in Section 6, we formulate the gauged TDGL equation as an abstract evolution equation in a Hilbert space. Moreover, this section is devoted to present examples of application of previously obtained theorems for TDGL equations.

In order to convert the systems equations (1)–(5) to the Cauchy problem (6)-(7) we are based on the papers and monographs (see [1, 30, 36–69]).

2. Preliminaries

Let (𝔛,‖⋅‖) be a real Banach space with the zero element 𝜃. Denote by 𝐵(𝑥,𝑟) the closed ball in 𝔛 centered at 𝑥 and with radius 𝑟. If 𝑋 is a subset of a linear topological space, then the symbols 𝑋 and Conv𝑋 stand for the closure and the convex closure of 𝑋, respectively.

Let 𝜒 denote the Hausdorff measure of noncompactness in 𝔛, defined on bounded subsets 𝑋 of 𝔛 in the following way (see [27]): 𝜒(𝑋){}.∶=inf𝑟>0∶𝑋canbecoveredwithfinitenumbersofballsofradius𝑟 (8)

Further, denote by 𝐶(ℝ₊,𝔛) the Fréchet space consisting of all functions defined and continuous on ℝ₊ with values in a Banach space 𝔛. The space 𝐶(ℝ₊,𝔛) is furnished with the family of seminorms; ‖𝑢‖_𝑛𝑢(𝑡[ℝ∶=sup{‖)‖∶𝑡∈0,𝑛]}for𝑛=1,2,…,𝑢∈𝐶₊,𝔛. (9) Let us recall two facts: (*) a sequence {𝑢_𝑛}^∞_𝑛=1 is convergent to 𝑢 in 𝐶(ℝ₊,𝔛) if and only if {𝑢_𝑛}^∞_𝑛=1 is uniformly convergent to 𝑥 on compact subsets of ℝ₊;(**) a subset 𝑈⊂𝐶(ℝ₊,𝔛) is relatively compact if and only if the restrictions to [0,𝑇] of all functions from 𝑈 form an equicontinuous set for each 𝑇>0 and 𝑈(𝑡) is relatively compact in 𝔛 for each 𝑡∈ℝ₊, where 𝑈(𝑡)={𝑢(𝑡)∶𝑢∈𝑈}.

Moreover, we recall that a nonempty subset 𝑈⊂𝐶(ℝ₊,𝔛) is said to be bounded if sup‖𝑢‖_𝑛∶𝑢∈𝑈<∞for𝑛=1,2,…. (10) Further, the family of all nonempty and bounded subsets of 𝐶(ℝ₊,𝔛) will be denoted by 𝔐_𝐶, while the family of all nonempty and relatively compact subsets of 𝐶(ℝ₊,𝔛) is denoted by 𝔑_𝐶. Obviously 𝔑_𝐶⊂𝔐_𝐶.

We will use a family of measures of noncompactness {𝜇_𝑇}_𝑇≥0 in the Fréchet space 𝐶(ℝ₊,𝔛) which was introduced in [20, 35]. In order to define these measures, recall some quantities [25–27]. Let us fix a nonempty bounded subset 𝑈 of the space 𝐶(ℝ₊,𝔛). For 𝑢∈𝑈, 𝜀≥0, and 𝑇≥0 denote by 𝜔^𝑇(𝑢,𝜀) the modulus of continuity of the function 𝑢 on the interval [0,𝑇], that is, 𝜔^𝑇(𝑢,𝜀)∶=sup𝑢𝑡₂−𝑢𝑡₁∶𝑡₁,𝑡₂∈[0,𝑇],𝑡₂−𝑡₁≤𝜀. (11) Further, let us put 𝜔^𝑇(𝑈,𝜀)∶=sup𝜔^𝑇(),𝜔𝑢,𝜀∶𝑢∈𝑈_𝑇0(𝑈)∶=lim_𝜀→0+𝜔^𝑇(𝑈,𝜀). (12) Obviously, the set 𝑈 is equicontinuous on [0,𝑇] if and only if 𝜔^𝑇₀(𝑈)=0.

Next, let us define the functions {𝜇_𝑇}_𝑇≥0 on the family of bounded subsets of 𝐶(ℝ₊,𝔛) by the following formula: 𝜇_𝑇(𝑈)∶=𝜔^𝑇₀(𝑈)+sup𝜒(𝑈(𝑡))∶𝑡∈[0,𝑇]. (13) It can be shown that the family {𝜇_𝑇}_𝑇≥0 has the following properties: 1^𝑜 the family ker{𝜇_𝑇}∶={𝑈∈𝔐_𝐶∶𝜇_𝑇(𝑈)=0for𝑇≥0} is nonempty and ker{𝜇_𝑇}=𝔑_𝐶;2^𝑜𝑈⊂𝑉⇒𝜇_𝑇(𝑈)≤𝜇_𝑇(𝑉) for 𝑇≥0;3^𝑜𝜇_𝑇(Conv𝑈)=𝜇_𝑇(𝑈) for 𝑇≥0;4^𝑜 if {𝑈_𝑛}^∞_𝑛=1 is a sequence of closed sets from 𝔐_𝐶 such that 𝑈_𝑛+1⊂𝑈_𝑛(𝑛=1,2,…) and if lim_𝑛→∞𝜇_𝑇(𝑈_𝑛)=0 for each 𝑇≥0, then the intersection set 𝑈_∞∶=⋂^∞_𝑛=1𝑈_𝑛 is nonempty.

Remark 1. Observe that in contrast to the definition of the concept of a measure of noncompactness given in [27], our mapping 𝜇_𝑇 may take the value ∞. Moreover, a single mapping 𝜇_𝑇 is not the measure of noncompactness in 𝐶(ℝ₊,𝔛) but the whole family {𝜇_𝑇}_𝑇≥0 can be called family of measures of noncompactness.

Remark 2. Let us notice that the intersection set 𝑈_∞ described in axiom 4^𝑜 is a member of the kernel of the family of measure of noncompactness {𝜇_𝑇}_𝑇≥0, and therefore, 𝑈_∞ is compact in 𝐶(ℝ₊,𝔛). In fact, the inequality 𝜇_𝑇(𝑈_∞)≤𝜇_𝑇(𝑈_𝑛) for 𝑛=1,2,… and 𝑇≥0 implies that 𝜇_𝑇(𝑈_∞)=0. Hence, 𝑈_∞∈ker{𝜇_𝑇}. This property of the set 𝑈_∞ will be very important in our further investigations.

Definition 3. A set {𝑒^−𝒜𝑡}_𝑡≥0 of bounded linear operators on 𝔛 is called 𝐶₀-semigroup if (i)𝑒^−𝒜0=𝐼, 𝑒^{−𝒜(𝑡+𝑠)}=𝑒^−𝒜𝑡𝑒^−𝒜𝑠 for 𝑡,𝑠≥0;(ii)for all 𝑥∈𝔛, the function ℝ₊∋𝑡↦𝑒^−𝒜𝑡𝑥 is continuous in ℝ₊.

Further, denote 𝑁(𝑇)∶=sup𝑒^−𝒜𝑡∶𝑡≤𝑇for𝑇≥0. (14)

Definition 4. A function 𝑥∈𝐶(ℝ₊,𝔛) is said to be a mild solution of the nonlocal initial-value problem (6)-(7) if for every 𝑡∈ℝ₊, 𝑢(𝑡)=𝑒^−𝒜𝑡𝑢₀+^𝑡₀𝑒^{−𝒜(𝑡−𝑠)}ℱ(𝑠,𝑢(𝑠))𝑑𝑠. (15)

To prove the existence results in this paper, we need the following lemmas.

Lemma 5 (see [21]). If 𝑌 is a bounded subset of Banach space 𝔛, then for each 𝜀>0 there is a sequence {𝑦_𝑘}^∞_𝑘=1⊂𝑌 such that 𝜒(𝑌)≤2𝜒𝑦_𝑘^∞_𝑘=1+𝜀. (16)

We call a set 𝑈⊂𝐿¹([0,𝑇],𝔛) uniformly integrable if there exists 𝜂∈𝐿¹([0,𝑇],ℝ₊) such that ‖𝑢(𝑠)‖≤𝜂(𝑠) for 𝑢∈𝑈 and a.e. 𝑠∈[0,𝑇].

Lemma 6 (see [70]). If {𝑢_𝑘}^∞_𝑘=1⊂𝐿¹([0,𝑇],𝔛) is uniformly integrable, then 𝜒({𝑢_𝑘(𝑡)}^∞_𝑘=1) is measurable and 𝜒^𝑡₀𝑢_𝑘(𝑠)𝑑𝑠^∞_𝑘=1≤2^𝑡₀𝜒𝑢_𝑘(𝑠)^∞_𝑘=1𝑑𝑠𝑓𝑜𝑟𝑡∈[0,𝑇]. (17)

Lemma 7 (see [34]). Assume that a set 𝑈⊂𝐶(ℝ₊,𝔛) is bounded. Then,(i)sup_{𝑡∈[0,𝑇]}𝜒(𝑈(𝑡))≤𝜒(𝑈([0,𝑇]))≤𝜔^𝑇₀(𝑈)+sup_{𝑡∈[0,𝑇]}𝜒(𝑈(𝑡)), (ii)sup_{𝑡∈[0,𝑇]}𝜒(𝑈(𝑡))≤𝜒_𝐶(𝑈)≤𝜔^𝑇₀(𝑈)+sup_{𝑡∈[0,𝑇]}𝜒(𝑈(𝑡)), for 𝑇≥0, where 𝑈([0,𝑇])∶={𝑢(𝑡)∶𝑡∈[0,𝑇],𝑢∈𝑈}.

Lemma 8 (Gronwall’s inequality [71]). Assume that the function 𝜙₁∶ℝ₊→ℝ₊ is measurable and the function 𝜙₂∶ℝ₊→ℝ₊ is locally integrable. Moreover, we assume that ∫^𝑡₀𝜙₁(𝑠)𝜙₂(𝑠)𝑑𝑠<∞ for 𝑡≥0 and a measurable function 𝜓∶ℝ₊→ℝ₊ satisfies the following inequality: 𝜓(𝑡)≤𝜙₁(𝑡)+^𝑡₀𝜙₂(𝑠)𝜓(𝑠)𝑑𝑠𝑓𝑜𝑟𝑡≥0. (18) Then, 𝜓(𝑡)≤𝜙₁(𝑡)+^𝑡₀𝜙₁(𝑠)𝜙₂(𝑠)exp^𝑡_𝑠𝜙₂(𝜏)𝑑𝜏𝑑𝑠𝑓𝑜𝑟𝑡≥0. (19)

Next, we consider the operator 𝐹∶𝐶(ℝ₊,𝔛)→𝐶(ℝ₊,𝔛) defined by the following formula: (𝐹𝑢)(𝑡)∶=^𝑡₀𝑒^{−𝒜(𝑡−𝑠)}ℱ(𝑠,𝑢(𝑠))𝑑𝑠,𝑡∈ℝ₊,𝑢∈𝐶ℝ₊,𝔛. (20)

Lemma 9 (see [20]). Assume that conditions (𝐻₁), (𝐻₂), and (𝐻₄) are satisfied (see Section 3), a set 𝑈⊂𝐶(ℝ₊,𝔛) is bounded, 𝑇>0, a function ℎ∶[0,𝑇]→ℝ₊ is measurable, and 2𝑁(𝑇)𝜒(ℱ(𝑠,𝑈(𝑠)))≤ℎ(𝑠) (21) for a.e. 𝑠∈[0,𝑇]. Then, 𝜔^𝑇₀(𝐹𝑈)≤^𝑇₀ℎ(𝑠)𝑑𝑠. (22)

3. Main Result

In this section, we give an existence result for the semilinear equation of evolution (6)-(7), and we describe the asymptotic behaviour of those solutions.

First, we will assume that the functions involved in (6) satisfy the following conditions:(𝐻₁)𝒜 is a linear operator acting from 𝐷(𝒜)⊂𝔛 and 𝒜 generates 𝐶₀-semigroup {𝑒^{−𝒜(𝑡−𝑠)}}_𝑡≥0;(𝐻₂)the mapping ℱ∶ℝ₊×𝔛→𝔛 satisfies the Carathéodory condition; that is, ℱ(⋅,𝑢) is measurable for 𝑢∈𝔛 and ℱ(𝑡,⋅) is continuous for a.e. 𝑡∈ℝ₊; (𝐻₃)𝑎∶ℝ₊→ℝ₊ and 𝑏∶ℝ₊→ℝ₊ are locally integrable functions such that ‖ℱ(𝑡,𝑢)‖≤𝑎(𝑡)+𝑏(𝑡)‖𝑢‖ (23) for any 𝑢∈𝔛 and a.e. 𝑡∈ℝ₊. (𝐻₄) There exists a locally integrable function 𝐿∶ℝ₊→ℝ₊ such that for any bounded 𝐷⊂𝔛, 𝜒(ℱ(𝑡,𝐷))≤𝐿(𝑡)𝜒(𝐷) (24) for a.e. 𝑡∈ℝ₊.

Theorem 10. Under assumptions (𝐻₁)–(𝐻₄), (6) with initial condition (7) has, for every 𝑢₀∈𝔛, at least one mild solution 𝑢=𝑢(𝑡) which satisfied the following estimate: ‖𝑢(𝑡(𝑡)(𝑡))‖≤𝑀+𝑁^𝑡₀𝑏(𝑠)𝑀(𝑠)exp^𝑡_𝑠𝑏(𝜏)𝑁(𝜏)𝑑𝜏𝑑𝑠, (25) where 𝑀(𝑡)=𝑁(𝑡)𝑢₀+^𝑡₀𝑎(𝑠)𝑑𝑠. (26)

Proof. Consider the operator 𝐺∶𝐶(ℝ₊,𝔛)→𝐶(ℝ₊,𝔛) defined by the following formula: 𝐺𝑢(𝑡)∶=𝑒^−𝒜𝑡𝑢₀+^𝑡₀𝑒^{−𝒜(𝑡−𝑠)}ℱ(𝑠,𝑢(𝑠))𝑑𝑠. (27) Now, let us observe that for any continuous function 𝑢∶ℝ₊→𝔛, in view of condition (𝐻₃), we get the following estimate: ‖𝐺𝑢(𝑡)‖≤𝑁(𝑡)𝑢₀+𝑁(𝑡)^𝑡₀(𝑎(𝑠)+𝑏(𝑠)‖𝑢(𝑠)‖)𝑑𝑠, (28) which yields ‖𝐺𝑢(𝑡)‖≤𝑀(𝑡)+𝑁(𝑡)^𝑡₀𝑏(𝑠)‖𝑢(𝑠)‖𝑑𝑠. (29) Next, consider the following integral equation: 𝒫(𝑡)=𝑀(𝑡)+𝑁(𝑡)^𝑡₀𝑏(𝑠)𝒫(𝑠)𝑑𝑠. (30) Solving this equation by standard methods, we get 𝒫(𝑡)=𝑀(𝑡)+𝑁(𝑡)^𝑡₀𝑏(𝑠)𝑀(𝑠)exp^𝑡_𝑠𝑏(𝜏)𝑁(𝜏)𝑑𝜏𝑑𝑠. (31) The function 𝒫(𝑡) is continuous, nonnegative, and nondecreasing. Observe that the following implication is true: ‖𝑢(𝑡)‖≤𝒫(𝑡)⟹‖𝐺𝑢(𝑡)‖≤𝒫(𝑡)for𝑡≥0. (32) Indeed, linking (29) and (32), we have ‖𝐺𝑢(𝑡)‖≤𝑀(𝑡)+𝑁(𝑡)^𝑡₀𝑏(𝑠𝑢(𝑠(𝑡)(𝑡))‖)‖𝑑𝑠≤𝑀+𝑁^𝑡₀𝑏(𝑠)𝒫(𝑠)𝑑𝑠=𝒫(𝑡). (33)
In the space 𝐶(ℝ₊,𝔛), let us consider the following set: Γ∶=𝑢∈𝐶ℝ₊,𝔛∶‖𝑢(𝑡)‖≤𝒫(𝑡),for𝑡≥0. (34) Obviously, the set Γ is convex and closed. Moreover, in view of (32), we have that 𝐺 is a self-mapping of Γ.
Using the criterion of convergence (*) in 𝐶(ℝ₊,𝔛) and standard techniques (see [31–33, 35]), we can show that the operator 𝐺 is continuous on 𝐶(ℝ₊,𝔛).
Now, we consider the sequence of sets {Γ_𝑛} defined by induction as follows: Γ₀∶=Γ,Γ_𝑛+1∶=Conv𝐺Γ_𝑛for𝑛=0,1,…. (35) This sequence is decreasing, that is, Γ_𝑛⊃Γ_𝑛+1 for 𝑛=0,1,2….
Further, let us fix 𝑇>0 and for 𝑡∈[0,𝑇], let us put 𝑔_𝑛(𝑡)∶=sup𝜒Γ_𝑛(𝑠)[,𝑤∶𝑠∈0,𝑡]_𝑛∶=𝜔^𝑇₀Γ_𝑛. (36)(37) The sequences {𝑤_𝑛} and {𝑔_𝑛(𝑡)} are nonincreasing for all 𝑡∈[0,𝑇], so they have limits 𝑔_∞(𝑡)∶=lim_𝑛→∞𝑔_𝑛(𝑡)[],𝑤,𝑡∈0,𝑇_∞∶=lim_𝑛→∞𝑤_𝑛. (38) Moreover, each function 𝑔_𝑛 is nondecreasing; therefore 𝑔_𝑛, and 𝑢_∞ are measurable on [0,𝑇] for 𝑛=1,2,….
Now, we apply the family of measures of noncompactness {𝜇_𝑇}_𝑇≥0 defined in 𝐶(ℝ₊,𝔛) by formula (13). In view of the above notation, we have 𝜇_𝑇Γ_𝑛=𝑤_𝑛+𝑔_𝑛(𝑇). (39) We show that 𝑤_∞=𝑔_∞(𝑇)=0. (40) To fix a number 𝑡∈[0,𝑇],  𝑛∈ℕ and take an arbitrary number 𝑠∈[0,𝑡].
We know from Lemma 5 that for any 𝜀>0, there is a sequence {𝑦_𝑘}^∞_𝑘=1⊂𝐺Γ_𝑛(𝑠), such that 𝜒𝐺Γ_𝑛(𝑠)≤2𝜒𝑦_𝑘^∞_𝑘=1+𝜀. (41) This implies that there is a sequence {𝑢_𝑘}^∞_𝑘=1⊂Γ_𝑛, such that 𝑦_𝑘=𝐺𝑢_𝑘(𝑠)for𝑘=1,2,…. (42) Hence, in view of Lemma 6, (𝐻₄), and (36), we obtain 𝜒𝐺Γ_𝑛(𝑠)≤2𝜒𝑦_𝑘^∞_𝑘=1+𝜀=2𝜒𝐺𝑢_𝑘(𝑠)^∞_𝑘=1+𝜀≤2𝜒𝑒^−𝒜𝑠𝑢₀+2𝜒^𝑠₀𝑒^{−𝒜(𝑠−𝜏)}ℱ𝜏,𝑢_𝑘(𝜏)𝑑𝜏^∞_𝑘=1+𝜀≤4^𝑠₀𝜒𝑒^{−𝒜(𝑠−𝜏)}ℱ𝜏,𝑢_𝑘(𝜏)^∞_𝑘=1(𝑇)𝑑𝜏+𝜀≤4𝑁^𝑠₀𝐿(𝜏)𝜒𝑢_𝑘(𝜏)^∞_𝑘=1(𝑇)𝑑𝜏+𝜀≤4𝑁^𝑡₀𝐿(𝜏)𝑔_𝑛(𝜏)𝑑𝜏+𝜀. (43) Since 𝜀>0 is arbitrary, it follows from the above inequalities that 𝜒𝐺Γ_𝑛(𝑠)≤4𝑁(𝑇)^𝑡₀𝐿(𝜏)𝑔_𝑛(𝜏)𝑑𝜏,𝑠∈[0,𝑡]. (44) Using (36), we have 𝑔_𝑛+1(𝑡)=sup𝜒𝐺Γ_𝑛(𝑠)[(𝑇)∶𝑠∈0,𝑡]≤4𝑁^𝑡₀𝐿(𝜏)𝑔_𝑛(𝜏)𝑑𝜏,𝑡∈[0,𝑇]. (45) Letting 𝑛→∞, we derive the following inequality: 𝑔_∞(𝑡)≤4𝑁(𝑇)^𝑡₀𝐿(𝜏)𝑔_∞(𝜏)𝑑𝜏,𝑡∈[0,𝑇]. (46) This inequality, together with Gronwall’s Lemma 8, implies that 𝑔_∞(𝑡)=0,𝑡∈[0,𝑇]. (47)
Next let us notice that in view of (𝐻₄) and (36), we have 𝜒ℱ𝑠,Γ_𝑛(𝑠)≤𝐿(𝑠)𝜒Γ_𝑛(𝑠)≤𝐿(𝑠)𝑔_𝑛(𝑠) (48) for a.e. 𝑠∈[0,𝑇], and the function [0,𝑇]∋𝑠⟼2𝑁(𝑇)𝐿(𝑠)𝑔_𝑛(𝑠) (49) is measurable. Then, in virtue of equality 𝜔^𝑇₀(𝐺Γ_𝑛)=𝜔^𝑇₀(𝐹Γ_𝑛) and Lemma 9, we get 𝜔^𝑇₀𝐺Γ_𝑛≤2𝑁(𝑇)^𝑇₀𝐿(𝑠)𝑔_𝑛(𝑠)𝑑𝑠. (50) Hence, we derive 𝑤_𝑛+1=𝜔^𝑇₀Γ_𝑛+1=𝜔^𝑇₀𝐺Γ_𝑛≤2𝑁(𝑇)^𝑇₀𝐿(𝑠)𝑔_𝑛(𝑠)𝑑𝑠. (51) Letting 𝑛→∞, we get 𝑤_∞≤2𝑁(𝑇)^𝑇₀𝐿(𝑠)𝑔_∞(𝑠)𝑑𝑠. (52) Keeping in mind (47), we deduce that 𝑤_∞=0. (53) This together with (39) and (47) yields lim_𝑛→∞𝜇_𝑇Γ_𝑛=lim_𝑛→∞𝑤_𝑛+𝑔_𝑛(𝑇)=𝑤_∞+𝑔_∞=0. (54)
Finally, using Remark 2 for the measure 𝜇_𝑇, we deduce that the set Γ_∞∶=⋂^∞_𝑛=0Γ_𝑛 is nonempty, convex, and compact. Then, by the Schauder-Tychonoff theorem, we conclude that operator 𝐹∶Γ_∞→Γ_∞ has at least one fixed-point 𝑢=𝑢(𝑡). Obviously, the function 𝑢=𝑢(𝑡) is a solution of problem (6)-(7), and, in view of the definition of the set Γ_∞, the estimate ‖𝑢(𝑡)‖≤𝒫(𝑡) holds to be true. This completes the proof.

4. Local and Global Attractivity

Following the concepts introduced in [36], we introduce first a few definitions of various kinds of the concept of attractivity of mild solution of (6).

Definition 11. The mild solution 𝑢=𝑢(𝑡) of (6) with initial condition (7) is said to be globally attractive if for each mild solution 𝑣=𝑣(𝑡) of (6) with initial condition 𝑣(0)=𝑣₀ we have that lim_𝑡→∞(𝑢(𝑡)−𝑣(𝑡))=𝜃. (55)

In other words, we may say that solutions of (6) are globally attractive if for arbitrary solutions 𝑢(𝑡) and 𝑣(𝑡) of this equation condition (55) are satisfied.

Definition 12. We say that mild solution 𝑢=𝑢(𝑡) of (6) with initial condition (7) is locally attractive if there exists a ball 𝐵(𝑢(0),𝑟) in the space 𝔛 such that for arbitrary solution 𝑣(𝑡) of (1) with initial-value 𝑣(0)∈𝐵(𝑢(0),𝑟), condition (55) does hold.

In the case when the limit (55) is uniform with respect to all solutions 𝑣(𝑡), that is, when for each 𝜀>0 there exist 𝑇>0 such that ‖𝑢(𝑡)−𝑣(𝑡)‖≤𝜀 (56) for all 𝑣(𝑡) being solutions of (6) with initial-value 𝑣(0)∈𝐵(𝑢(0),𝑟), and for 𝑡≥𝑇, we will say that solution 𝑢=𝑢(𝑡) is uniformly locally attractive on ℝ₊.

Now, we formulate the main result of this section. We will consider (6) under the following conditions:(𝐻^′₁)𝒜 is the infinitesimal generator of an exponentially stable 𝐶₀-semigroup {𝑒^−𝒜𝑡}_𝑡≥0; that is, there exist 𝑀>0, 𝜌>0 such that ‖𝑒^−𝒜𝑡‖≤𝑀𝑒^−𝜌𝑡 for all 𝑡≥0; (𝐻₅) there exist locally integrable functions 𝑚∶ℝ₊→ℝ₊, such that ‖ℱ(𝑡,𝑢)−ℱ(𝑡,𝑣)‖≤𝑚(𝑡)‖𝑢−𝑣‖ (57) for 𝑡≥0 and 𝑢,𝑣∈𝔛. Moreover, we assume that lim_𝑡→∞^𝑡₀𝑚(𝑠)𝜌𝑑𝑠−𝑀𝑡=−∞. (58)

Remark 13. The property (𝐻^′₁) is generally satisfied in diffusion problem. A necessary and sufficient condition for (𝐻^′₁) is presented in [72].

The main result of this section is shown in the given theorem below.

Theorem 14. Under assumptions (𝐻^′₁) and (𝐻₂)–(𝐻₅), problem (6)-(7) has a mild solution 𝑢=𝑢(𝑡) for each 𝑢₀∈𝔛, which is globally attractive and locally uniformly attractive.

Proof. Existence of a solution 𝑢=𝑢(𝑡) is a consequence of Theorem 10. Let us fix 𝑟>0 and 𝑣₀∈𝐵(𝑢(0),𝑟).
Let 𝑣=𝑣(𝑡) denote a mild solution of (6) with the initial condition 𝑣(0)=𝑣₀. Using (𝐻^′₁) and (𝐻₃), we get ‖𝑣(𝑡)‖≤𝑒^−𝒜𝑡𝑣₀+^𝑡₀𝑒^{−𝒜(𝑡−𝑠)}⋅‖ℱ((𝑠𝑢𝑠,𝑣))‖𝑑𝑠≤𝑀₀+𝑟+𝑀^𝑡₀𝑒^𝜌𝑠𝑎(𝑠)+𝑑𝑠_𝑡0𝑀𝑒^𝜌𝑠𝑏(𝑠)‖𝑣(𝑠)‖𝑑𝑠. (59) Now, let us put 𝒫₁(𝑡)=𝑀𝑣₀+𝑟+𝑀^𝑡₀𝑒^𝜌𝑠𝑎(𝑠)𝒫𝑑𝑠,₂(𝑡)=𝑀𝑒^𝜌𝑠𝑏(𝑡). (60) Taking into account Lemma 5, we obtain ‖𝑣(𝑡)‖≤𝒫₁(𝑡)+^𝑡₀𝒫₂(𝑠)𝒫₁(𝑠)exp^𝑠₀𝒫₂(𝜏)𝑑𝜏𝑑𝑠. (61) Further, let 𝑆 be the set of all mild solutions 𝑣(𝑡) of (6) with the initial-value 𝑣₀=𝑣(0)∈𝐵(𝑢(0),𝑟).
Put 𝑔(𝑡)=sup{‖𝑢(𝑡)−𝑣(𝑡)‖∶𝑣∈𝑆}. (62) The estimate (61) implies that the function 𝑔(𝑡) is well defined. Applying (𝐻₅) and (𝐻^′₁), we get ‖𝑢(𝑡)−𝑣(𝑡)‖≤𝑒^−𝒜𝑡⋅𝑢₀−𝑣₀+^𝑡₀𝑒^{−𝒜(𝑡−𝑠)}𝑚(𝑠)‖𝑢(𝑠)−𝑣(𝑠)‖𝑑𝑠≤𝑀𝑒^−𝜌𝑡𝑟+^𝑡₀𝑀𝑒^{−𝑔(𝑡−𝑠)}𝑚(𝑠)𝑔(𝑠)𝑑𝑠, (63) or, equivalently, 𝑒^𝜌𝑡𝑔(𝑡)≤𝑀𝑟+^𝑡₀𝑀𝑚(𝑠)𝑒^𝜌𝑠𝑔(𝑠)𝑑𝑠. (64) Using again Lemma 5 for the above estimate (where ℎ(𝑡)=𝑒^𝜌𝑡𝑔(𝑡)), we obtain 𝑒^𝜌𝑡𝑔(𝑡)≤𝑀𝑟+^𝑡₀𝑀²𝑟𝑚(𝑠)exp^𝑡_𝑠𝑀𝑚(𝜏)𝑑𝜏𝑑𝑠. (65) Elementary calculations lead to the following equality: ^𝑡₀𝑀²𝑟𝑚(𝑠)exp^𝑡_𝑠𝑀𝑚(𝜏)𝑑𝜏𝑑𝑠=𝑀𝑟exp^𝑡₀𝑀𝑚(𝑠)𝑑𝑠−1. (66) Hence, 𝑔(𝑡)≤𝑀𝑟exp−𝜌𝑡+𝑀𝑟exp^𝑡₀(𝑠).𝑀𝑚𝑑𝑠−𝜌𝑡−𝑀𝑟exp−𝜌𝑡 (67) Applying assumption (𝐻₅), we derive lim_𝑡→∞𝑔(𝑡)=0, (68) and this proves that 𝑢(𝑡) is locally attractive. Finally, this equality together with definition of the function 𝑔(𝑡) implies that 𝑢(𝑡) is globally attractive. The proof is complete.

Remark 15. In the case when 𝑚(𝑡)≡𝑚 is constant, the following condition lim_𝑡→∞^𝑡₀𝑚(𝑠)𝜌𝑑𝑠−𝑀𝑡=−∞ (69) means that 𝑚<𝜌𝑀. Observe that this condition cannot be weakened. This observation is illustrated by the following exampled.

Example 16. Let 𝔛=ℝ, 𝑚(𝑡)≡𝑚, ℱ(𝑡,𝑢)=sin𝑡+𝑚𝑢, 𝑀=1, and 𝑒^−𝒜𝑡=𝑒^−𝜌𝑡. Then the equation 𝑢(𝑡)=𝑒^−𝒜𝑡𝑢₀+^𝑡₀𝑒^{−𝒜(𝑡−𝑠)}ℱ(𝑠,𝑢(𝑠))𝑑𝑠 (70) (for any fixed 𝑢₀∈ℝ) has the solution 𝑢(𝑡) expressed by the following formula: 𝑢(𝑡)=𝜌−𝑚sin𝑡−cos𝑡𝜌−𝑚²+𝑢+1⁰+1𝜌−𝑚²𝑒+1^{(𝑚−𝜌)𝑡}. (71) Notice that for 𝑚≥𝜌, the solution 𝑢(𝑡) is neither globally attractive nor locally uniformly attractive, because for each other solution 𝑣(𝑡) with initial condition 𝑣(0)=𝑣₀, obviously described by similar formula as 𝑢(𝑡), we would have a contradiction: |𝑢(𝑡)(𝑡=𝑢−𝑣)|⁰−𝑣₀𝑒^{(𝑚−𝜌)𝑡}𝑢⟶∞or₀−𝑣₀. (72)

5. Asymptotic Behaviour

In this section, we will give a theorem describing asymptotic behaviour of mild solutions of (6) with condition (7). This theorem generalizes the result included in [72, Theorem  4.4]. First, we formulate the assumptions.(𝐻^′₃) This condition is almost identical with (𝐻₃) and the only difference is that we assume the functions 𝑎 and 𝑏 are locally essentially bounded on ℝ₊. (𝐻^′₅) There exists 𝑢_∞∈𝔛 such that there exists the limit lim_𝑡→∞ℱ(𝑡,𝑢_∞) and lim_𝑡→∞ℱ𝑡,𝑢_∞=𝒜⁻¹𝑢_∞. (73) Moreover, there exists a number 𝑚<𝜌/𝑀 such that ℱ(𝑡,𝑢)−ℱ𝑡,𝑢_∞≤𝑚𝑢−𝑢_∞ (74) for 𝑡≥0 and 𝑢∈𝔛.

Remark 17. The condition (𝐻^′₅) in conjunction with (𝐻^′₁) ensures the existence of 𝒜⁻¹ (see [72]). Clearly, (𝐻^′₃) implies (𝐻₃).

Theorem 18. Under assumptions (𝐻^′₁), (𝐻₂), (𝐻^′₃), (𝐻₄), and (𝐻^′₅), (6) with condition (7) has a mild solution 𝑢=𝑢(𝑡) for each 𝑢₀∈𝔛 such that lim_𝑡→∞𝑢(𝑡)=𝑢_∞.

Proof. The existence of a mild solution 𝑢(𝑡) is guaranteed by Theorem 10. Let us put 𝑔(𝑡)=𝑢(𝑡)−𝑢_∞,ℱ_∞=lim_𝑡→∞ℱ𝑡,𝑢_∞. (75) We show that lim_𝑡→∞𝑔(𝑡)=0. (76) Recall that if assumption (𝐻^′₁) is fulfilled, then for each 𝑧∈𝔛 we have (see [72]) ^∞₀𝑒^−𝒜𝑠𝑧𝑑𝑠=−𝒜⁻¹𝑧. (77) Using the above fact and (𝐻^′₅), we get 𝑢_∞=−𝒜⁻¹−𝒜⁻¹𝑢_∞=−𝒜⁻¹ℱ_∞=^∞₀𝑒^−𝒜𝑠ℱ_∞𝑑𝑠. (78) Linking the above equality with (𝐻^′₁), we obtain 𝑔(𝑡)=𝑒^−𝒜𝑡𝑢₀+^𝑡₀𝑒^{−𝒜(𝑡−𝑠)}ℱ(𝑠,𝑢(𝑠))𝑑𝑠−^∞₀𝑒^−𝒜𝑠ℱ_∞𝑑𝑠≤𝑀𝑒^−𝜌𝑡𝑢₀+^∞_𝑡𝑒^−𝒜𝑠ℱ_∞+𝑑𝑠_𝑡0𝑒^{−𝒜(𝑡−𝑠)}ℱ(𝑠,𝑢(𝑠))𝑑𝑠−^𝑡₀𝑒^−𝒜𝑠ℱ_∞𝑑𝑠≤𝑀𝑒^−𝜌𝑡𝑢₀+𝑀𝑒^−𝜌𝑡𝜌ℱ_∞+^𝑡₀𝑒^{−𝒜(𝑡−𝑠)}ℱ(𝑠,𝑢(𝑠))𝑑𝑠−^𝑡₀𝑒^{−𝒜(𝑡−𝑠)}ℱ_∞𝑑𝑠≤𝑀𝑒^−𝜌𝑡𝑢₀+ℱ_∞𝜌+^𝑡₀𝑀𝑒^{−𝜌(𝑡−𝑠)}ℱ(𝑠,𝑢(𝑠))−ℱ_∞𝑑𝑠≤𝑀𝑒^−𝜌𝑡𝑢₀+ℱ_∞𝜌+𝑀𝑒^{−𝜌𝑡𝑡}₀𝑒^𝜌𝑠ℱ𝑠,𝑢_∞−ℱ_∞𝑑𝑠+𝑀𝑒^{−𝜌𝑡𝑡}₀𝑒^𝜌𝑠ℱ(𝑠,𝑢(𝑠))−ℱ𝑠,𝑢_∞𝑑𝑠. (79) Next, putting 𝑐(𝑡)=𝑀𝑒^{−𝜌(𝑡)}𝑢₀+ℱ_∞𝜌+𝑀𝑒^{−𝜌𝑡𝑡}₀𝑒^𝜌𝑠ℱ𝑠,𝑢_∞−ℱ_∞𝑑𝑠 (80) and applying (𝐻^′₅), we derive the following inequality: 𝑔(𝑡)≤𝑐(𝑡)+𝑚𝑀𝑒^{−𝜌𝑡𝑡}₀𝑒^𝜌𝑠𝑔(𝑠)𝑑𝑠. (81) Hence, 𝑒^𝜌𝑡𝑔(𝑡)≤𝑒^𝜌(𝑡)𝑐(𝑡)+𝑚𝑀^𝑡₀𝑒^𝜌𝑠𝑔(𝑠)𝑑𝑠. (82) The above inequality in conjunction with Lemma 5 gives 𝑒^𝜌𝑡𝑔(𝑡)≤𝑒^𝜌𝑡𝑐(𝑡)+^𝑡₀𝑒^𝜌𝑠𝑐(𝑠)𝑚𝑀⋅exp(𝑚𝑀(𝑡−𝑠))𝑑𝑠=𝑒^𝜌𝑡𝑐(𝑡)+𝑚𝑀𝑒^{𝑚𝑀𝑡𝑡}₀𝑐(𝑠)exp𝜌−𝑚𝑀𝑠𝑑𝑠. (83) Hence, 𝑔(𝑡)≤𝑐(𝑡)+𝑚𝑀∫^𝑡₀𝑐(𝑠)𝑒^{(𝑔−𝑚𝑀)𝑠}𝑑𝑠𝑡.exp𝜌−𝑚𝑀 (84) Before proving (76), we first show that lim_𝑡→∞𝑐(𝑡)=0. (85) To prove this equality, it is sufficient to show that the last component in the formula expressing 𝑐(𝑡), that is, 𝑀𝑒^−𝜌𝑡∫^𝑡₀𝑒^𝜌𝑠‖ℱ(𝑠,𝑢_∞)−ℱ_∞‖𝑑𝑠, tends to zero as 𝑡→∞.
To this end, let us fix 𝜀>0. Assumption (𝐻^′₅) implies that there exists 𝑡₀ such that supℱ𝑠,𝑢_∞−ℱ_∞∶𝑠≥𝑡₀≤𝜀𝜌.2𝑀 (86) This inequality together with (𝐻^′₃) implies esssupℱ𝑠,𝑢_∞−ℱ_∞∶𝑠∈ℝ<∞. (87) Now, fix 𝑡₁∈ℝ such that 𝑡₁≥𝑡 and esssupℱ𝑠,𝑢_∞−ℱ_∞≤∶𝑠∈ℝ𝜀𝜌exp𝜌𝑡₁2𝑀exp𝜌𝑡₀.−1 (88) Then using (86) and (88), we conclude that for 𝑡≥𝑡₁ we have 𝑀𝑒^{−𝜌𝑡𝑡}₀𝑒^𝜌𝑠ℱ𝑠,𝑢_∞−ℱ_∞𝑑𝑠≤𝑀𝑒^{−𝜌𝑡𝑡0}₀𝑒^𝜌𝑠ℱ𝑠,𝑢_∞−ℱ_∞𝑑𝑠+𝑀𝑒^{−𝜌𝑡𝑡}_𝑡0𝑒^𝜌𝑠ℱ𝑠,𝑢_∞−ℱ_∞𝜀𝑑𝑠≤2+𝜀2=𝜀. (89) This fact proves (85).
Further, using (𝐻^′₅) and (85) and employing de L’Hospital’s rule for the fraction on the right-hand side of inequality (84), we obtain that condition (76) is satisfied. This fact completes the proof.

6. An Application to the Ginzburg-Landau Equations of Superconductivity

In this section, we formulate the gauged time-dependent Ginzburg-Landau (TDGL) equations as an abstract evolution equation in a Hilbert space. Moreover, we show applications of the above theorems to TDGL equations.

We assume that Ω is a bounded domain in ℝ^𝑛 with boundary 𝜕Ω of class 𝐶^1,1. That is, Ω is an open and connected set whose boundary 𝜕Ω is a compact (𝑛−1)-manifold described by Lipschitz continuous differentiable charts. We consider two- and three-dimensional problems (𝑛=2 and 𝑛=3, resp.). Assume that the vector potential 𝐀 takes its values in ℝ^𝑛. The vector 𝐇 will represent the (externally) applied magnetic field, which is a function of space and time; similarly to 𝐀, it takes its values in ℝ^𝑛. The function 𝛾 is defined and satisfies Lipschitz condition on 𝜕Ω, and 𝛾(𝑥)≥0 for 𝑥∈𝜕Ω. The parameters in the TDGL equations are 𝜂, a (dimensionless) friction coefficient, and 𝜅, the (dimensionless) Ginzburg-Landau parameter.

The order parameter should be thought of as the wave function of the center-of-mass motion of the “superelectrons” (Cooper pairs), whose density is 𝑛_𝑠=|𝜓|² and whose flux is 𝐉_𝑠. The vector potential 𝐀 determines the electromagnetic field; 𝐄=−𝜕_𝑡𝐀−∇𝜙 is the electric field and 𝐁=∇×𝐀 is the magnetic induction, where 𝐉, the total current, is the sum of a “normal” current 𝐉_𝑛=𝐄, the supercurrent 𝐉_𝑠, and the transport current 𝐉_𝑡=∇×𝐇. The normal current obeys Ohm’s law 𝐉_𝑛=𝜎_𝑛𝐄; the “normal conductivity” coefficient 𝜎_𝑛 is equal to one in the adopted system of units. The difference 𝐌=𝐁−𝐇 is known as the magnetization. The trivial solution (𝜓=0, 𝐁=𝐇, 𝐄=𝟎) represents the normal state, where all superconducting properties have been lost.

Now we accept the following notion: all Banach spaces are real; the (real) dual of a Banach space 𝑋 is denoted by 𝑋^′. The symbol 𝐿^𝑝(Ω), for 1≤𝑝≤∞, denotes the usual Lebesgue space, with norm ‖⋅‖_𝐿^𝑝; (⋅,⋅) is the inner product in 𝐿²(Ω). 𝑊^𝑚,2(Ω), for nonnegative integer 𝑚, is the usual Sobolev space, with norm ‖⋅‖_𝑊^𝑚,2; 𝑊^𝑚,2(Ω) is a Hilbert space for the inner product (⋅,⋅)_𝑚,2 given by (𝑢,𝑣)_𝑚,2=∑_{|𝛼|≤𝑚}(𝜕^𝛼𝑢,𝜕^𝛼𝑣) for 𝑢,𝑣∈𝑊^𝑚,2(Ω). Fractional Sobolev space 𝑊^𝑠,2(Ω), with a fractional 𝑠, is defined by interpolation ([40, Chap. VII], and [41, 49, 50]). 𝐶^𝜈(Ω), for 𝜈≥0, 𝜈=𝑚+𝜆 with 0≤𝜆<1, is the space of 𝑚 times continuous differentiable functions on Ω; those 𝑚th-order derivatives satisfy the Hölder condition with exponent 𝜆 if 𝜈 is a proper fraction; the norm ‖⋅‖_𝐶^𝜈 is defined in the usual way.

The definitions extend to the space of vector-valued functions in the standard way, with the caveat that the inner product in [𝐿²(Ω)]^𝑛 is defined by (𝑢,𝑣)=∫_Ω𝑢⋅𝑣, where the symbol (𝑢,𝑣) indicates the scalar product in ℝ^𝑛. Complex-valued functions are interpreted as vector-valued functions with two real components.

Functions that vary in space and time, like the order parameter and the vector potential, are considered as mappings from the time domain, which is a subinterval of [0,∞), into spaces of complex- or vector-valued functions defined in Ω. Let 𝑋=(𝑋,‖⋅‖_𝑋) be a Banach space of functions defined in Ω. Then, the functions are defined in Ω. Then, the functions of space and time defined on Ω×(0,𝑇), for 𝑇>0, may be considered as elements of 𝐿^𝑝(0,𝑇;𝑋), for 1≤𝑝≤∞, or 𝑊^2,𝑚(0,𝑇;𝑋), for nonnegative 𝑚, or 𝐶^𝜈(0,𝑇;𝑋), for 𝜈≥0, 𝜈=𝑚+𝜆 with 0≤𝜆<1. Detailed definitions can be found, for example, in [43].

Obviously, function spaces of ordered pairs (𝜓,𝐀), where 𝜓∶Ω→ℝ² and 𝐀∶Ω→ℝ^𝑛(𝑛=2,3), play an important role in the study of the gauged TDGL equations. We therefore adapt the following special notation: ℋ=[𝑋(Ω)]²×[𝑋(Ω)]^𝑛, for any Banach space for the order parameter 𝜓 and the vector potential 𝐀, respectively. A suitable framework for the functional analysis of the gauged TDGL equations is the Cartesian product 𝒲^1+𝛼,2=[𝑊^1+𝛼,2(Ω)]²×[𝑊^1+𝛼,2(Ω)]^𝑛, where 𝛼∈(1/2,1). This space is continuously imbedded in 𝒲^1,2∩ℒ^∞.

Assume 𝐇≠𝟎 and 𝐇∈[𝐿²(Ω)]^𝑛. Let 𝐀_𝐇 be a minimizer of the convex quadratic form 𝐽_𝜔≡𝐉_𝜔[𝐴], 𝐽_𝜔[𝐀]=_Ω𝜔(∇⋅𝐀)²+|∇×𝐀−𝐇|²𝑑𝑥, (90) on the domain 𝒟𝐽_𝜔=𝐀∈𝑊^1,2(Ω)^𝑛∶𝐧⋅𝐀=0on𝜕Ω. (91)

We now introduce the reduced vector potential 𝐀^′: 𝐀^′=𝐀−𝐀_𝐇. (92) In terms of 𝜓 and 𝐀^′ the gauged TDGL equations have the following form: 𝜕𝜓−1𝜕𝑡𝜂𝜅²Δ𝜓=𝜑inΩ×(0,∞),𝜕𝐀^′𝜕𝑡+∇×∇×𝐀^′−𝜔∇∇⋅𝐀^′=𝐅inΩ×(0,∞)𝐧⋅∇+𝛾𝜓=0,𝐧⋅𝐀^′=0,𝐧×∇⋅𝐀^′=𝟎on𝜕Ω×(0,∞). (93) Here, 𝜑 and 𝐅 are nonlinear functions of 𝜓 and 𝐀^′: 𝜑≡𝜑𝑡,𝜓,𝐀^′=1𝜂−2𝑖𝜅∇𝜓⋅𝐀^′+𝐀_𝐇−𝑖𝜅1−𝜂𝜅²𝜔𝜓∇⋅𝐀^′−𝜓𝐀^′+𝐀_𝐇²+1−𝜓²𝜓,𝐅≡𝐅𝑡,𝜓,𝐀^′=1𝜓2𝑖𝑘^∗∇𝜓−𝜓∇𝜓^∗−𝜓²𝐀^′+𝐀_𝐇−𝜕𝐀_𝐇.𝜕𝑡 (94)(95) The equations are supplemented by initial data, which is in the followimg form: 𝜓=𝜓₀,𝐀=𝐀₀onΩ×{0}, (96) where 𝜓₀ and 𝐀₀ are given, and by (92), we have 𝜓=𝜓₀,𝐀^′=𝐀₀−𝐀_𝐇(0)onΩ×{0}. (97)

In the next part we connect the evolution of the solution (𝜓,𝐀^′) of the system of (93) with the initial data (𝜓₀,𝐀₀−𝐀_𝐇(0)) with the dynamics of a vector 𝑢 in a Hilbert, space ℒ²=𝐿^𝑝(Ω)²×𝐿^𝑝(Ω)^𝑛. (98)

The following analysis is restricted to the case 𝜔>0 and the case 𝜔=0 (see [73]).

Let vector 𝑢∶[0,∞)→ℒ² represent the pair (𝜓,𝐀^′), 𝑢=𝜓,𝐀^′≡𝜓,𝐀−𝐀_𝐇, (99) and let 𝒜 be the linear self-adjoint operator in ℒ² associated with the quadratic form 𝑄_𝜔≡𝑄_𝜔[𝑢] given by the following formula: 𝑄_𝜔[𝑢]=_Ω1𝜂𝜅²∇𝜓²+𝜔∇⋅𝐀^′2+∇×𝐀^′2+𝑑𝑥_𝜕Ω𝛾𝜂𝜅²𝜓²𝑑𝜎(𝑥), (100) on the domain 𝒟𝑄_𝜔=𝒟𝒜^1/2=𝑢=𝜓,𝐀^′∈𝒲^1,2∶𝐧⋅𝐀^′=0on𝜕Ω. (101) The quadratic form 𝑄_𝜔 is nonnegative. Furthermore, since 𝜔>0, 𝑄_𝜔[𝜓,𝐀^′]+𝑐‖𝜓‖_𝐿² is coercive on 𝒲^1,2 for any constant 𝑐>0. Hence, 𝒜 is positively definite in ℒ² ([44, Chap. 1], equation (5,45)). If it does not lead to confusion, we use the same symbol 𝒜 for the restriction 𝒜_𝜓 and 𝒜_𝐀 of 𝒜 to the respective linear subspace [𝐿²(Ω)]²=[𝐿²(Ω)]²×{0} (for 𝜓) and [𝐿²(Ω)]^𝑛≡{0}×[𝐿²(Ω)]^𝑛 (for 𝐀) of ℒ².

Now, consider the initial-value problems (6) and (7) in ℒ², where ℱ(𝑡,𝑢)=(𝜑,𝐅), 𝜑 and 𝐅 is given by (94) and (95) and 𝑢₀=(𝜓₀,𝐀₀−𝐀_𝐇(0)).

With 𝜆∈(1/2,1) and 𝑢₀∈𝒲^1+𝜆,2, we say that 𝑢 are a solution of (6) and (7) on the interval [0,𝑇], for some 𝑇>0, if 𝑢∶[0,𝑇]→𝒲^1+𝜆,2 is continuous and 𝑢(𝑡)=𝑒^{−𝒜(𝑡)}+^𝑡₀𝑒^{−𝒜(𝑡−𝑠)}ℱ(𝑠,𝑢(𝑠))𝑑𝑠for𝑡∈[0,𝑇], (102) in ℒ². A mild solution of the initial-value problems (6) and (7) defines a weak solution (𝜓,𝐀^′) of the boundary value problem (93), which in turn defines a weak solution (𝜓,𝐀) of the gauged TDGL equations, provided 𝐀_𝐇 is sufficiently regular.

Namely, let us assume that 𝔛=ℒ² and 𝑢₀∈ℒ², and 𝑢∶ℝ₊×ℝ^𝑛→ℝ^𝑛 for 𝑛∈ℕ is an unknown function, 𝑢=𝑢(𝑡,𝑥). In order to apply Theorems 10 and 14, we are not going to consider 𝑢 as a function of 𝑡 and 𝑥 together, but rather as a mapping 𝑢 of variable 𝑡 into the space 𝔛=ℒ² of functions 𝑥, that is, 𝑢∶ℝ₊→ℒ², 𝑢(𝑡)(𝑥)=𝑢(𝑡,𝑥), 𝑥∈ℝ^𝑛, 𝑡∈ℝ₊.

Remark 19. Since 𝒜 is the linear self-adjoint operator in ℒ² associated with the quadratic form (100), then 𝒜 generates a semigroup {𝑒^−𝒜𝑡}_𝑡≥0 (see [74]).

Below, we formulate the principal theorem of this paper. This theorem is a simple consequence of Theorem 10.

Theorem 20. Under assumptions (𝐻₁)–(𝐻₄), problem (6)-(7) is equivalent to the system of (1)–(5) (TDGL equations) and has for each 𝑢₀∈ℒ² at least one mild solution in sense of Definition 4  𝑢=𝑢(𝑡). Moreover, each solution 𝑢(𝑡) satisfies estimate (25).

We observe that in the case when {𝑒^−𝒜𝑡}_𝑡≥0 is the semigroup of contractions and 𝑎(𝑡)≡𝑎, 𝑏(𝑡)≡𝑏 for 𝑡∈ℝ₊ are constant, after simple calculations based on estimate (25), we get that the solution 𝑢=𝑢(𝑡) has the asymptotic characterization: ‖𝑢(𝑡)‖=𝑂𝑡³,as𝑡⟶∞. (103)

For further purposes, let us formulate the following assumption:(A)𝒜 is the infinitesimal generator of an exponentially stable 𝐶₀-semigroup {𝑒^−𝒜𝑡}_𝑡≥0.

The next result of this chapter is shown in the given theorem below.

Theorem 21. Under assumptions (A) and (𝐻₂)–(𝐻₄), problem (6)-(7), which is equivalent to the systems equations (1)–(5) (TDGL equations), has a mild solution 𝑢=𝑢(𝑡) for each 𝑢₀∈ℒ² which is globally attractive and locally uniformly attractive.