Regularity for Variational Evolution Integrodifferential Inequalities

Abstract

We deal with the regularity for solutions of nonlinear functional integrodifferential equations governed by the variational inequality in a Hilbert space. Moreover, by using the simplest definition of interpolation spaces and the known regularity result, we also prove that the solution mapping from the set of initial and forcing data to the state space of solutions is continuous, which very often arises in application. Finally, an example is also given to illustrate our main result.

1. Introduction

In this paper, we deal with the regularity for solutions of nonlinear functional integrodifferential equations governed by the variational inequality in a Hilbert space 𝐻: 𝑥^′(𝑡)(𝑡)(𝑡)(𝑥(𝑡(𝑧)≤+𝐴𝑥,𝑥−𝑧+𝜙))−𝜙_𝑡0𝑘()𝑔((𝑠(𝑡)(𝑡)𝑥(0)𝑡−𝑠𝑠,𝑥))𝑑𝑠+ℎ,𝑥−𝑧,a.e.,0<𝑡≤𝑇,𝑧∈𝐻,=𝑥₀,(VIP) where 𝐴 is a unbounded linear operator associated with a sesquilinear form satisfying Gårding’s inequality and 𝜙∶𝐻→(−∞,+∞] is a lower semicontinuous, proper convex function. The nonlinear mapping 𝑔 is a Lipschitz continuous from ℝ×𝑉 into 𝐻 in the second coordinate, where 𝑉 is a dense subspace of 𝐻.

The background of these problems has emerged vigorously in such applied fields as automatic control theory, network theory, and the dynamic systems.

By using the subdifferential operator 𝜕𝜙, the control system (VIP) is represented by the following nonlinear functional differential equation on 𝐻: 𝑥^′(𝑡)+𝐴𝑥(𝑡)+𝜕𝜙(𝑥(𝑡))∋^𝑡₀𝑘()𝑔((𝑠(𝑡)𝑥(0)𝑡−𝑠𝑠,𝑥))𝑑𝑠+ℎ,0<𝑡≤𝑇,=𝑥₀.(NDE)

In Section 4.3.2 of Barbu [1] (also see Section  4.3.1 in [2]) is widely developed the existence of solutions for the case 𝑔≡0. Recently, the regular problem for solutions of the nonlinear functional differential equations with a nonlinear hemicontinuous and coercive operator 𝐴 was studied in [3]. Some results for solutions of a class of semilinear equations with the nonlinear terms have been dealt with in [3–7]. As for nontrivial physical examples from the field of visco-elastic materials modeled by integrodifferential equations on Banach spaces, we refer to [8].

In this paper, we will define 𝜙_𝜖∶𝐻→𝐻(𝜖>0) such that the function 𝜙_𝜖 is Fréchet differentiable on 𝐻 and its Frećhet differential 𝜕𝜙_𝜖 is a single valued and Lipschitz continuous on 𝐻 with Lipschitz constant 𝜖⁻¹, where 𝜕𝜙_𝜖=𝜖⁻¹(𝐼−(𝐼+𝜖𝜕𝜙)⁻¹) as is seen in Corollary  2.2 in [1, Chapter II]. It is also well-known results that lim_𝜖→0𝜙_𝜖=𝜙 and lim_𝜖→0𝜕𝜙_𝜖(𝑥)=(𝜕𝜙)⁰(𝑥) for every 𝑥∈𝐷(𝜕𝜙), where (𝜕𝜙)⁰ is the minimal segment of 𝜕𝜙. Now, we introduce the smoothing system corresponding to (NDE) as follows: 𝑥^′(𝑡)+𝐴𝑥(𝑡)+𝜕𝜙_𝜖(𝑥(𝑡))=^𝑡₀𝑘()𝑔((𝑠(𝑡)𝑥(0)𝑡−𝑠𝑠,𝑥))𝑑𝑠+ℎ,0<𝑡≤𝑇,=𝑥₀.(SDE1)

First we recall some regularity results and a variation of constant formula for solutions of the semilinear functional differential equation (in the case 𝑔≡0 in (SDE1): 𝑥^′(𝑡)+𝐴𝑥(𝑡)+𝜕𝜙_𝜖(𝑥(𝑡))=ℎ(𝑡)(1.1) in a Hilbert space 𝐻.

Next, based on the regularity results for (1.1), we intend to establish the regularity for solutions of (NDE). Here, our approach is that results of a class of semilinear equations as (1.1) on 𝐿²-regularity remain valid under the above formulation perturbed of nonlinear terms. Here, we note that sine 𝐴 is not bounded operator 𝐻 into itself, the Lipschitz continuity of nonlinear terms must be defined on some adjusted spaces (see Section 3). Moreover, using the simplest definition of interpolation spaces and known regularity, we have that the solution mapping from the set of initial and forcing data to the state space of solutions is continuous, which very often arises in application. Finally, an example is also given to illustrate our main result.

2. Preliminaries

Let 𝑉 and 𝐻 be complex Hilbert spaces forming Gelfand triple 𝑉⊂𝐻⊂𝑉^∗ with pivot space 𝐻. The norms of 𝑉, 𝐻 and 𝑉^∗ are denoted by ||⋅||, |⋅|, and ||⋅||_∗, respectively. The inner product in 𝐻 is defined by (⋅,⋅). The embeddings 𝑉↪𝐻↪𝑉^∗(2.1) are continuous. Then the following inequality easily follows: ‖𝑢‖_∗≤|𝑢|≤‖𝑢‖,∀𝑢∈𝑉.(2.2)

Let 𝑎(⋅,⋅) be a bounded sesquilinear form defined in 𝑉×𝑉 and satisfying Gårding’s inequality Re𝑎(𝑢,𝑢)≥𝜔₁‖𝑢‖²−𝜔₂|𝑢|²,𝜔₁>0,𝜔₂≥0.(2.3) Let 𝐴 be the operator associated with the sesquilinear form 𝑎(⋅,⋅): (𝐴𝑢,𝑣)=𝑎(𝑢,𝑣),𝑢,𝑣∈𝑉.(2.4) Then 𝐴 is a bounded linear operator from 𝑉 to 𝑉^∗ and −𝐴 generates an analytic semigroup in both of 𝐻 and 𝑉^∗ as is seen in [9, Theorem 6.1]. The realization for the operator 𝐴 in 𝐻 which is the restriction of 𝐴 to 𝐷(𝐴)={𝑢∈𝑉;𝐴𝑢∈𝐻}(2.5) is also denoted by 𝐴. From the following inequalities: 𝜔₁‖𝑢‖²≤Re𝑎(𝑢,𝑢)+𝜔₂|𝑢|²≤𝐶|𝐴𝑢||𝑢|+𝜔₂|𝑢|²≤max𝐶,𝜔₂‖𝑢‖_𝐷(𝐴)|𝑢|,(2.6) where ‖𝑢‖_𝐷(𝐴)=|𝐴𝑢|²+|𝑢|^21/2(2.7) is the graph norm of 𝐷(𝐴), it follows that there exists a constant 𝐶₁>0 such that ‖𝑢‖≤𝐶₁‖𝑢‖^1/2_𝐷(𝐴)|𝑢|^1/2.(2.8) Thus, we have the following sequence: 𝐷(𝐴)⊂𝑉⊂𝐻⊂𝑉^∗⊂𝐷(𝐴)^∗,(2.9) where each space is dense in the next one and continuous injection.

Lemma 2.1. With the notations (2.8), (2.9), one has (𝐷(𝐴),𝐻)_1/2,2=𝑉,(2.10) where (𝐷(𝐴),𝐻)_1/2,2 denotes the real interpolation space between 𝐷(𝐴) and 𝐻 (Section 2.4 of [10] or [11]).

The following abstract linear parabolic equation: 𝑥^′(𝑡)(𝑡)(𝑡)𝑥(0)+𝐴𝑥=ℎ,0<𝑡≤𝑇,=𝑥₀,(LE) has a unique solution 𝑥∈𝐿²(0,𝑇;𝐷(𝐴))∩𝑊^1,2(0,𝑇,𝐻) for each 𝑇>0 if 𝑥₀∈𝑉≡(𝐷(𝐴),𝐻)_1/2,2 and ℎ∈𝐿²(0,𝑇;𝐻). Moreover, one has ‖𝑥‖_𝐿²_{(0,𝑇;𝐷(𝐴))∩𝑊}^1,2_{(0,𝑇,𝐻)}≤𝐶₂𝑥_{0(𝐷(𝐴),𝐻)1/2,2}+‖ℎ‖_𝐿²_{(0,𝑇;𝐻)},(2.11) where 𝐶₂ depends on 𝑇 and 𝑀 (see [12, Theorem 2.3], [13]).

In order to substitute 𝐻 for the intermediate space 𝑉 considering 𝐴 as an operator in 𝐵(𝑉,𝑉^∗) instead of 𝐵(𝐷(𝐴),𝐻) one proves the following result.

Lemma 2.2. Let 𝑇>0. Then 𝐻=𝑥∈𝑉^∗∶^𝑇₀𝐴𝑒^𝑡𝐴𝑥²_∗𝑑𝑡<∞.(2.12) Hence, it implies that 𝐻=(𝑉,𝑉^∗)_1/2,2 in the sense of intermediate spaces generated by an analytic semigroup.

Proof. Put 𝑢(𝑡)=𝑒^𝑡𝐴𝑥 for 𝑥∈𝐻. From the result of Theorem  2.3 in [12] it follows 𝑢∈𝐿²(0,𝑇;𝑉)∩𝑊^1,20,𝑇;𝑉^∗,(2.13) hence ^𝑇₀𝐴𝑒^𝑡𝐴𝑥²_∗𝑑𝑡=^𝑇₀𝑢^′(𝑡)²_∗𝑑𝑡<∞.(2.14)
Conversely, suppose that 𝑥∈𝑉^∗ and ∫^𝑇₀||𝐴𝑒^𝑡𝐴𝑥||²_∗𝑑𝑡<∞. Put 𝑢(𝑡)=𝑒^𝑡𝐴𝑥. Then since 𝐴 is an isomorphism from 𝑉 to 𝑉^∗ there exists a constant 𝑐>0 such that ^𝑇₀‖𝑢(𝑡)‖²𝑑𝑡≤𝑐^𝑇₀‖𝐴𝑢(𝑡)‖²_∗𝑑𝑡=𝑐^𝑇₀𝐴𝑒^𝑡𝐴𝑥²_∗𝑑𝑡.(2.15) Thus, we have 𝑢∈𝐿²(0,𝑇;𝑉)∩𝑊^1,2(0,𝑇;𝑉^∗). By using the definition of real interpolation spaces by trace method, it is known that the embedding 𝐿²(0,𝑇;𝑉)∩𝑊^1,2(0,𝑇;𝑉^∗)↪𝐶([0,𝑇];𝐻) is continuous. Hence, it follows 𝑥=𝑢(0)∈𝐻.

In view of Lemma 2.2 we can apply (2.11) to (LE) in the space 𝑉^∗ as follows.

Proposition 2.3. Let 𝑥₀∈𝐻 and ℎ∈𝐿²(0,𝑇;𝑉^∗), 𝑇>0. Then there exists a unique solution 𝑥 of (LE) belonging to 𝐿²(0,𝑇;𝑉)∩𝑊^1,20,𝑇;𝑉^∗↪𝐶([0,𝑇];𝐻)(2.16) and satisfying ‖𝑥‖_𝐿²_{(0,𝑇;𝑉)∩𝑊}^1,2_(0,𝑇;𝑉^∗₎≤𝐶₂𝑥₀+‖ℎ‖_𝐿²_(0,𝑇;𝑉^∗₎,(2.17) where 𝐶₂ is a constant depending on 𝑇.

Let 𝜙∶𝑉→(−∞,+∞] be a lower semicontinuous, proper convex function. Then the subdifferential operator 𝜕𝜙 of 𝜙 is defined by 𝜕𝜙(𝑥)=𝑥^∗∈𝑉^∗;𝜙(𝑥)≤𝜙𝑦+𝑥^∗,𝑥−𝑦,𝑦∈𝑉.(2.18) First, let us concern with the following perturbation of subdifferential operator: 𝑥^′(𝑡)(𝑡)(𝑥(𝑡(𝑡)𝑥(0)+𝐴𝑥+𝜕𝜙))∋ℎ,0<𝑡≤𝑇,=𝑥₀.(VE)

Using the regularity for the variational inequality of parabolic type in case where 𝜙∶𝑉→(−∞,+∞] is a lower semicontinuous, proper convex function as is seen in [1, Section 4.3] one has the following result on (VE).

Proposition 2.4. (1) Let ℎ∈𝐿²(0,𝑇;𝑉^∗) and 𝑥₀∈𝑉 satisfying that 𝜙(𝑥₀)<∞. Then (VE) has a unique solution: 𝑥∈𝐿²(0,𝑇;𝑉)∩𝑊^1,20,𝑇;𝑉^∗↪𝐶([0,𝑇];𝐻),(2.19) which satisfies 𝑥^′(𝑡)=ℎ(𝑡)−𝐴𝑥(𝑡)−𝜕𝜙(𝑥(𝑡))⁰,‖𝑥‖_𝐿²_∩𝑊^1,2_∩𝐶≤𝐶₃1+𝑥₀+‖ℎ‖_𝐿²_(0,𝑇;𝑉^∗₎,(2.20) where 𝐶₃ is a constant and 𝐿²∩𝑊^1,2∩𝐶=𝐿²(0,𝑇;𝑉)∩𝑊^1,2(0,𝑇;𝑉^∗)∩𝐶([0,𝑇];𝐻).
(2) Let 𝐴 be symmetric and let us assume that there exist 𝑔∈𝐻 such that for every 𝜖>0 and any 𝑦∈𝐷(𝜙)𝐽_𝜖𝑦+𝜖𝑔∈𝐷𝜙,𝜙𝐽_𝜖𝑦+𝜖𝑔≤𝜙𝑦.(2.21) Then for ℎ∈𝐿²(0,𝑇;𝐻) and 𝑥₀∈𝐷(𝜙)∩𝑉,  (VE) has a unique solution: 𝑥∈𝐿²(0,𝑇;𝐷(𝐴))∩𝑊^1,2(0,𝑇;𝐻)↪𝐶([0,𝑇];𝐻),(2.22) which satisfies ‖𝑥‖_𝐿²_∩𝑊^1,2_∩𝐶≤𝐶₃1+𝑥₀+‖ℎ‖_𝐿²_{(0,𝑇;𝐻)}.(2.23)

Remark 2.5. When the principal operator 𝐴 is bounded from 𝐻 to itself, we assume that 𝜙∶𝐻→(−∞,+∞] is a lower semicontinuous, proper convex function and 𝑔∶[0,𝑇]×𝐻→𝐻 be a nonlinear mapping satisfying the following: 𝑔𝑡,𝑥₁−𝑔𝑡,𝑥₂≤𝐿𝑥₁−𝑥₂,∀𝑥₁,𝑥₂∈𝐻.(2.24) Then it is easily seen that the result of (2) of Proposition 2.4. is immediately obtained.

Remark 2.6. Here, we remark that if 𝑉 is compactly embedded in 𝐻 and 𝑥∈𝐿²(0,𝑇;𝑉)) (or the semigroup operator 𝑆(𝑡) is compact), the following embedding: 𝐿²(0,𝑇;𝑉)∩𝑊^1,20,𝑇;𝑉^∗↪𝐿²(0,𝑇;𝐻)(2.25) is compact in view of Theorem 2 of Aubin [14]. Hence, the mapping (𝑥₀,𝑓)↦𝑥 is compact from 𝑉×𝐿²(0,𝑇;𝑉^∗) to 𝐿²(0,𝑇;𝐻), which is also applicable to optimal control problem.

3. Regularity for Solutions

We start with the following assumption.

Assumption (F). Let 𝑔∶[0,𝑇]×𝑉→𝐻 be a nonlinear mapping satisfying the following: 𝑔(𝑡,𝑥)−𝑔𝑡,𝑦≤𝐿𝑥−𝑦,𝑔(𝑡,0)=0∀𝑥,𝑦∈𝑉(3.1) for a positive constant 𝐿.
For 𝑥∈𝐿²(0,𝑇;𝑉) we set 𝑓(𝑡,𝑥)=^𝑡₀𝑘(𝑡−𝑠)𝑔(𝑠,𝑥(𝑠))𝑑𝑠,(3.2) where 𝑘 belongs to 𝐿²(0,𝑇).

Lemma 3.1. Let 𝑥∈𝐿²(0,𝑇;𝑉), 𝑇>0. Then 𝑓(⋅,𝑥)∈𝐿²(0,𝑇;𝐻). And 𝑓(⋅,𝑥)_{𝐿2(0,𝑇;𝐻)}≤𝐿‖𝑘‖_𝐿²√𝑇‖𝑥‖_𝐿²_{(0,𝑇;𝑉)}.(3.3) Moreover, if 𝑥₁,𝑥₂∈𝐿²(0,𝑇;𝐻), then 𝑓⋅,𝑥₁−𝑓⋅,𝑥_{2𝐿2(0,𝑇;𝐻)}≤𝐿‖𝑘‖√𝑇𝑥₁−𝑥_{2𝐿2(0,𝑇;𝑉)}.(3.4)

The proof is immediately obtained from Assumption (F).

For every 𝜖>0, define 𝜙_𝜖(𝑥)=inf𝑥−𝐽_𝜖𝑥²_∗𝐽2𝜖+𝜙_𝜖𝑥∶𝑥∈𝐻,(3.5) where 𝐽_𝜖=(𝐼+𝜖𝜕𝜙)⁻¹. Then the function 𝜙_𝜖 is Frećhet differentiable on 𝐻 and its Frećhet differential 𝜕𝜙_𝜖 is Lipschitz continuous on 𝐻 with Lipschitz constant 𝜖⁻¹ where 𝜕𝜙_𝜖=𝜖⁻¹(𝐼−(𝐼+𝜖𝜕𝜙)⁻¹) as is seen in Corollary 2.2 in [1, Chapter II]. It is also well-known results that lim_𝜖→0𝜙_𝜖=𝜙 and lim_𝜖→0𝜕𝜙_𝜖(𝑥)=(𝜕𝜙)⁰(𝑥) for every 𝑥∈𝐷(𝜕𝜙), where (𝜕𝜙)⁰ is the minimal segment of 𝜕𝜙.

Now, one introduces the smoothing system corresponding to (NDE) as follows: 𝑥^′(𝑡)+𝐴𝑥(𝑡)+𝜕𝜙_𝜖(𝑥(𝑡()(𝑡)𝑥(0)))=𝑓𝑡,𝑥+ℎ,0<𝑡≤𝑇,=𝑥₀.(SDE2) Since −𝐴 generates a semigroup 𝑆(𝑡) on 𝐻, the mild solution of (SDE2) can be represented by 𝑥_𝜖(𝑡)=𝑆(𝑡)𝑥₀+^𝑡₀𝑆(𝑡−𝑠)𝑓𝑠,𝑥_𝜖+ℎ(𝑠)−𝜕𝜙_𝜖𝑥_𝜖(𝑠)𝑑𝑠.(3.6)

One will use a fixed point theorem and a step and step method to get the global solution for (NDE). Then one needs the following hypothesis.

Assumption (A). (𝜕𝜙)⁰ is uniformly bounded, that is, 𝜕𝜙⁰𝑥≤𝑀₁,𝑥∈𝑉.(3.7)

Lemma 3.2. For given 𝜖, 𝜆>0, let 𝑥_𝜖 and 𝑥_𝜆 be the solutions of (SDE2) corresponding to 𝜖 and 𝜆, respectively. Then there exists a constant 𝐶 independent of 𝜖 and 𝜆 such that 𝑥_𝜖−𝑥_{𝜆𝐶([0,𝑇];𝐻)∩𝐿2(0,𝑇;𝑉)}≤𝐶(𝜖+𝜆),0<𝑇.(3.8)

Proof. From (SDE2) we have 𝑥^′_𝜖(𝑡)−𝑥^′_𝜆(𝑡)+𝐴𝑥_𝜖(𝑡)−𝑥_𝜆(𝑡)+𝜕𝜙_𝜖𝑥_𝜖(𝑡)−𝜕𝜙_𝜆𝑥_𝜆(𝑡)=𝑓𝑡,𝑥_𝜖−𝑓𝑡,𝑥_𝜆,(3.9) and hence, from (2.3) and multiplying by 𝑥_𝜖(𝑡)−𝑥_𝜆(𝑡), it follows that 12𝑑𝑥𝑑𝑡^𝜖(𝑡)−𝑥_𝜆(𝑡)²+𝜔₁𝑥_𝜖(𝑡)−𝑥_𝜆(𝑡)²+𝜕𝜙_𝜖𝑥_𝜖(𝑡)−𝜕𝜙_𝜆𝑥_𝜆(𝑡),𝑥_𝜖(𝑡)−𝑥_𝜆(𝑡)≤𝑓𝑡,𝑥_𝜖−𝑓𝑡,𝑥_𝜆,𝑥_𝜖(𝑡)−𝑥_𝜆(𝑡)+𝜔₂𝑥_𝜖(𝑡)−𝑥_𝜆(𝑡)².(3.10) Here, we note 𝑓𝑡,𝑥_𝜖−𝑓𝑡,𝑥_𝜆≤𝐿‖𝑘‖_𝐿²𝑥_𝜖(⋅)−𝑥_𝜆(⋅)_{𝐿2(0,𝑡;𝑉)𝑇0}𝑥_𝜖(⋅)−𝑥_𝜆(⋅)²_{𝐿2()0,𝑡;𝑉}𝑑𝑡=𝑇^𝑇₀𝑥_𝜖(𝑡)−𝑥_𝜆(𝑡)²𝑑𝑡.(3.11) Thus, we have 𝑓𝑡,𝑥_𝜖−𝑓𝑡,𝑥_𝜆,𝑥_𝜖(𝑡)−𝑥_𝜆(𝑡)≤𝑓𝑡,𝑥_𝜖−𝑓𝑡,𝑥_𝜆⋅𝑥_𝜖(𝑡)−𝑥_𝜆(𝑡≤𝜔₁)2𝑇𝐿‖𝑘‖_𝐿²²𝑓𝑡,𝑥_𝜖−𝑓𝑡,𝑥_𝜆²+𝑇𝐿‖𝑘‖_𝐿²²2𝜔₁𝑥_𝜖(𝑡)−𝑥_𝜆(𝑡)²≤𝜔₁𝑥2𝑇^𝜖(⋅)−𝑥_𝜆(⋅)²_{𝐿2()0,𝑡;𝐻}+𝑇𝐿‖𝑘‖_𝐿²²2𝜔₁𝑥_𝜖(𝑡)−𝑥_𝜆(𝑡)².(3.12) Therefore, by using the monotonicity of 𝜕𝜙 and integrating (3.10) over [0,𝑇] it holds 12𝑥_𝜖(𝑡)−𝑥_𝜆(𝑡)²+𝜔₁2^𝑇₀𝑥_𝜖(𝑡)−𝑥_𝜆(𝑡)²≤𝑑𝑡_𝑇0𝜕𝜙_𝜖𝑥_𝜖(𝑡)−𝜕𝜙_𝜆𝑥_𝜆(𝑡),𝜆𝜕𝜙_𝜆𝑥_𝜆(𝑡)−𝜖𝜕𝜙_𝜖𝑥_𝜖(𝑡+𝑇)𝑑𝑡𝐿||𝑘||_𝐿²²2𝜔₁+𝜔₂^𝑇₀𝑥_𝜖(𝑡)−𝑥_𝜆(𝑡)²𝑑𝑡.(3.13) Here, we used that 𝜕𝜙_𝜖𝑥_𝜖(𝑡)=𝜖⁻¹𝑥_𝜖(𝑡)−𝐼+𝜖𝜕𝜙⁻¹𝑥_𝜖(𝑡).(3.14) Since |𝜕𝜙_𝜖(𝑥)|≤|(𝜕𝜙)⁰𝑥| for every 𝑥∈𝐷(𝜕𝜙) it follows from Assumption (A) and using Gronwall’s inequality that 𝑥_𝜖−𝑥_{𝜆𝐶([0,𝑇];𝐻)∩𝐿2(0,𝑇;𝑉)}≤𝐶(𝜖+𝜆),0<𝑇.(3.15)

Let 𝑥∈𝐿¹(0,𝑇;𝑉). Then it is well known that lim_ℎ→0ℎ^−1ℎ₀‖𝑥(𝑡+𝑠)−𝑥(𝑡)‖𝑑𝑠=0(3.16) for almost all point of 𝑡∈(0,𝑇).

Definition 3.3. The point 𝑡 which permits (3.16) to hold is called the Lebesgue point of 𝑥.

We establish the following results on the solvability of (NDE).

Theorem 3.4. Let Assumptions (F) and (A) be satisfied. Then for every (𝑥₀,ℎ)∈𝑉×𝐿²(0,𝑇;𝑉^∗), (NDE) has a unique solution: 𝑥∈𝐿²(0,𝑇;𝑉)∩𝑊^1,20,𝑇;𝑉^∗∩𝐶([0,𝑇];𝐻),(3.17) and there exists a constant 𝐶₄ depending on 𝑇 such that ‖𝑥‖_𝐿²_∩𝑊^1,2_∩𝐶≤𝐶₄1+𝑥₀+‖ℎ‖_𝐿²_(0,𝑇;𝑉^∗₎.(3.18)

Proof. Let us fix 𝑇₀>0 such that 𝐶₁𝐶₂𝜖⁻¹+𝑇₀𝐿‖𝑘‖_𝐿²𝑇₀√2^1/2<1.(3.19) Let 𝑦∈𝐿²(0,𝑇₀;𝑉). Then 𝑓(⋅,𝑦(⋅))∈𝐿²(0,𝑇₀;𝐻) from Assumption (F). Set (𝐹𝑥)(𝑡)=𝑓(𝑡,𝑥(𝑡))−𝜕𝜙_𝜖(𝑥(𝑡)),0≤𝑡≤𝑇₀.(3.20) Then from Lemma 3.1 it follows that 𝐹𝑥₁(𝑡)−𝐹𝑥₂(𝑡)≤𝜖⁻¹+𝑇₀𝐿‖𝑘‖_𝐿²𝑥₁(𝑡)−𝑥₂(𝑡).(3.21) For 𝑖=1,2, we consider the following equation: 𝑥^′_𝑖(𝑡)+𝐴𝑥_𝑖(𝑡)=𝐹𝑦_𝑖(𝑡)+ℎ(𝑡),0<𝑡≤𝑇₀,𝑥_𝑖(0)=𝑥₀.(3.22) Then 𝑑𝑥𝑑𝑡¹(𝑡)−𝑥₂(𝑡)+𝐴𝑥₁(𝑡)−𝑥₂(𝑡)=𝐹𝑦₁(𝑡)−𝐹𝑦₂(𝑡)𝑥,𝑡>0,₁(0)−𝑥₂(0)=0.(3.23) From (2.11) it follows that 𝑥₁−𝑥_{2𝐿2(0,𝑇0;𝐷(𝐴0))∩𝑊1,2(0,𝑇0;𝐻)}≤𝐶₂𝐹𝑦₁−𝐹𝑦_{2𝐿2(0,𝑇0;𝐻)}.(3.24) Using the Hölder inequality we also obtain that 𝑥₁−𝑥_{2𝐿2(0,𝑇0;𝐻)}=^𝑇0₀𝑥₁(𝑡)−𝑥₂(𝑡)²𝑑𝑡^1/2=^𝑇0₀^𝑡₀𝑥̇₁(𝜏)−𝑥̇₂(𝜏)𝑑𝜏²𝑑𝑡^1/2≤^𝑇0₀𝑡^𝑡₀𝑥̇₁(𝜏)−𝑥̇₂(𝜏)²𝑑𝜏𝑑𝑡^1/2≤√𝑇₀2𝑥₁−𝑥_{2𝑊1,2(0,𝑇0;𝐻)}.(3.25) Therefore, in terms of (2.8) and (3.25) we have 𝑥₁−𝑥_{2𝐿2(0,𝑇0;𝑉)}≤𝐶₁𝑥₁−𝑥_2𝐿^1/2_{20,𝑇0𝐴;𝐷0}𝑥₁−𝑥_2𝐿^1/2_{20,𝑇0;𝐻}≤𝐶₁𝑥₁−𝑥_2𝐿^1/2_{20,𝑇0𝐴;𝐷0}𝑇₀√2^1/2𝑥₁−𝑥_2𝑊^1/2_{1,20,𝑇0;𝐻}≤𝐶₁𝑇₀√2^1/2𝑥₁−𝑥_{2𝐿2(0,𝑇0;𝐷(𝐴0))∩𝑊1,2(0,𝑇0;𝐻)}≤𝐶₁𝐶₂𝑇₀√2^1/2𝐹𝑦₁−𝐹𝑦_{2𝐿2(0,𝑇0∶𝐻)}≤𝐶₁𝐶₂𝜖⁻¹+𝑇₀𝐿‖𝑘‖_𝐿²𝑇₀√2^1/2𝑦₁−𝑦_{2𝐿2(0,𝑇0;𝑉)}.(3.26) So by virtue of the condition (3.19) the contraction principle gives that (SDE2) has a unique solution in [0,𝑇₀]. Thus, letting 𝜆→0 in Lemma 3.1 we can see that there exists a constant 𝐶 independent of 𝜖 such that 𝑥_𝜖−𝑥_{𝐶([0,𝑇0];𝐻)∩𝐿2(0,𝑇0;𝑉)}≤𝐶𝜖,0<𝑇₀,(3.27) and hence, lim_𝜖→0𝑥_𝜖(𝑡)=𝑥(𝑡) exists in 𝐻. From Assumption (F) and (3.27) it follows that 𝑓⋅,𝑥_𝜖→𝑓(⋅,𝑥),stronglyin𝐿²0,𝑇₀;𝐻,𝐴𝑥_𝑛→𝐴𝑥,stronglyin𝐿²0,𝑇₀;𝑉^∗.(3.28) Since 𝜕𝜙_𝜖(𝑥_𝜖) is uniformly bounded by Assumption (A), from (3.27), (3.28) we have that 𝑑𝑥𝑑𝑡^𝜖→𝑑𝑑𝑡𝑥,weaklyin𝐿²0,𝑇₀;𝑉^∗,(3.29) therefore 𝜕𝜙_𝜖𝑥_𝜖→𝑓(⋅,𝑥)+ℎ−𝑥^′−𝐴𝑥,weaklyin𝐿²0,𝑇₀;𝑉^∗.(3.30) Since (𝐼+𝜖𝜕𝜙)⁻¹𝑥_𝜖→𝑥strongly and 𝜕𝜙 is demiclosed, we have that 𝑓(⋅,𝑥)+ℎ−𝑥′−𝐴𝑥∈𝜕𝜙(𝑥)in𝐿²0,𝑇₀;𝑉^∗.(3.31) Thus we have proved that 𝑥(𝑡) satisfies a.e. on (0,𝑇₀) the equation  (NDE).
Let 𝑦 be the solution of 𝑦^′(𝑡)+𝐴𝑦(𝑡)+𝜕𝜙𝑦(𝑡)∋0,0<𝑡≤𝑇₀,𝑦(0)=𝑥₀,(3.32) then, it implies 𝑑𝑥(𝑡)(𝑡)𝑥(𝑡)(𝑡)(𝑥(𝑡𝑦(𝑡)()(𝑡).𝑑𝑡−𝑦+𝐴−𝑦+𝜕𝜙))−𝜕𝜙∋𝑓𝑡,𝑥+ℎ(3.33) Noting that ||⋅||≤|⋅|≤||⋅||, by multiplying by 𝑥(𝑡)−𝑦(𝑡) and using the monotonicity of 𝜕𝜙 and (2.3), we obtain 12𝑑𝑥(𝑡)(𝑡𝑑𝑡−𝑦)²+𝜔₁𝑥(𝑡)−𝑦(𝑡)²≤𝜔₂𝑥(𝑡)−𝑦(𝑡)²+𝑓(𝑡,𝑥)+ℎ(𝑡)⋅𝑥(𝑡)−𝑦(𝑡).(3.34) Since 𝑓()(𝑡⋅𝑥(𝑡)(𝑡≤1𝑡,𝑥+ℎ)−𝑦)2𝜔₁𝑓(𝑡,𝑥)+ℎ(𝑡))²+𝜔₁2𝑥(𝑡)−𝑦(𝑡)²(3.35) for every 𝑐>0 and by integrating on (3.34) over (0,𝑡) we have 𝑥(𝑡)−𝑦(𝑡)²+𝜔₁^𝑡₀𝑥(𝑠)−𝑦(𝑠)²≤1𝑑𝑠𝜔₁𝑓(⋅,𝑥)+ℎ_{𝐿2(0,𝑇0;𝑉∗)}+2𝜔₂^𝑡₀𝑥(𝑠)−𝑦(𝑠)²𝑑𝑠(3.36) and by Gronwall’s inequality: 𝑥(𝑡)−𝑦(𝑡)²+𝜔₁^𝑡₀𝑥(𝑠)−𝑦(𝑠)²𝑑𝑠≤𝜔₁⁻¹𝑒^2𝜔2𝑇0𝑓(⋅,𝑥)+ℎ²_{𝐿20,𝑇0;𝑉∗}.(3.37) Let us fix 𝑇₀>𝑇₁>0 so that 𝑇₁ is a Lebesgue point of 𝑥, 𝜙(𝑥(𝑇₁))<∞, and 𝜔₁⁻¹𝑒^2𝜔2𝑇1𝑇₁𝐿‖𝑘‖_𝐿²<𝜔₁.(3.38) Put 𝑁=𝜔₁⁻²𝑒^𝜔2𝑇1,(3.39) then from Assumption (F) it follows 𝑥−𝑦_{𝐿2(0,𝑇1;𝑉)}≤𝑁𝑓(⋅,𝑥)+ℎ_{𝐿2(0,𝑇1;𝑉∗)}≤𝑁𝑇₁𝐿‖𝑘‖_𝐿²‖𝑥‖_𝐿²_{(0,𝑇1;𝑉)}+𝑁‖ℎ‖_𝐿²_{(0,𝑇1∶𝑉}^∗₎(3.40) and hence, from (2.17) in Proposition 2.3, we have that ‖𝑥‖_𝐿²_{(0,𝑇1;𝑉)}≤11−𝑁𝑇₁𝐿‖𝑘‖_𝐿²𝑦_{𝐿2(0,𝑇1;𝑉)}+𝑁‖ℎ‖_𝐿²_{(0,𝑇1∶𝑉}^∗₎≤11−𝑁𝑇₁𝐿‖𝑘‖_𝐿²𝐶₂1+𝑥₀+𝑁‖ℎ‖_𝐿²_{(0,𝑇1∶𝑉}^∗₎≤𝐶₄1+𝑥₀+‖ℎ‖_𝐿²_{(0,𝑇1∶𝑉}^∗₎(3.41) for some positive constant 𝐶₄. Since the condition (3.38) is independent of initial values, noting the Assumption (A), the solution of (NDE) can be extended to the internal [0,𝑛𝑇₁] for natural number 𝑛, that is, for the initial 𝑥(𝑛𝑇₁) in the interval [𝑛𝑇₁,(𝑛+1)𝑇₁], as analogous estimate (3.41) holds for the solution in [0,(𝑛+1)𝑇₁]. The norm estimate of 𝑥 in 𝑊^1,2(0,𝑇;𝐻) can be obtained by acting on both side of (NDE) by 𝑥′(𝑡) and by using 𝑑𝜙(𝑥(𝑡=𝑔(𝑡),𝑑𝑑𝑡))𝑥(𝑡)𝑑𝑡,a.e.,0<𝑡,(3.42) for all 𝑔(𝑡)∈𝜕𝜙(𝑥(𝑡)). Furthermore, the estimate (3.18) is immediately obtained from (3.41).

Theorem 3.5. Let Assumptions (F) and (A) be satisfied and (𝑥₀,ℎ)∈𝑉×𝐿²(0,𝑇;𝑉^∗), then the solution 𝑥 of (NDE) belongs to 𝑥∈𝐿²(0,𝑇;𝑉)∩𝑊^1,2(0,𝑇;𝑉^∗) and the mapping: 𝑉×𝐿²0,𝑇;𝑉^∗∋𝑥₀,ℎ↦𝑥∈𝐿²(0,𝑇;𝑉)∩𝐶([0,𝑇];𝐻)(3.43) is continuous.

Proof. If (𝑥₀,ℎ)∈𝑉×𝐿²(0,𝑇;𝑉^∗) then 𝑥 belongs to 𝐿²(0,𝑇;𝑉)∩𝑊^1,2(0,𝑇;𝑉^∗) form Theorem 3.4. Let (𝑥_0𝑖,ℎ_𝑖)∈𝑉×𝐿²(0,𝑇;𝑉^∗) and 𝑥_𝑖 be the solution of (NDE) with (𝑥_0𝑖,ℎ_𝑖) in place of (𝑥₀,ℎ) for 𝑖=1,2. Multiplying on (NDE) by 𝑥₁(𝑡)−𝑥₂(𝑡), we have 12𝑑𝑥𝑑𝑡¹(𝑡)−𝑥₂(𝑡)²+𝜔₁𝑥₁(𝑡)−𝑥₂(𝑡)²≤𝜔₂𝑥₁(𝑡)−𝑥₂(𝑡)²+𝑓𝑡,𝑥₁−𝑓𝑡,𝑥₂𝑥₁(𝑡)−𝑥₂(𝑡+ℎ₁)(𝑡)−ℎ₂(𝑡)_∗𝑥₁(𝑡)−𝑥₂(𝑡).(3.44) Let us fix 𝑇₁>𝑇₂>0 so that 𝑇₂ is a Lebesgue point of 𝑥, 𝜙(𝑥(𝑇₂)<∞, and 𝜔₁−𝜔₁⁻¹𝑒^2𝜔2𝑇2𝑇₂𝐿‖𝐾‖_𝐿²>0.(3.45) Since ℎ₁(𝑡)−ℎ₂(𝑡))_∗𝑥₁(𝑡)−𝑥₂(𝑡≤1)𝜔₁ℎ₁(𝑡)−ℎ₂(𝑡)²_∗+𝜔₁4𝑥₁(𝑡)−𝑥₂(𝑡)²,(3.46) by integrating on (3.44) over [0,𝑇₂] where 𝑇₂<𝑇 and as is seen in (3.37), it follows 𝑥₁−𝑥₂²_{𝐶0,𝑇2;𝐻}+𝜔₁2𝑥₁−𝑥₂²_{𝐿20,𝑇2;𝑉}≤𝑥₀₁−𝑥₀₂²+1𝜔₁𝑓𝑡,𝑥₁−𝑓𝑡,𝑥₂²_{𝐿2(0,𝑇2;𝐻)}+2𝜔₁ℎ₁−ℎ_{2𝐿2(0,𝑇2;𝑉∗)}≤𝑥₀₁−𝑥₀₂²+𝜔₁⁻¹𝑇₂𝐿‖𝐾‖_𝐿²𝑥₁−𝑥₂²_{𝐿20,𝑇2;𝑉}+2𝜔₁ℎ₁−ℎ_{2𝐿2(0,𝑇2;𝑉∗)}.(3.47) Putting that 𝑁₁𝜔≡min1,¹2−𝜔₁⁻¹√𝑇₂𝐿‖𝐾‖_𝐿^21/2,𝑁₂2≡max1,𝜔₁,(3.48) we have 𝑥₁−𝑥_{2𝐿2∩𝐶}≤2𝑁₂1−𝑁₁𝑥₀₁−𝑥₀₂+ℎ₁−ℎ₂.(3.49) Suppose (𝑥_0𝑛,ℎ_𝑛)→(𝑥₀,ℎ) in 𝑉×𝐿²(0,𝑇;𝑉^∗), and let 𝑥_𝑛 and 𝑥 be the solutions (SDE2) with (𝑥_0𝑛,ℎ_𝑛) and (𝑥₀,ℎ), respectively. Then, by virtue of (3.44) and (3.49), we see that 𝑥_𝑛→𝑥 in 𝐿²(0,𝑇₂,𝑉)∩𝑊^1,2(0,𝑇₂,𝑉^∗)↪𝐶([0,𝑇₂];𝐻). This implies that 𝑥_𝑛(𝑇₂)→𝑥(𝑇₂) in 𝐻. Therefore the same argument shows that 𝑥_𝑛→𝑥 in 𝐿²𝑇₂,min2𝑇₂,𝑇;𝑉∩𝐶𝑇₂,min2𝑇₂,𝑇;𝐻.(3.50) Repeating this process, we conclude that 𝑥_𝑛→𝑥 in 𝐿²(0,𝑇;𝑉)∩𝑊^1,2(0,𝑇₂,𝑉^∗)↪𝐶([0,𝑇₂];𝐻).

4. Example

Let Ω be bounded domain in ℝ^𝑛 with smooth boundary 𝜕Ω. We define the following spaces: 𝐻¹(Ω)=𝑢∶𝑢,𝜕𝑢𝜕𝑥_𝑖∈𝐿²(Ω),𝐻,𝑖=1,2,…,𝑛₂(Ω)=𝑢∶𝑢,𝜕𝑥𝜕𝑥_𝑖,𝜕²𝑢𝜕𝑥_𝑖𝜕𝑥_𝑗∈𝐿²(Ω),𝐻,𝑖,𝑗=1,2,…,𝑛₁₀(Ω)=𝑢∶𝑢∈𝐻¹(Ω),𝑢_𝜕Ω=0=theclosureof𝐶^∞₀(Ω)in𝐻¹(Ω),(4.1) where 𝜕/𝜕𝑥_𝑖𝑢 and 𝜕²/𝜕𝑥_𝑖𝜕𝑥_𝑗𝑢 are the derivative of 𝑢 in the distribution sense. The norm of 𝐻¹₀(Ω) is defined by ‖𝑢‖=_Ω^𝑛_𝑖=1𝜕𝑢(𝑥)𝜕𝑥_𝑖²𝑑𝑥^1/2.(4.2) Hence 𝐻¹₀(Ω) is a Hilbert space. Let 𝐻⁻¹(Ω) = 𝐻¹₀(Ω)^∗ be a dual space of 𝐻¹₀(Ω). For any 𝑙∈𝐻⁻¹(Ω) and 𝑣∈𝐻¹₀(Ω), the notation (𝑙,𝑣) denotes the value 𝑙 at 𝑣. In what follows, we consider the regularity for given equations in the spaces: 𝑉=𝐻¹₀(Ω)=𝑢∈𝐻¹(Ω);𝑢=0on𝜕Ω,𝐻=𝐿²(Ω),𝑉^∗=𝐻⁻¹(Ω)(4.3) as introduced in Section 2. We deal with the Dirichlet condition’s case as follows.

Assume that 𝑎_𝑖𝑗=𝑎_𝑗𝑖 are continuous and bounded on Ω and {𝑎_𝑖𝑗(𝑥)} is positive definite uniformly in Ω, that is, there exists a positive number 𝛿 such that ^𝑛_𝑖,𝑗=1𝑎_𝑖𝑗(𝑥)𝜉_𝑖𝜉_𝑗≥𝛿𝜉²,∀𝜉∈Ω.(4.4) Let 𝑏_𝑖∈𝐿^∞(Ω),𝑐∈𝐿^∞(Ω),𝛽_𝑖=^𝑛_𝑗=1𝜕𝑎_𝑖𝑗𝜕𝑥_𝑗+𝑏_𝑖.(4.5) For each 𝑢,𝑣∈𝐻¹₀(Ω), let us consider the following sesquilinear form: 𝑎(𝑢,𝑣)=_Ω^𝑛_𝑖,𝑗=1𝑎_𝑖𝑗𝜕𝑢𝜕𝑥_𝑖𝜕𝑣𝜕𝑥_𝑗+^𝑛_𝑗=1𝛽_𝑖𝜕𝑢𝜕𝑥_𝑖𝑣+𝑐𝑢𝑣𝑑𝑥.(4.6) Since {𝑎_𝑖𝑗} is real symmetric, by (4.4) the inequality: ^𝑛_𝑖,𝑗=1𝑎_𝑖𝑗(𝑥)𝜉_𝑖𝜉_𝑗≥𝛿𝜉²(4.7) holds for all complex vectors 𝜉=(𝜉₁,…,𝜉_𝑛). By hypothesis, there exists a constant 𝐾 such that |𝛽_𝑖(𝑥)|≤𝐾 and 𝑐(𝑥)≤𝐾 hold a.e., hence Re𝑎(𝑢,𝑢)≥_Ω𝛿^𝑛_𝑖=1𝜕𝑢𝜕𝑥_𝑖²𝑑𝑥−𝐾_Ω^𝑛_𝑖=1𝜕𝑢𝜕𝑥_𝑖|𝑢|𝑑𝑥−𝐾_Ω|𝑢|²𝑑𝑥≥𝛿_Ω^𝑛_𝑖=1𝜕𝑢𝜕𝑥_𝑖²𝑑𝑥−𝐾_Ω^𝑛_𝑖=1𝜖2𝜕𝑢𝜕𝑥_𝑖²+1|𝑢|2𝜖²𝑑𝑥−𝐾_Ω|𝑢|²=𝜖𝑑𝑥𝛿−2𝐾^𝑛_𝑖=1Ω𝜕𝑢𝜕𝑥_𝑖²𝑑𝑥−𝑛𝐾2𝜖+𝐾_Ω|𝑢|²𝑑𝑥.(4.8) By choosing 𝜖=𝛿𝐾⁻¹, we have ()≥𝛿Re𝑎𝑢,𝑢2^𝑛_𝑖=1Ω𝜕𝑢𝜕𝑥_𝑖²𝑑𝑥−𝑛𝐾²2𝛿+𝐾_Ω|𝑢|²=𝛿𝑑𝑥2‖𝑢‖²₁−𝑛𝐾²𝛿2𝛿+𝐾+2‖𝑢‖².(4.9) By virtue of Lax-Milgram theorem, we know that for any 𝑣∈𝑉 there exists 𝑓∈𝑉^∗ such that 𝑎(𝑢,𝑣)=𝑓,𝑣.(4.10) Therefore, we know that the associated operator 𝐴∶𝑉→𝑉^∗ defined by (𝐴𝑢,𝑣)=−𝑎(𝑢,𝑣),𝑢,𝑣∈𝑉(4.11) is bounded and satisfies conditions (2.3) in Section 2.

Let 𝑔∶[0,𝑇]×𝑉→𝐻 be a nonlinear mapping defined by 𝑔(𝑡,𝑢(𝑡,𝑥))=^𝑡₀^𝑛_𝑖=1𝜕𝜕𝑥_𝑖𝜎_𝑖(𝑠,∇𝑢(𝑠,𝑥))𝑑𝑠.(4.12)

We assume the following.

Assumption (F1). The partial derivatives 𝜎_𝑖(𝑠,𝜉), 𝜕/𝜕𝑡𝜎_𝑖(𝑠,𝜉) and 𝜕/𝜕𝜉_𝑗𝜎_𝑖(𝑠,𝜉), exist and continuous for 𝑖=1,2, 𝑗=1,2,…,𝑛, and 𝜎_𝑖(𝑠,𝜉) satisfies an uniform Lipschitz condition with respect to 𝜉, that is, there exists a constant 𝐿>0 such that 𝜎_𝑖(𝑠,𝜉)−𝜎_𝑖𝑠,𝜉̂≤𝐿𝜉−𝜉̂,(4.13) where |⋅| denotes the norm of 𝐿²(Ω).

Lemma 4.1. If Assumption (F1) is satisfied, then the mapping 𝑡↦𝑔(𝑡,⋅) is continuously differentiable on [0,𝑇] and 𝑢↦𝑔(⋅,𝑢) is Lipschitz continuous on 𝑉.

Proof. Put 𝑔₁(𝑠,𝑢)=^𝑛_𝑖=1𝜕𝜕𝑥_𝑖𝜎_𝑖(𝑠,∇𝑢),(4.14) then we have 𝑔₁(𝑠,𝑢)∈𝐻⁻¹(Ω). For each 𝑤∈𝐻¹₀(Ω), we satisfy the following that 𝑔₁(𝑠,𝑢),𝑤=−^𝑛_𝑖=1𝜎_𝑖(),𝜕𝑠,∇𝑢𝜕𝑥_𝑖𝑤.(4.15) The nonlinear term is given by 𝑔(𝑡,𝑢)=^𝑡₀𝑔₁(𝑠,𝑢)𝑑𝑠.(4.16) For any 𝑤∈𝐻¹₀(Ω), if 𝑢 and 𝑢 belong to 𝐻¹₀(Ω), by Assumption (F1) we obtain 𝑔(𝑡,𝑢)−𝑔(𝑡,𝑢),𝑤≤𝐿𝑇‖𝑢−𝑢‖‖𝑤‖.(4.17)

We set 𝑓(𝑡,𝑢)=^𝑡₀𝑘(𝑡−𝑠)^𝑠₀^𝑛_𝑖=1𝜕𝜕𝑥_𝑖𝜎_𝑖(𝜏,∇𝑢(𝜏,𝑥))𝑑𝜏𝑑𝑠,(4.18) where 𝑘 belongs to 𝐿²(0,𝑇). Let 𝜙∶𝐻¹₀(Ω)→(−∞,+∞] be a lower semicontinuous, proper convex function. Now in virtue of Lemma 4.1, we can apply the results of Theorem 3.4 as follows.

Theorem 4.2. Let Assumption (F1) be satisfied. Then for any 𝑢₀∈𝐻¹₀(Ω) and ℎ∈𝐿²(0,𝑇;𝐻⁻¹(Ω)), the following nonlinear problem: 𝑢^′(𝑡)(𝑡)(𝑡)(𝑢(𝑡(𝑧)≤𝑓()(𝑡)(𝑡)+𝐴𝑢,𝑢−𝑧+𝜙))−𝜙𝑡,𝑢+ℎ,𝑢−𝑧,a.e.,0<𝑡≤𝑇,𝑧∈𝐿²(Ω),𝑢(0)=𝑢₀(4.19) has a unique solution: 𝑢∈𝐿²0,𝑇;𝐻¹₀(Ω)∩𝑊^1,20,𝑇;𝐻⁻¹(Ω)↪𝐶[0,𝑇];𝐿²(Ω).(4.20)
Furthermore, the following energy inequality holds: there exists a constant 𝐶_𝑇 depending on 𝑇 such that ‖𝑢‖_𝐿²_∩𝑊^1,2≤𝐶_𝑇1+𝑢₀+‖ℎ‖_𝐿²_(0,𝑇;𝐻⁻¹_(Ω)).(4.21)

Acknowledgment

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011-0026609).