On Parabolic Variational Inequalities with Multivalued Terms and Convex Functionals

Vy Khoi Le; Klaus Schmitt

doi:10.1515/ans-2018-0004

Article Open Access

On Parabolic Variational Inequalities with Multivalued Terms and Convex Functionals

Vy Khoi Le and Klaus Schmitt

Published/Copyright: March 8, 2018

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Advanced Nonlinear Studies Volume 18 Issue 2

Abstract

In this paper, we consider the following parabolic variational inequality containing a multivalued term and a convex functional: Find u∈Lp⁢(0,T;W01,p⁢(Ω)) and f∈F⁢(⋅,⋅,u) such that u⁢(⋅,0)=u0 and

〈 u t + A ⁢ u , v - u 〉 + Ψ ⁢ ( v ) - Ψ ⁢ ( u ) ≥ ∫ Q f ⁢ ( v - u ) ⁢ 𝑑 x ⁢ 𝑑 t for all ⁢ v ∈ L p ⁢ ( 0 , T ; W 0 1 , p ⁢ ( Ω ) ) ,

where A is the principal term; F is a multivalued lower-order term; Ψ⁢(u)=∫0Tψ⁢(t,u)⁢𝑑t is a convex functional. Moreover, we study the existence and other properties of solutions of this inequality assuming certain growth conditions on the lower-order term F.

Keywords: Variational Inequalities; Multivalued Term; Convex Functional; Sub-supersolutions; Extremal Solutions

MSC 2010: 58E35; 47J20; 47J25; 35J87

1 Introduction

In this paper, we are concerned with the following parabolic variational inequality with a multivalued term defined on a cylindrical domain Q=Ω×(0,T):

Find u∈X0:=Lp⁢(0,T;W01,p⁢(Ω)) and f∈F⁢(⋅,⋅,u) such that u⁢(⋅,0)=u0 and

(1.1) 〈 u t + A ⁢ u , v - u 〉 + Ψ ⁢ ( v ) - Ψ ⁢ ( u ) ≥ ∫ Q f ⁢ ( v - u ) ⁢ 𝑑 x ⁢ 𝑑 t for all ⁢ v ∈ X 0 ,

where A is the time-dependent principal term, F is a multivalued lower-order term and Ψ⁢(u)=∫0Tψ⁢(t,u)⁢𝑑t is a convex functional from X0 to ℝ∪{∞}.

Our goal here is to develop a systematic study of the existence and other properties of solutions of this inequality under certain growth conditions on the lower-order term F. This investigation is a further step in the study of parabolic variational inequalities with multivalued terms following some previous papers [3, 16]. We also refer to [5, 6, 9, 10, 18, 19, 20, 21] and the references therein for various approaches and different aspects of the general theory of parabolic variational inequalities.

The above inequality (1.1) was studied in [3] in the particular case where Ψ=IK is the indicator functional of a closed convex set K. In that case, (1.1) becomes the following variational inequality:

Find u∈K such that u⁢(⋅,0)=u0 and

〈 u t + A ⁢ u , v - u 〉 ≥ ∫ Q f ⁢ ( v - u ) ⁢ 𝑑 x ⁢ 𝑑 t for all ⁢ v ∈ K .

The method employed in [3], in the coercive case, is a penalty method where the penalty functional β of K is assumed to satisfy a certain growth condition, in addition to the usual conditions associated with penalty functionals of closed and convex sets (cf. [3, Definition 3.1 and condition (P)] and [17, Section 5.2 in Chapter 3]). Since penalty methods do not generally apply to variational inequalities containing convex functionals, the approach and arguments in [3] are no longer suitable for studying (1.1) where convex functionals, instead of convex sets, are involved.

To study the solvability of (1.1) in the case where the multivalued term F satisfies an appropriate global growth condition, we use here a combination of Rothe’s method, via an existence result in [10] for linear single-valued lower-order terms, with fixed point/topological arguments and appropriate a priori estimates. When such global conditions are not available, we follow a sub-supersolution (lattice) approach to investigate the existence and some other properties of the solution set of (1.1).

As in [3], we establish here existence and comparison/enclosure results for solutions of parabolic variational inequalities with upper semicontinuous multivalued functions, now in the more general situation where the constraints are given by convex functionals rather than by convex sets. Furthermore, with the estimates on the time derivative ut established in Sections 2 and 3, we are able to establish here, in Section 4, additional qualitative behavior of the solutions of (1.1), such as compactness properties of the solution set that imply the existence of extremal (maximal and minimal) solutions. Such qualitative results were not generally available with the method used in [3] (cf. [3, Remark 3.10 (c)]). On the other hand, the principal operator in (1.1), of the present paper, is assumed to have a variational structure, i.e., to have a potential functional, a requirement which is not needed in [3].

We note that the existence of extremal solutions to parabolic partial differential equations, between sub- and supersolutions, was established in [1] for classical solutions, and in [2] for weak solutions. On the other hand, the existence of extremal solutions to elliptic variational inequalities was investigated in [12] for inequalities defined on convex sets and in [11] for inequalities containing general convex functionals. These results naturally lead to the question about the existence of extremal solutions to parabolic variational inequalities, between sub- and supersolutions. As mentioned above, this question will be addressed in Section 4 for solutions of (1.1).

The discussion in this paper is also motivated by the recent work [16], where a parabolic variational inequality related to a sandpile problem was investigated. The topological degree arguments used in that paper do not seem to work here in the case where F is multivalued. A simple but essential reason is that, contrary to the case of single-valued functions, the composition of a convex-valued multivalued mapping or even a single-valued mapping is, in general, a multivalued mapping with non-convex values. The arguments in Sections 3 and 4 may therefore be potentially applied to study variational inequalities for sandpile problems with multivalued terms.

The paper is organized as follows. We introduce, in Section 2, notations and conditions needed on the mappings, except those on the multivalued function F, involved in (1.1) together with the corresponding function spaces, and prove a preparatory existence result. In Section 3, we present the assumptions on F and prove a main existence theorem for (1.1) when F satisfies a certain global growth condition. The main tools for such result are a priori estimates and a topological fixed point theorem for multivalued mappings (the fixed point theorem of Fan–Glicksberg, cf. [8] or [23]). Section 4 is concerned with existence and enclosure theorems for (1.1) in the case where F does not satisfy the global growth condition as in Section 3. We adopt here a sub-supersolution approach and apply some ideas and arguments developed earlier for elliptic variational inequalities containing convex functionals (cf. [14]). After introducing appropriate concepts of sub- and supersolutions for (1.1), we prove in Section 4.1 the existence of solutions to (1.1) whose values lie between a number of sub- and supersolutions. Some further qualitative properties of these solutions are given in Section 4.2.

2 Setting of the Problem – Auxiliary Results – Examples

2.1 Assumptions and Notations

Let Ω be a bounded domain in ℝN (N≥1) with Lipschitz boundary ∂⁡Ω and let Q=Ω×(0,T) be the space-time cylindrical domain with T∈(0,∞). For p∈(2,∞), we denote by W1,p⁢(Ω) and W01,p⁢(Ω) the standard first order Sobolev spaces on Ω, equipped with the usual norms

∥ u ∥ W 1 , p ⁢ ( Ω ) = ∥ u ∥ L p ⁢ ( Ω ) + ∥ | ∇ ⁡ u | ∥ L p ⁢ ( Ω ) and ∥ u ∥ W 0 1 , p ⁢ ( Ω ) = ∥ | ∇ ⁡ u | ∥ L p ⁢ ( Ω ) .

We also consider the Banach spaces X=Lp⁢(0,T;W1,p⁢(Ω)) and X0=Lp⁢(0,T;W01,p⁢(Ω)) with the norms

∥ u ∥ X = ( ∫ 0 T ∥ u ⁢ ( t ) ∥ W 1 , p ⁢ ( Ω ) p ) 1 / p and ∥ u ∥ X 0 = ( ∫ 0 T ∥ u ⁢ ( t ) ∥ W 0 1 , p ⁢ ( Ω ) p ) 1 / p .

Let W={u∈X:ut∈X*} and W0={u∈X0:ut∈X0*} where ut is the time derivative in the sense of distributions. It is known that the imbeddings W,W0↪C⁢([0,T],L2⁢(Ω)) are continuous and the imbeddings W,W0↪Lp⁢(Q) are compact. We use 〈⋅,⋅〉 for the dual pairing between Z and Z*, its topological dual, where Z is one of the spaces W1,p⁢(Ω), W01,p⁢(Ω), X, or X0. However, in the cases where clarification is needed, we shall use subscripts such as 〈⋅,⋅〉Z*,Z instead. We also use (⋅,⋅) for the standard inner products in L2⁢(Ω) and L2⁢(Q)≡L2⁢(0,T;L2⁢(Ω)). Because of the evolution triples W01,p⁢(Ω)⊂L2⁢(Ω)⊂W-1,p′⁢(Ω) and X0⊂L2⁢(Q)⊂X0*, we see, for example, that if u∈X0 and u*∈L2⁢(Q) then

〈 u * , u 〉 X 0 * , X 0 = ∫ 0 T 〈 u * ⁢ ( t ) , u ⁢ ( t ) 〉 W - 1 , p ′ ⁢ ( Ω ) , W 0 1 , p ⁢ ( Ω ) ⁢ 𝑑 t = ( u * , u ) = ∫ 0 T ( u * ⁢ ( t ) , u ⁢ ( t ) ) ⁢ 𝑑 t .

Assume

a = ( a 1 , … , a n ) : Ω × [ 0 , T ] × ℝ N → ℝ N , ( x , t , ξ ) ↦ a ⁢ ( x , t , ξ )

satisfies the following conditions:

For all (t,ξ)∈[0,T]×ℝN, a⁢(⋅,t,ξ) is measurable on Ω, and for a.e. x∈Ω, a⁢(x,⋅,⋅) is continuous on [0,T]×ℝN.
There are c1∈(0,∞) and c2∈C⁢([0,T],L+p′⁢(Ω)) such that

(2.1) | a ⁢ ( x , t , ξ ) | ≤ c 1 ⁢ | ξ | p - 1 + c 2 ⁢ ( x , t ) for a.e. ⁢ x ∈ Ω , all ⁢ t ∈ [ 0 , T ] , ξ ∈ ℝ N ,

where p′ is the Hölder conjugate of p and L+p′⁢(Ω)={u∈Lp′⁢(Ω):u≥0⁢ a.e. on ⁢Ω}.

Suppose that a has a potential function, that is, there exists a function

α : Ω × [ 0 , T ] × ℝ N → ℝ

with the following properties:

For all (t,ξ)∈[0,T]×ℝN, α⁢(⋅,t,ξ) is measurable; for a.e. x∈Ω and all ξ∈ℝN, α⁢(x,⋅,ξ) is continuous; and for a.e. x∈Ω and all t∈[0,T], α⁢(x,t,⋅) is strictly convex and differentiable. Moreover,

(2.2) ∂ ⁡ α ∂ ⁡ ξ i ⁢ ( x , t , ξ ) = a i ⁢ ( x , t , ξ ) for a.e. ⁢ x ∈ Ω , all ⁢ t ∈ [ 0 , T ] , ξ ∈ ℝ N ,

and

(2.3) α ⁢ ( ⋅ , ⋅ , 0 ) ∈ C ⁢ ( [ 0 , T ] , L 1 ⁢ ( Ω ) ) .

As a consequence of (A3), we see that for a.e. (x,t)∈Q, a⁢(x,t,⋅) is strictly monotone on ℝN, that is, if ξ,ξ′∈ℝN and ξ≠ξ′ then

(2.4) ( a ⁢ ( x , t , ξ ) - a ⁢ ( x , t , ξ ′ ) ) ⋅ ( ξ - ξ ′ ) > 0 .

It also follows from (2.1) and (2.3) that α satisfies the growth condition

(2.5) | α ⁢ ( x , t , ξ ) | ≤ c 3 ⁢ | ξ | p + c 4 ⁢ ( x , t ) for a.e. ⁢ x ∈ Ω , all ⁢ t ∈ [ 0 , T ] , ξ ∈ ℝ N ,

with c3∈(0,∞) and c4∈C⁢([0,T],L+1⁢(Ω)). We also assume the following coercivity condition:

There are c5∈(0,∞) and c6∈C⁢([0,T],L+1⁢(Ω)) such that

(2.6) α ⁢ ( x , t , ξ ) ≥ c 5 ⁢ | ξ | p - c 6 ⁢ ( x , t ) for a.e. ⁢ x ∈ Ω , all ⁢ t ∈ [ 0 , T ] , ξ ∈ ℝ N .

Note that a sufficient condition for (A4) is the following:

There are c~5∈(0,∞) and c~6∈C⁢([0,T],L+1⁢(Ω)) such that

a ⁢ ( x , t , ξ ) ⁢ ξ ≥ c ~ 5 ⁢ | ξ | p - c ~ 6 ⁢ ( x , t ) for a.e. ⁢ x ∈ Ω , all ⁢ t ∈ [ 0 , T ] , ξ ∈ ℝ N .

We further need the following continuity condition on α⁢(x,⋅,ξ):

There exists c7>0 such that

(2.7) | α ⁢ ( x , t , ξ ) - α ⁢ ( x , s , ξ ) | ≤ c 7 ⁢ ( 1 + | ξ | p ) ⁢ | t - s | for a.e. ⁢ x ∈ Ω , all ⁢ t , s ∈ [ 0 , T ] , ξ ∈ ℝ N .

In view of (2.2) and the Mean Value Theorem, we see that a sufficient condition for (2.7) in terms of a is the following condition:

| a ⁢ ( x , t , ξ ) - a ⁢ ( x , s , ξ ) | ≤ c ~ 7 ⁢ ( 1 + | ξ | p - 1 ) ⁢ | t - s | for a.e. ⁢ x ∈ Ω , all ⁢ t , s ∈ [ 0 , T ] , ξ ∈ ℝ N .

Next, let us define the necessary operators and functionals associated with a and α. For u∈X, we denote by ∇⁡u⁢(x,t)=∇x⁡u⁢(x,t) the gradient of u with respect to x, which is a function in [Lp⁢(Q)]N. By (A2), the mapping A:X0→X0* given by

〈 A ⁢ u , v 〉 = ∫ Q a ⁢ ( x , t , ∇ ⁡ u ) ⁢ ∇ ⁡ v ⁢ d ⁢ x ⁢ d ⁢ t

is well defined and it follows from (2.1), that there is a constant c8>0 such that

∥ A ⁢ u ∥ X 0 * ≤ c 8 ⁢ ( 1 + ∥ u ∥ X 0 p - 1 ) for all ⁢ u ∈ X 0 .

For t∈[0,T] and u∈W01,p⁢(Ω), (2.5) implies that the function α⁢(⋅,t,∇⁡u⁢(⋅)) belongs to L1⁢(Ω). Thus, the function ψ0:[0,T]×W01,p⁢(Ω)→ℝ given by

ψ 0 ⁢ ( t , u ) = ∫ Ω α ⁢ ( x , t , ∇ ⁡ u ⁢ ( x ) ) ⁢ 𝑑 x

is well defined. Furthermore, for each t∈[0,T], the function ψ0⁢(t,⋅) is convex on W01,p⁢(Ω) with effective domain Dom⁡(ψ0⁢(t,⋅))=W01,p⁢(Ω). Moreover, it follows from (2.1)–(2.2) that for all t∈[0,T], ψ0⁢(t,⋅) is differentiable on W01,p⁢(Ω) and its (Fréchet) derivative with respect to u is given by

〈 D ⁢ ψ 0 ⁢ ( t , u ) , v 〉 = 〈 D u ⁢ ψ 0 ⁢ ( t , u ) , v 〉 = ∫ Ω a ⁢ ( x , t , ∇ ⁡ u ) ⁢ ∇ ⁡ v ⁢ d ⁢ x for all ⁢ u , v ∈ W 0 1 , p ⁢ ( Ω ) .

As a consequence, the subdifferential of ψ0⁢(t,⋅) at u (in the sense of Convex Analysis) is the same as its Fréchet derivative, ∂⁡ψ0⁢(t,u)={D⁢ψ0⁢(t,u)}. Similarly, (A2) and (A3) imply that the functional Ψ0 defined on X0 by

Ψ 0 ⁢ ( u ) := ∫ 0 T ψ 0 ⁢ ( t , ∇ ⁡ u ⁢ ( t ) ) ⁢ 𝑑 t = ∫ Q α ⁢ ( x , t , ∇ ⁡ u ⁢ ( x , t ) ) ⁢ 𝑑 x ⁢ 𝑑 t for all ⁢ u ∈ X 0

is convex and differentiable on X0 with

(2.8) ∂ ⁡ Ψ 0 ⁢ ( t , u ) = { D ⁢ Ψ 0 ⁢ ( u ) } = { A ⁢ u } for all ⁢ u ∈ X 0 .

Next, it follows from (2.5) and (2.6) that there are positive constants c9, c10, c11, c12 such that

(2.9) c 9 ⁢ ∥ u ∥ W 0 1 , p ⁢ ( Ω ) p - c 9 - 1 ≤ ψ 0 ⁢ ( t , u ) ≤ c 10 ⁢ ( ∥ u ∥ W 0 1 , p ⁢ ( Ω ) p + 1 ) for all ⁢ t ∈ [ 0 , T ] , u ∈ W 0 1 , p ⁢ ( Ω ) ,

and

(2.10) c 11 ⁢ ∥ u ∥ X 0 p - c 11 - 1 ≤ Ψ 0 ⁢ ( u ) ≤ c 12 ⁢ ( ∥ u ∥ X 0 p + 1 ) for all ⁢ u ∈ X 0 .

On the other hand, it follows from (2.7) and direct calculations that there are constants c13,c14>0 such that for all s,t∈[0,T] and all u,v∈W01,p⁢(Ω),

(2.11) | ψ 0 ⁢ ( t , u ) - ψ 0 ⁢ ( s , v ) | ≤ c 13 ⁢ ( 1 + ∥ u ∥ W 0 1 , p ⁢ ( Ω ) p ) ⁢ | t - s | + c 14 ⁢ ( 1 + ∥ u ∥ W 0 1 , p ⁢ ( Ω ) p - 1 + ∥ v ∥ W 0 1 , p ⁢ ( Ω ) p - 1 ) ⁢ ∥ u - v ∥ W 0 1 , p ⁢ ( Ω ) .

To represent the constraint, let ψ:[0,T]×W01,p⁢(Ω)→ℝ∪{∞} be a function satisfying the following conditions:

For all t∈[0,T], ψ⁢(t):=ψ⁢(t,⋅) is a proper, convex, and lower semicontinuous functional from W01,p⁢(Ω) to ℝ∪{∞}.
There exist r∈[0,p) and c15,c16≥0 such that

(2.12) ψ ⁢ ( t , u ) ≥ - c 15 ⁢ ∥ u ∥ W 0 1 , p ⁢ ( Ω ) r - c 16 for all ⁢ t ∈ [ 0 , T ] , u ∈ W 0 1 , p ⁢ ( Ω ) .
ψ ⁢ ( ⋅ , 0 ) ∈ L 1 ⁢ ( 0 , T ) .
There exists a constant c17>0 such that for each t∈[0,T], u∈Dom⁡(ψ⁢(t,⋅)), and each s∈[t,T], there exists v∈Dom⁡(ψ⁢(s,⋅)) such that

(2.13) ∥ u - v ∥ W 0 1 , p ⁢ ( Ω ) ≤ c 17 ⁢ ( s - t ) ,

and

(2.14) ψ ⁢ ( s , v ) ≤ ψ ⁢ ( t , u ) + c 17 ⁢ ( s - t ) ⁢ ( 1 + ∥ u ∥ W 0 1 , p ⁢ ( Ω ) p + | ψ ⁢ ( t , u ) | ) .

It follows from (2.12) and Young’s inequality with ε, that for any ε>0, there is c16⁢(ε)>0 such that

(2.15) ψ ⁢ ( t , u ) ≥ - ε ⁢ ∥ u ∥ W 0 1 , p ⁢ ( Ω ) p - c 16 ⁢ ( ε ) for all ⁢ t ∈ [ 0 , T ] , u ∈ W 0 1 , p ⁢ ( Ω ) .

In particular, for u∈X0,

ψ ⁢ ( t , u ⁢ ( t ) ) = ψ ⁢ ( t , u ⁢ ( ⋅ , t ) ) ≥ - ∥ u ⁢ ( ⋅ , t ) ∥ W 0 1 , p ⁢ ( Ω ) p - c 16 ⁢ ( 1 ) for all ⁢ t ∈ [ 0 , T ] .

Since ∥u⁢(⋅,t)∥W01,p⁢(Ω)p∈L1⁢(0,T), we see that the integral

Ψ ⁢ ( u ) := ∫ 0 T ψ ⁢ ( t , u ⁢ ( t ) ) ⁢ 𝑑 t

exists in ℝ∪{∞}. Moreover, Ψ is a convex, lower semicontinuous functional from X0 to ℝ∪{∞} and u∈Dom⁡(Ψ), if and only if ψ⁢(⋅,u⁢(⋅))∈L1⁢(0,T).

2.2 Examples

In this section, we present some examples of the mappings a and ψ considered above. A classical example for a satisfying (A1)–(A5) is the p-Laplacian, where

〈 A ⁢ u , v 〉 = ∫ Q | ∇ ⁡ u | p - 2 ⁢ ∇ ⁡ u ⋅ ∇ ⁡ v ⁢ d ⁢ x ⁢ d ⁢ t .

In this case, a is given by a⁢(x,t,ξ)=|ξ|p-2⁢ξ and α⁢(x,t,ξ)=1p⁢|ξ|p for t∈[0,T],x∈Ω, and ξ∈ℝN. A simple time-dependent variant of the p-Laplacian is given by a⁢(x,t,ξ)=γ⁢(x,t)⁢|ξ|p-2⁢ξ and α⁢(x,t,ξ)=γ⁢(x,t)p⁢|ξ|p, where γ is a bounded Carathéodory function on Ω×[0,T]. The conditions (A1)–(A5) still hold whenever ess⁢inf⁡{γ⁢(x,t):(x,t)∈Ω×[0,T]}>0 and γ⁢(x,t) is uniformly Lipschitz continuous with respect to t, i.e., |γ⁢(x,t)-γ⁢(x,s)|≤γ0⁢|t-s| for all s,t∈[0,T], almost all x∈Ω, with some positive constant γ0.

As an example of convex functionals ψ and Ψ that were not covered in [3], let us consider the integral functional ψ⁢(t,u)=∫Γj⁢(u)⁢𝑑x for u∈W01,p⁢(Ω), where Γ is a measurable subset of Ω and j:ℝ→[0,∞] is a convex, lower semicontinuous function such that j⁢(0)<∞. For u∈W01,p⁢(Ω), since j⁢(u) is measurable and nonnegative on Ω, the integral ∫Ωj⁢(u)⁢𝑑x exists and belongs to [0,∞]. The effective domain of ψ⁢(t,⋅) consists of functions u∈W01,p⁢(Ω) such that j⁢(u)|Γ∈L1⁢(Γ). Conditions (C2) and (C3) follow directly from the definition of ψ and condition (C1) is a straightforward consequence of Fatou’s lemma. Condition (C4) holds trivially, choosing v=u.

A time-dependent variant of the above example is given by

ψ ⁢ ( t , u ) = η ⁢ ( t ) ⁢ ∫ Γ j ⁢ ( u ) ⁢ 𝑑 x ,

where j is as above and η:[0,T]→ℝ is a Lipschitz continuous function such that

η 0 := min ⁡ { η ⁢ ( t ) : t ∈ [ 0 , T ] } > 0 .

Conditions (C1)–(C4) still hold in this more general situation. For example, let us verify condition (C4). Let η1 be a Lipschitz constant for η, i.e., |η⁢(s)-η⁢(t)|≤η1⁢|s-t| for all s,t∈[0,T]. For any s,t∈[0,T] with s≥t and u∈Dom⁡(ψ⁢(t,⋅)), we have ∫Γj⁢(u)⁢𝑑x<∞ and ψ⁢(t,u)≥η0⁢∫Γj⁢(u)⁢𝑑x. By choosing again v=u, we have u∈Dom⁡(ψ⁢(s,⋅)) and (2.13) is satisfied immediately. Moreover,

ψ ⁢ ( s , u ) = ψ ⁢ ( t , u ) + [ η ⁢ ( s ) - η ⁢ ( t ) ] ⁢ ∫ Γ j ⁢ ( u ) ⁢ 𝑑 x

≤ ψ ⁢ ( t , u ) + η 1 ⁢ ( s - t ) ⁢ ∫ Γ j ⁢ ( u ) ⁢ 𝑑 x

≤ ψ ⁢ ( t , u ) + η 1 η 0 ⁢ ( s - t ) ⁢ ψ ⁢ ( t , u ) .

Hence, (2.14) also holds. We note that these arguments can be extended, using straightforward modifications, to the case where ψ⁢(t,u)=η⁢(t)⁢∫Γj⁢(x,u)⁢𝑑x, and where j also depends on x∈Γ.

2.3 An Existence Lemma

Let us conclude this section with a preparatory existence theorem of solutions of parabolic variational inequalities with linear lower-order terms.

Lemma 2.1.

Assume conditions (A1)–(A5) and (C1)–(C4), let u0∈Dom⁡(ψ⁢(⋅,0)). Then, for each f∈L2⁢(Q), there exists a unique u∈W0∩Dom⁡(Ψ) such that

(2.16) 〈 u t , v - u 〉 + 〈 A ⁢ u , v - u 〉 + Ψ ⁢ ( v ) - Ψ ⁢ ( u ) ≥ ( f , v - u ) for all ⁢ v ∈ X 0 ,

and

(2.17) u ⁢ ( ⋅ , 0 ) = u 0 .

Moreover, u∈L∞⁢(0,T,W01,p⁢(Ω)), ut∈L2⁢(Q), and there exists C>0 depending only on ∥f∥L2⁢(Q) such that

(2.18) ∥ u ∥ L ∞ ⁢ ( 0 , T , W 0 1 , p ⁢ ( Ω ) ) , ∥ u t ∥ L 2 ⁢ ( Q ) ≤ C .

Proof.

For t∈[0,T] and u∈W01,p⁢(Ω), let us define

ψ 1 ⁢ ( t , u ) = ψ 0 ⁢ ( t , u ) + ψ ⁢ ( t , u ) ,

and for u∈X0, define

Ψ 1 ⁢ ( u ) = ∫ 0 T ψ 1 ⁢ ( t , u ⁢ ( t ) ) ⁢ 𝑑 t = Ψ 0 ⁢ ( u ) + Ψ ⁢ ( u ) .

We have Dom⁡(ψ1⁢(t,⋅))=Dom⁡(ψ⁢(t,⋅)) and Dom⁡(Ψ1)=Dom⁡(Ψ). It follows from the corresponding properties of ψ0,Ψ0, and ψ,Ψ, that ψ1 satisfies condition (C1). From (2.11), (2.13), (2.14), we see that ψ1 also satisfies (C4) (with a different constant c17).

In fact, for s,t∈[0,T],s≥t, and u∈W01,p⁢(Ω), let v satisfy (2.13). We need to prove that ψ1 satisfies an estimate similar to (2.14). Note from (2.11) and (2.13) that

ψ 0 ⁢ ( s , v ) ≤ ψ 0 ⁢ ( t , u ) + c 13 ⁢ ( 1 + ∥ u ∥ W 0 1 , p ⁢ ( Ω ) p ) ⁢ | t - s | + c 14 ⁢ c 17 ⁢ ( 1 + ∥ u ∥ W p - 1 + ∥ v ∥ W p - 1 ) ⁢ | t - s |

(2.19) ≤ ψ 0 ⁢ ( t , u ) + c ⁢ ( 1 + ∥ u ∥ p + ∥ v ∥ p ) ⁢ | t - s | ,

where c denotes a generic positive constant that does not depend on u,v,s,t. Since

∥ v ∥ ≤ c 17 ⁢ | t - s | ≤ ∥ u ∥ + c 17 ⁢ T ,

we have ∥v∥p≤c⁢(1+∥u∥p) and according to (2.19),

(2.20) ψ 0 ⁢ ( s , v ) ≤ ψ 0 ⁢ ( t , u ) + c ⁢ ( 1 + ∥ u ∥ p ) ⁢ | t - s | .

On the other hand, we have from (2.9) that |ψ0⁢(t,u)|≤c⁢(1+∥u∥p) and thus

(2.21) | ψ ⁢ ( t , u ) | ≤ | ψ 1 ⁢ ( t , u ) | + c ⁢ ( 1 + ∥ u ∥ p ) .

Combining (2.14) and (2.20) and noting (2.21), we obtain

ψ 1 ⁢ ( s , v ) ≤ ψ 1 ⁢ ( t , u ) + c ⁢ | t - s | ⁢ ( 1 + ∥ u ∥ p + | ψ ⁢ ( t , u ) | ) + c ⁢ ( 1 + ∥ u ∥ p ) ⁢ | t - s |

(2.22) ≤ ψ 1 ⁢ ( t , u ) + c ⁢ | t - s | ⁢ ( 1 + ∥ u ∥ W 0 1 . p ⁢ ( Ω ) p + | ψ 1 ⁢ ( t , u ) | )

for some constant c>0 independent of s,t,u,v, i.e., ψ1 satisfies (C4).

As a consequence of (2.9) and (2.15) with ε=c9/2, we see that for all t∈[0,T] and all u∈W01,p⁢(Ω),

(2.23) ψ 1 ⁢ ( t , u ) ≥ c 9 2 ⁢ ∥ u ∥ W 0 1 , p ⁢ ( Ω ) p - c 18 ,

with c18>0.

We note that (2.16) is equivalent to u∈Dom⁡(Ψ1) and

(2.24) 〈 u t , v - u 〉 + Ψ 1 ⁢ ( v ) - Ψ 1 ⁢ ( u ) ≥ ( f , v - u ) for all ⁢ v ∈ X 0 .

In fact, assume u satisfies (2.16). Since Dom⁡(Ψ1)=Dom⁡(Ψ), we have from (2.8) that

〈 A ⁢ u , v - u 〉 ≤ Ψ 0 ⁢ ( v ) - Ψ 0 ⁢ ( u )

for all v∈X0, showing that u satisfies (2.24). Conversely, let u satisfy (2.24). For w∈X0 and t∈(0,1), putting v=(1-t)⁢u+t⁢w in (2.24) yields

(2.25) t ⁢ 〈 u t , w - u 〉 + Ψ 0 ⁢ ( u + t ⁢ ( w - u ) ) - Ψ 0 ⁢ ( u ) + Ψ ⁢ ( ( 1 - t ) ⁢ u + t ⁢ w ) - Ψ ⁢ ( u ) ≥ t ⁢ ( f , w - u ) .

Together with the convexity of Ψ, this implies that

〈 u t , w - u 〉 + t - 1 ⁢ [ Ψ 0 ⁢ ( u + t ⁢ ( w - u ) ) - Ψ 0 ⁢ ( u ) ] + Ψ ⁢ ( w ) - Ψ ⁢ ( u ) ≥ ( f , w - u ) .

Letting t→0+ in this inequality and noting (2.8), we see that u satisfies (2.16) with w instead of v.

For the existence of solutions of (2.24) and the estimate in (2.18), we apply [10, Theorem 7.2]. Note that conditions (2.22) and (2.23) imply that the assumptions (H1) and (H2) in that theorem are satisfied. The existence of a solution u of (2.24)–(2.17) and thus of (2.16)–(2.17) with the estimate (2.18) is a direct consequence of [10, Theorem 7.2].

To see the uniqueness, we note that the strict convexity of α yields that Ψ0 is strictly convex on X0. Hence Ψ1 is also strictly convex on W0. Let u1,u2∈W0 satisfy (2.24) and u1⁢(⋅,0)=u2⁢(⋅,0)=u0. Putting v=12⁢(u1+u2) in (2.24) with u replaced by u1,u2 and adding the inequalities thus obtained, we get

(2.26) - 1 2 ⁢ 〈 ( u 1 - u 2 ) t , u 1 - u 2 〉 + 2 ⁢ Ψ 1 ⁢ ( u 1 + u 2 2 ) - Ψ 1 ⁢ ( u 1 ) - Ψ 1 ⁢ ( u 2 ) ≥ 0 .

Since

〈 ( u 1 - u 2 ) t , u 1 - u 2 〉 = ∥ ( u 1 - u 2 ) ⁢ ( T ) ∥ L 2 ⁢ ( Ω ) 2 ≥ 0 ,

inequality (2.26) and the strict convexity of Ψ1 show that u1=u2. ∎

3 Existence of Solutions to Inequalities with Multivalued Lower-Order Terms and Global Growth Conditions

In this section, we focus on the existence of solutions of (1.1) in the case where F is a multivalued function that also depends on u and satisfies some global growth condition in terms of u.

3.1 Assumptions – Auxiliary Considerations

For a normed vector space Z, we use the notation

𝒦 ⁢ ( Z ) = { A ∈ 2 Z : A ≠ ∅ , A ⁢ is closed and convex } .

Also, for A,B∈2Z∖{∅}, hZ*⁢(A,B) will denote the Hausdorff semi-distance from A to B,

h Z * ⁢ ( A , B ) = sup a ∈ A ⁡ dist Z ⁡ ( a , B ) = sup a ∈ A ⁡ ( inf b ∈ B ⁡ ∥ a - b ∥ Z ) .

Let L0⁢(Q,ℝ) be the set of all (equivalence classes of) single-valued measurable functions from Q to ℝ and let F:Q×ℝ→𝒦⁢(ℝ) satisfy the following conditions:

F is superpositionally measurable, i.e., if u∈L0⁢(Q,ℝ) then (x,t)↦F⁢(x,t,u⁢(x,t)) is a (multivalued) measurable function on Q.
For a.e. (x,t)∈Q, F⁢(x,t,⋅) is upper semicontinuous from ℝ to 𝒦⁢(ℝ).
There exist h0∈L2⁢(Q) and h1∈[0,∞) such that

(3.1) sup ⁡ { | f | : f ∈ F ⁢ ( x , t , s ) } ≤ h 0 ⁢ ( x , t ) + h 1 ⁢ | s | p / 2 for a.e. ⁢ ( x , t ) ∈ Q , all ⁢ s ∈ ℝ .

Let u∈L0⁢(Q,ℝ). It follows from (F1) that the set

F ~ ⁢ ( u ) = { f ∈ L 0 ⁢ ( Q , ℝ ) : f ⁢ ( x , t ) ∈ F ⁢ ( x , t , u ⁢ ( x , t ) ) ⁢ for a.e. ⁢ ( x , t ) ∈ Q }

of all measurable selections of F⁢(x,t,u⁢(x,t)) is nonempty. Moreover, (F3) implies that, if u∈Lp⁢(Q), then F~⁢(u)⊂L2⁢(Q). This allows us to define the mapping

F ~ : L p ⁢ ( Q ) → 2 L 2 ⁢ ( Q ) ∖ { ∅ } , u ↦ F ~ ⁢ ( u ) .

Further, since F⁢(x,t,s)∈𝒦⁢(ℝ) for a.e. (x,t)∈Q and all s∈ℝ, straightforward arguments show that F~⁢(u) is convex and closed in L2⁢(Q) whenever u∈Lp⁢(Q).

For each f∈L2⁢(Q), let u=uf be the unique solution of (2.16)–(2.17). Given u0∈Dom⁡(ψ⁢(0,⋅)), in view of the compact imbedding of W0 in Lp⁢(Q), we see that the mapping S:L2⁢(Q)→Lp⁢(Q), f↦S⁢(f)=uf, is well defined. A useful property of S is given in the following lemma.

Lemma 3.1.

S is completely continuous from L2⁢(Q) to Lp⁢(Q) in the following sense: If {fn} is a sequence in L2⁢(Q) and fn⇀f in L2⁢(Q) then S⁢(fn)→S⁢(f) in Lp⁢(Q).

Proof.

Let un=S⁢(fn), n∈ℕ. Since {fn} is a bounded sequence in L2⁢(Q), we see that the sequences {∥un∥X0} and {∥un⁢t∥L2⁢(Q)} are both bounded. Hence, by passing to subsequences, if necessary, we can assume that

(3.2) u n ⇀ u in ⁢ X 0 and u n ⁢ t ⇀ g in ⁢ L 2 ⁢ ( Q ) ( ⊂ X 0 * ) .

Since un⁢t converges to both ut and g in the sense of distributions on (0,T), we have g=ut and thus un⁢t⇀ut in L2⁢(Q). From the theorem of Aubin–Lions (see [22]) and the compactness of the imbedding W01,p⁢(Ω)↪Lp⁢(Ω), it follows that

(3.3) u n → u ∈ L p ⁢ ( Q ) ( ⊂ L 2 ⁢ ( Q ) ) .

Let us prove that u is a solution of (2.16)–(2.17), i.e., u=S⁢(f). In fact, since un⁢(0)→u⁢(0) in L2⁢(Ω), we have u⁢(0)=u0. Moreover, for all v∈X0 and all n∈ℕ, we have

(3.4) 〈 u n ⁢ t , v - u n 〉 + Ψ 1 ⁢ ( v ) - Ψ 1 ⁢ ( u ) ≥ ( f n , v - u n ) .

The limits in (3.2) and (3.3) imply that

(3.5) 〈 u n ⁢ t , v - u n 〉 = ( u n ⁢ t , v - u n ) → ( u t , v - u ) = 〈 u t , v - u 〉

and

(3.6) ( f n , v - u n ) → ( f , v - u ) as ⁢ n → ∞ .

On the other hand, since Ψ1 is convex and lower semicontinuous on X0, it is weakly lower semicontinuous there and, as a consequence of (3.2), we see that u∈Dom⁡(Ψ1) and

(3.7) Ψ 1 ⁢ ( u ) ≤ lim inf ⁡ Ψ 1 ⁢ ( u n ) .

Letting n→∞ in (3.4) and noting (3.5)–(3.7), we immediately obtain that u satisfies (2.24), i.e., u=S⁢(f). ∎

3.2 An Existence Theorem

Let us consider the case where F:Q×ℝ→𝒦⁢(ℝ) is a multivalued function that depends on u as well. A function u∈W0 is called a solution of the evolutionary variational inequality (with the multivalued function F)

(3.8) 〈 u t , v - u 〉 + 〈 A ⁢ u , v - u 〉 + Ψ ⁢ ( v ) - Ψ ⁢ ( u ) ≥ ∫ Q F ⁢ ( x , t , u ⁢ ( x , t ) ) ⁢ ( v - u ) ⁢ 𝑑 x ⁢ 𝑑 t for all ⁢ v ∈ X 0 ,

if and only if u satisfies (1.1) for some f∈L2⁢(Q) such that f⁢(x,t)∈F⁢(x,t,u⁢(x,t)) for a.e. (x,t)∈Q.

Theorem 3.2.

Under assumptions (A1)–(A5), (C1)–(C4), and (F1)–(F3), the variational inequality (3.8) has a solution u∈W0∩L∞⁢(0,T;W01,p⁢(Ω)) such that ut∈L2⁢(Q).

Proof.

For the sake of clarity, we divide the proof into several steps.

Step 1. We derive in this step an estimate for S⁢(f) in Lp⁢(Q). Let f∈L2⁢(Q) and u=S⁢(f). Letting v=0 in (2.24) yields

(3.9) - ( u t , u ) + Ψ 1 ⁢ ( 0 ) - Ψ 1 ⁢ ( u ) ≥ - ( f , u ) .

On the other hand,

(3.10) ( u t , u ) = 1 2 ⁢ ∥ u ⁢ ( T ) ∥ L 2 ⁢ ( Ω ) 2 - 1 2 ⁢ ∥ u ⁢ ( 0 ) ∥ L 2 ⁢ ( Ω ) 2 ≥ - 1 2 ⁢ ∥ u 0 ∥ L 2 ⁢ ( Ω ) 2 .

From (2.10) and (C3), we have

(3.11) Ψ 1 ⁢ ( 0 ) ≤ c 12 + ∥ ψ 1 ⁢ ( ⋅ , 0 ) ∥ L 1 ⁢ ( 0 , T ) .

It thus follows from (2.23) that Ψ1⁢(u)≥c92⁢∥u∥X0p-c18⁢T. Using this estimate and (3.10)–(3.11) in (3.9) yields

c 9 2 ⁢ ∥ u ∥ X 0 p ≤ 1 2 ⁢ ∥ u 0 ∥ L 2 ⁢ ( Ω ) 2 + c 12 + ∥ ψ 1 ⁢ ( ⋅ , 0 ) ∥ L 1 ⁢ ( 0 , T ) + ∥ f ∥ L 2 ⁢ ( Q ) ⁢ ∥ u ∥ L 2 ⁢ ( Q ) .

Hence, by Young’s inequality and the continuous imbeddings X0↪Lp⁢(Q)↪L2⁢(Q), we see that there are positive constants c19 and c20 independent of f such that

(3.12) c 9 4 ⁢ ∥ u ∥ X 0 p ≤ c 19 + c 20 ⁢ ∥ f ∥ L 2 ⁢ ( Q ) p ′ .

Thus, for some constant c21>0, independent of f,

(3.13) ∥ S ⁢ ( f ) ∥ L p ⁢ ( Q ) = ∥ u ∥ L p ⁢ ( Q ) ≤ c 21 ⁢ ( 1 + ∥ f ∥ L 2 ⁢ ( Q ) 1 p - 1 ) .

Step 2. Let BR2={f∈L2⁢(Q):∥f∥L2⁢(Q)≤R} be the closed ball in L2⁢(Q) with center at 0 and radius R. We show in this step that for R>0, sufficiently large, if f∈BR2, then F~⁢(S⁢(f))⊂BR2.

In fact, let u∈Lp⁢(Q) and η∈F~⁢(u). We have from (3.1) that

(3.14) ∥ η ∥ L 2 ⁢ ( Q ) ≤ ∥ h 0 ∥ L 2 ⁢ ( Q ) + h 1 ⁢ ∥ u ∥ L p ⁢ ( Q ) p 2 .

Let f∈BR2 and η∈F~⁢(S⁢(f)). The estimates in (3.13) and (3.14) imply that

∥ η ∥ L 2 ⁢ ( Q ) ≤ ∥ h 0 ∥ L 2 ⁢ ( Q ) + h 1 ⁢ ∥ S ⁢ ( f ) ∥ L p ⁢ ( Q ) p 2

≤ ∥ h 0 ∥ L 2 ⁢ ( Q ) + h 1 ⁢ c 21 p 2 ⁢ ( 1 + R 1 p - 1 ) p 2

≤ c 22 ⁢ ( 1 + R p 2 ⁢ ( p - 1 ) )

for some constant c22>0 independent of R, f, and η.

Since p>2, we have p2⁢(p-1)<1. Hence, there is R>0 sufficiently large such that

c 22 ⁢ ( 1 + R p 2 ⁢ ( p - 1 ) ) ≤ R .

For such R, we have η∈BR2. Since this is true for all η∈F~⁢(S⁢(f)), we see that F~⁢(S⁢(f))⊂BR2.

Step 3. Let us consider L2⁢(Q) with the weak topology σ. Let R>0 be chosen in the previous step such that (F~∘S)⁢(BR2)⊂BR2. Note that BR2 is σ-compact. We shall prove in this step that F~∘S is upper semicontinuous from (BR2,σ) to (BR2,σ) and that (3.8) has a solution.

In fact, we first note that (L2⁢(Q),σ) is a locally convex topological vector space and (BR2,σ) is metrizable. For f∈BR2, F~⁢(S⁢(f)) is nonempty, convex, and closed in L2⁢(Q) and thus σ-closed. On the other hand, F~ is upper semicontinuous from Lp⁢(Q) to L2⁢(Q), where both spaces are endowed with their norm topology. In fact, it follows from (F2) that for a.e. (x,t)∈Q, F⁢(x,t) is h-upper semicontinuous from ℝ to 𝒦⁢(ℝ) (cf. [8, Chapter 1, Definition 2.60] for Hausdorff h-semicontinuity). Hence, the growth condition (3.1) and [8, Chapter 2, Theorem 7.26] imply that F~ is h-upper semicontinuous and thus upper semicontinuous from Lp⁢(Q) to L2⁢(Q) (both equipped with their norm topology).

For f∈BR2, let us prove the σ-upper semicontinuity of F~∘S at f. Suppose by contradiction that there are a weakly open set U in L2⁢(Q) and a sequence {fn} in BR2 such that

F ~ ⁢ ( S ⁢ ( f ) ) ⊂ U ,

(3.15) f n ⇀ f in ⁢ L 2 ⁢ ( Q ) ,

(3.16) F ~ ⁢ ( S ⁢ ( f ) ) ⊄ U for all ⁢ n ∈ ℕ .

From (3.15) and Lemma 3.1, we have that S⁢(fn)→S⁢(f) in Lp⁢(Q). Since U is also (strongly) open in L2⁢(Q), by the upper semicontinuity of F~ from Lp⁢(Q) to L2⁢(Q), we see that F~⁢(S⁢(f))⊂U for all n∈ℕ, sufficiently large, contradicting (3.16).

It now follows from the fixed point theorem of Fan–Glicksberg (see [8, 23]) that F~∘S has a fixed point in BR2, i.e., there is f∈BR2 such that f∈F~⁢(S⁢(f)). It is immediate that u=S⁢(f) is a solution of (3.8).

Step 4. We shall verify in this step that the set P of all solutions of (3.8) is a closed and bounded subset of W0.

First, let us prove that P is bounded in X0. In fact, let u∈P and η∈F~⁢(u) satisfy

(3.17) 〈 u t , v - u 〉 + Ψ 1 ⁢ ( v ) - Ψ 1 ⁢ ( u ) ≥ ( η , v - u ) for all ⁢ v ∈ X 0 .

As a consequence of (3.12) and (3.1), we obtain

c 9 4 ⁢ ∥ u ∥ X 0 p ≤ c 19 + c 20 ⁢ ( ∥ h 0 ∥ L 2 ⁢ ( Q ) + h 1 ⁢ ∥ u ∥ L p ⁢ ( Q ) p 2 ) p ′ ≤ c ⁢ ( 1 + ∥ u ∥ X 0 p ⁢ p ′ 2 )

for some constant c>0 independent of u. Since p>p⁢p′2, we see from this estimate that ∥u∥X0 is uniformly bounded for u∈P. It follows from (3.1) that the set {∥η∥L2⁢(Q):η∈F~⁢(u),u∈P} is bounded. Moreover, since u satisfies (3.17), we see that the set {∥ut∥L2⁢(Q):u∈P} is also bounded. Hence, P is a bounded set in W0.

To prove the closedness of P in W0, let {un} be a sequence in P such that

(3.18) u n → u in ⁢ W 0 .

For each n∈ℕ, there is ηn∈F~⁢(un) satisfying

(3.19) ( u n ⁢ t , v - u n ⁢ t ) + Ψ 1 ⁢ ( v ) - Ψ 1 ⁢ ( u n ) ≥ ( η n , v - u n ) for all ⁢ v ∈ X 0 .

We obtain from (3.18) that un→u in Lp⁢(Q) and thus hL2⁢(Q)*⁢(F~⁢(un),F~⁢(u))→0 as n→∞. Hence, there exists a sequence {η~n}⊂F~⁢(u) such that ∥ηn-η~n∥L2⁢(Q)→0. Since F~⁢(u) is a closed, convex, and bounded subset of L2⁢(Q), it is weakly compact there, and by passing to a subsequence, we can assume that η~⇀η in L2⁢(Q) for some η∈F~⁢(u). Consequently, ηn⇀η in L2⁢(Q). For v∈X0, we have (ηn,v-un)→(η,v-u). Moreover, (un⁢t,v-un)=〈un⁢t,v-un〉→〈ut,v-u〉, and lastly, because of the weak lower semicontinuity of Ψ1 on X0, we have Ψ1⁢(u)≤lim inf⁡Ψ1⁢(un).

Letting n→∞ in (3.19) and taking into account the above limits, we see that u and η satisfy (3.17) for all v∈X0 with η∈F~⁢(u), that is, u∈P. We also note that the arguments presented above are still valid if the strong convergence in (3.18) is replaced by the weak convergence, which implies that P is weakly compact in W0. ∎

4 Existence of Solutions to Inequalities with Multivalued Lower-Order Terms and Local Growth Conditions

Let us consider now the case where the growth condition (3.1) is valid only locally, that is, when s belongs to certain bounded intervals. If sub- and supersolutions of (3.8), defined in an appropriate sense, exist, we still obtain in that case the existence and certain qualitative properties of solutions of (3.8).

4.1 An Existence and Enclosure Theorem

For u,v∈Lp⁢(Ω) (resp. u,v∈Lp⁢(Q)), we use the usual pointwise ordering: u≤v, if and only if u⁢(x)≤v⁢(x) for a.e. x∈Ω (resp. x∈Q). For ui∈Lp⁢(Ω) or ui∈Lp⁢(Q) (i=1,…,m), we define

⋀ i = 1 m u i = min ⁡ { u 1 , … , u m } and ⋁ i = 1 m u i = max ⁡ { u 1 , … , u m } ,

where these operations are taken pointwise in Ω or Q. Note that if u1,…,um∈Z where Z is either W1,p⁢(Ω), W01,p⁢(Ω), X, X0,Lp⁢(Ω) or Lp⁢(Q), then ⋀i=1mui,⋁i=1mui∈Z. Also, for A,B⊂Z, a∈Z, and ⋆∈{∧,∨}, we use the notation

A ⋆ B = { u ⋆ v : u ∈ A , v ∈ B } and a ⋆ A = { a } ⋆ A = { a ⋆ u : u ∈ A } .

Next, we introduce an ordering on subsets of X and on real-valued functions defined on X.

Definition 4.1.

(a) For A,B⊂X, A,B≠∅, we say that A⪯B (or B⪰A), if and only if a∈A and b∈B imply that a∧b∈A and a∨b∈B.

(b) For f,g:X→ℝ∪{∞}, we say that f⪯g (or g⪰f), if and only if, for all u,v∈X,

(4.1) f ⁢ ( u ∧ v ) + g ⁢ ( u ∨ v ) ≤ f ⁢ ( u ) + g ⁢ ( v ) .

Let Dom⁡(f) denote the effective domain of f, Dom⁡(f)={u∈X:f⁢(u)<∞}. We note that f⪯g, if and only if Dom⁡(f)⪯Dom⁡(g) and (4.1) holds for all u∈Dom⁡(f), v∈Dom⁡(g). With this remark, we can define the relation f⪯g for functions f and g that are defined only on subsets of X.

Suppose F:Q×ℝ→2ℝ satisfies (F1)–(F2) and let u0∈Dom⁡(ψ⁢(⋅,0)).

Definition 4.2.

(a) A function u¯∈W is called a subsolution of (3.8) if there exist Ψ¯:X→ℝ∪{∞} and η¯:Q→ℝ such that

u ¯ ∈ Dom ⁡ ( Ψ ¯ ) , η¯∈L2⁢(Q), and

(4.2) η ¯ ⁢ ( x , t ) ∈ F ⁢ ( x , t , u ¯ ⁢ ( x , t ) ) for a.e. ⁢ ( x , t ) ∈ Q ,
Ψ ¯ ⪯ Ψ ,
u ¯ ⁢ ( ⋅ , 0 ) ≤ u 0 a.e. on Ω and u¯≤0 a.e. on Γ:=∂⁡Ω×(0,T), and
〈 u ¯ t , v - u ¯ 〉 + 〈 A ⁢ u ¯ , v - u ¯ 〉 + Ψ ¯ ⁢ ( v ) - Ψ ¯ ⁢ ( u ) ≥ ( η ¯ , v - u ¯ ) for all v∈u¯∧Dom⁡(Ψ).

(b) Similarly, a function u¯∈W is called a supersolution of (3.8) if there exist Ψ¯:X→ℝ∪{∞} and η¯:Q→ℝ such that

u ¯ ∈ Dom ⁡ ( Ψ ¯ ) , η¯∈L2⁢(Q), and

(4.3) η ¯ ⁢ ( x , t ) ∈ F ⁢ ( x , t , u ¯ ⁢ ( x , t ) ) for a.e. ⁢ ( x , t ) ∈ Q ,
Ψ ¯ ⪰ Ψ ,
u ¯ ⁢ ( ⋅ , 0 ) ≥ u 0 a.e. on Ω and u¯≥0 a.e. on Γ, and
〈 u ¯ t , v - u ¯ 〉 + 〈 A ⁢ u ¯ , v - u ¯ 〉 + Ψ ¯ ⁢ ( v ) - Ψ ¯ ⁢ ( u ) ≥ ( η ¯ , v - u ¯ ) for all v∈u¯∨Dom⁡(Ψ).

Remark 4.3.

Direct calculations show that in the case Ψ=0 and f is single-valued, i.e., when (3.8) becomes a parabolic equation, then the above concepts of sub- and supersolutions in Definition 4.2 reduce to the concepts of sub- and supersolutions of equations.

With these definitions, we have the following existence and enclosure theorem for (3.8) in the presence of sub- and supersolutions and assuming a local growth condition on F on the set between them.

Theorem 4.4.

Suppose (A1)–(A5), (C1)–(C4), and (F1)–(F2) are satisfied. Let u¯1,…,u¯k be subsolutions of (3.8) and u¯1,…,u¯m supersolutions of (3.8) such that

u ¯ 0 := ⋁ i = 1 k u ¯ i ≤ ⋀ j = 1 m u ¯ j = : u ¯ 0 a.e. in Q .

Assume that F satisfies the following growth condition between u¯i (i=1,…,k) and u¯j (j=1,…,m): There is h0∈L2⁢(Q) such that

(4.4) sup ⁡ { | η | : η ∈ F ⁢ ( x , t , u ) } ≤ h 0 ⁢ ( x , t )

for a.e. (x,t)∈Q and all

u ∈ [ min ⁡ { u ¯ i ⁢ ( x , t ) : i ∈ { 1 , … , k } } , max ⁡ { u ¯ j ⁢ ( x , t ) : j ∈ { 1 , … , m } } ] .

Then (3.8) has a solution u such that u∈L∞⁢(0,T;W01,p⁢(Ω)), ut∈L2⁢(Q), and u¯0≤u≤u¯0.

Proof.

The proof follows the general ideas of the sub-supersolution method on elliptic variational inequalities with multivalued or single-valued terms, so only the main steps with some substantial differences in our current problem are presented here.

The first part of our proof uses arguments similar to those in [15] and [13, Theorem 3.7], so only an outline is given here together with necessary adaptations/modifications; we refer to [13] for more details and complete arguments.

Let u¯i,η¯i (1≤i≤k) and u¯j,η¯j (1≤j≤m) satisfy conditions (i)–(iv) in Definition 4.2 (a) and (b) of sub- and supersolutions. We define the functions η¯ and η¯ as follows. Let

Q 1 = { ( x , t ) ∈ Q : u ¯ 0 ⁢ ( x , t ) = u ¯ 1 ⁢ ( x , t ) } ,

Q i = { ( x , t ) ∈ Q ∖ ⋃ l = 1 i - 1 Q l : u ¯ 0 ⁢ ( x , t ) = u ¯ i ⁢ ( x , t ) } , i = 2 , … , k .

Similarly, let

Q 1 = { ( x , t ) ∈ Q : u ¯ 0 ⁢ ( x , t ) = u ¯ 1 ⁢ ( x , t ) } ,

Q j = { ( x , t ) ∈ Q ∖ ⋃ l = 1 j - 1 Q l : u ¯ 0 ⁢ ( x , t ) = u ¯ j ⁢ ( x , t ) } , j = 2 , … , m .

It follows from their definitions that Qi (1≤i≤k) (resp. Qj (1≤j≤m)) are disjoint measurable subsets of Q and

Q = ⋃ i = 1 k Q i = ⋃ j = 1 m Q j .

On the other hand, note that u¯0,u¯0∈X. We then define

η ¯ = ∑ i = 1 k η ¯ i ⁢ χ Q i , η ¯ = ∑ j = 1 m η ¯ j ⁢ χ Q j ,

where χS (S⊂Q) is the characteristic function of S. It is clear from (4.2) and (4.3) that

(4.5) η ¯ , η ¯ ∈ L 2 ⁢ ( Q ) ,

and

η ¯ ⁢ ( x , t ) ∈ F ⁢ ( x , t , u ¯ 0 ⁢ ( x , t ) ) , η ¯ ⁢ ( x , t ) ∈ F ⁢ ( x , t , u ¯ 0 ⁢ ( x , t ) ) for a.e. ⁢ ( x , t ) ∈ Q .

Next, we define the truncated function F0⁢(x,t,u) of F⁢(x,t,u) as in [13]: F0 is the function from Q×ℝ to 2ℝ given by

(4.6) F 0 ⁢ ( x , t , u ) = { { η ¯ ⁢ ( x , t ) } if ⁢ u < u ¯ 0 ⁢ ( x , t ) , F ⁢ ( x , t , u ) if ⁢ u ¯ 0 ⁢ ( x , t ) ≤ u ≤ u ¯ 0 ⁢ ( x , t ) , { η ¯ ⁢ ( x , t ) } if ⁢ u > u ¯ 0 ⁢ ( x , t ) .

It can be verified by direct calculations, using the conditions (F1) and (F2), assumed on F, that F0 also satisfies (F1) and (F2). Moreover, from (4.4) and (4.5),

(4.7) sup ⁡ { | ξ | : ξ ∈ F 0 ⁢ ( x , t , u ) } ≤ h 0 ⁢ ( x , t ) + | η ¯ ⁢ ( x , t ) | + | η ¯ ⁢ ( x , t ) | for a.e. ⁢ ( x , t ) ∈ Q , all ⁢ u ∈ ℝ ,

with h0+|η¯|+|η¯|∈L2⁢(Q). Hence, F0 also satisfies (F3).

Next, let b:Q×ℝ→ℝ be given by

(4.8) b ⁢ ( x , t , u ) = { [ u - u ¯ 0 ⁢ ( x , t ) ] p / 2 if ⁢ u > u ¯ 0 ⁢ ( x , t ) , 0 if ⁢ u ¯ 0 ⁢ ( x , t ) ≤ u ≤ u ¯ 0 ⁢ ( x , t ) , - [ u ¯ 0 ⁢ ( x , t ) - u ] p / 2 if ⁢ u < u ¯ 0 ⁢ ( x , t ) ⁢ for ⁢ ( x , t ) ∈ Q , u ∈ ℝ .

It is clear that b is a Carathéodory function and there exists a constant c23>0 such that

(4.9) | b ⁢ ( x , t , u ) | ≤ c 23 ⁢ ( | u | p / 2 + | u ¯ 0 ⁢ ( x , t ) | p / 2 + | u ¯ 0 ⁢ ( x , t ) | p / 2 ) for a.e. ⁢ ( x , t ) ∈ Q ⁢ all ⁢ u ∈ ℝ .

Next, we put, for (x,t)∈Q and u∈ℝ,

T i ⁢ ( x , t , u ) = | η ¯ i ⁢ ( x , t ) - η ¯ ⁢ ( x , t ) | ⁢ σ ⁢ ( u - u ¯ i ⁢ ( x , t ) u ¯ 0 ⁢ ( x , t ) - u ¯ i ⁢ ( x , t ) ) , 1 ≤ i ≤ k ,

and

T j ⁢ ( x , t , u ) = | η ¯ j ⁢ ( x , t ) - η ¯ ⁢ ( x , t ) | ⁢ [ 1 - σ ⁢ ( u - u ¯ 0 ⁢ ( x , t ) u ¯ j ⁢ ( x , t ) - u ¯ 0 ⁢ ( x , t ) ) ] , 1 ≤ j ≤ m ,

where

σ ⁢ ( s ) = { 1 if ⁢ s < 0 , 1 - s if ⁢ 0 ≤ s ≤ 1 , 0 if ⁢ s > 1 .

For i∈{1,…,k} and j∈{1,…,m}, it follows from their definitions, and some direct calculations, that Ti and Ti are Carathéodory functions and there exists H0∈L2⁢(Q) such that for all i∈{1,…,k}, j∈{1,…,m},

(4.10) | T i ⁢ ( x , t , u ) | , | T j ⁢ ( x , t , u ) | ≤ H 0 ⁢ ( x , t ) for a.e. ⁢ ( x , t ) ∈ Q , all ⁢ u ∈ ℝ .

Let us consider the following auxiliary variational inequality of finding u∈Dom⁡(Ψ) and η∈L2⁢(Q) such that

(4.11) u t ∈ L 2 ⁢ ( Q ) , η ⁢ ( x , t ) ∈ F 0 ⁢ ( x , t , u ⁢ ( x , t ) ) for a.e. ⁢ ( x , t ) ∈ Q ,

(4.12) u ⁢ ( ⋅ , 0 ) = u 0 ,

and

〈 u t , v - u 〉 + 〈 A ⁢ ( u ) , v - u 〉 + ∫ Q b ⁢ ( x , t , u ) ⁢ ( v - u ) ⁢ 𝑑 x ⁢ 𝑑 t - ∑ i = 1 k ∫ Q T i ⁢ ( x , t , u ) ⁢ ( v - u ) ⁢ 𝑑 x ⁢ 𝑑 t

(4.13) + ∑ j = 1 m ∫ Q T j ⁢ ( x , t , u ) ⁢ ( v - u ) ⁢ 𝑑 x ⁢ 𝑑 t + Ψ ⁢ ( v ) - Ψ ⁢ ( u ) ≥ ∫ Q η ⁢ ( x , t ) ⁢ ( v - u ) ⁢ 𝑑 x ⁢ 𝑑 t for all ⁢ v ∈ X 0 .

This inequality is of the form (3.8) with F replaced by

F 1 := F 0 - b + ∑ i = 1 k T i - ∑ j = 1 m T j .

Since F0 satisfies (F1)–(F2) and b,Ti,Tj (1≤i≤k, 1≤j≤m) are (single-valued) Carathéodory functions, we see that F1 also satisfies (F1)–(F2). Furthermore, it follows from (4.7) and the growth conditions satisfied by b,Ti,Tj (1≤i≤k, 1≤j≤m) that F1 satisfies (F3) as well. In fact, for a.e. (x,t)∈Q, all u∈ℝ, and all η∈F0⁢(x,t,u), we get, from (4.7), (4.9) and (4.10), that

| η ⁢ ( x , t ) - b ⁢ ( x , t , u ) + ∑ i = 1 k T i ⁢ ( x , t , u ) - ∑ j = 1 m T j ⁢ ( x , t , u ) |

≤ h 0 ⁢ ( x , t ) + | η ¯ ⁢ ( x , t ) | + | η ¯ ⁢ ( x , t ) | + c 23 ⁢ ( | u | p / 2 + | u ¯ ⁢ ( x , t ) | p / 2 + | u ¯ ⁢ ( x , t ) | p / 2 ) + ( k + m ) ⁢ H 0 ⁢ ( x , t )

= c 23 ⁢ | u | p / 2 + H 1 ⁢ ( x , t ) ,

with H1∈L2⁢(Q) independent of x,t, and u. This shows that F1 also satisfies condition (3.1) in (F3).

It follows from Theorem 3.2 that (4.11) has a solution u with ut∈L2⁢(Q) and u∈L∞⁢(0,T;W01,p⁢(Ω)). Let u be any such solution of (4.11). Let us prove that

(4.14) u ¯ ≤ u ≤ u ¯ a.e. on ⁢ Q .

We shall prove here the first inequality; the second is verified in a similar manner. For any s∈{1,…,k}, let us prove that

(4.15) u ¯ s ≤ u a.e. on ⁢ Q .

Note that u¯s∨u∈X, and moreover, u¯s⁢(x,t)∨u⁢(x,t)=0 for a.e. (x,t)∈∂⁡Ω×(0,T). Thus, u¯s∨u∈X0. Letting v=u¯s∨u=u+(u¯s-u)+ in (4.13) yields

〈 u t , ( u ¯ s - u ) + 〉 + ∫ Q a ⁢ ( x , t , ∇ ⁡ u ) ⁢ ∇ ⁡ [ ( u ¯ s - u ) + ] ⁡ d ⁢ x ⁢ d ⁢ t + ∫ Q b ⁢ ( x , t , u ) ⁢ ( u ¯ s - u ) + ⁢ 𝑑 x ⁢ 𝑑 t

- ∑ i = 1 k ∫ Q T i ⁢ ( x , t , u ) ⁢ ( u ¯ s - u ) + ⁢ 𝑑 x ⁢ 𝑑 t + ∑ j = 1 m ∫ Q T j ⁢ ( x , t , u ) ⁢ ( u ¯ s - u ) + ⁢ 𝑑 x ⁢ 𝑑 t

+ Ψ ⁢ ( u ¯ s ∨ u ) - Ψ ⁢ ( u ) - ∫ Q η ⁢ ( u ¯ s - u ) + ⁢ 𝑑 x ⁢ 𝑑 t ≥ 0 .

From (iv) in Definition 4.2 with u¯s, η¯s, and Ψ¯s instead of u¯, η¯, and Ψ¯, and

v = u ¯ s ∧ u = u ¯ s - ( u ¯ s - u ) + ∈ u ¯ s ∧ Dom ⁡ ( Ψ ) ,

we obtain

- 〈 ( u ¯ s ) t , ( u ¯ s - u ) + 〉 - ∫ Q a ⁢ ( x , t , ∇ ⁡ u ¯ s ) ⁢ ∇ ⁡ [ ( u ¯ s - u ) + ] ⁡ d ⁢ x ⁢ d ⁢ t + Ψ ¯ s ⁢ ( u ¯ s ∧ u ) - Ψ ¯ s ⁢ ( u ¯ s ) + ∫ Q η ¯ s ⁢ ( u ¯ s - u ) + ⁢ 𝑑 x ⁢ 𝑑 t ≥ 0 .

Adding these inequalities yields

〈 ( u - u ¯ s ) t , ( u ¯ s - u ) + 〉 + ∫ Q [ a ⁢ ( x , t , ∇ ⁡ u ) - a ⁢ ( x , t , u ¯ s ) ] ⁢ ∇ ⁡ [ ( u ¯ s - u ) + ] ⁡ d ⁢ x ⁢ d ⁢ t

- ∫ Q η ⁢ ( u ¯ s - u ) + ⁢ 𝑑 x ⁢ 𝑑 t + ∫ Q η ¯ s ⁢ ( u ¯ s - u ) + ⁢ 𝑑 x ⁢ 𝑑 t + ∫ Q b ⁢ ( x , t , u ) ⁢ ( u ¯ s - u ) + ⁢ 𝑑 x ⁢ 𝑑 t

- ∑ i = 1 k ∫ Q T i ⁢ ( x , t , u ) ⁢ ( u ¯ s - u ) + ⁢ 𝑑 x ⁢ 𝑑 t + ∑ j = 1 m ∫ Q T j ⁢ ( x , t , u ) ⁢ ( u ¯ s - u ) + ⁢ 𝑑 x ⁢ 𝑑 t

+ Ψ ⁢ ( u ¯ s ∨ u ) + Ψ ¯ s ⁢ ( u ¯ s ∧ u ) - Ψ ⁢ ( u ) - Ψ ¯ s ⁢ ( u ¯ s ) ≥ 0 .

Since Ψ¯s⪯Ψ and Ψ⁢(u),Ψ¯s⁢(u¯s)<∞, we have

Ψ ⁢ ( u ¯ s ∨ u ) + Ψ ¯ s ⁢ ( u ¯ s ∧ u ) - Ψ ⁢ ( u ) - Ψ ¯ s ⁢ ( u ¯ s ) ≤ 0 .

Hence,

〈 ( u - u ¯ s ) t , ( u ¯ s - u ) + 〉 + ∫ Q [ a ⁢ ( x , t , ∇ ⁡ u ) - a ⁢ ( x , t , u ¯ s ) ] ⁢ ∇ ⁡ [ ( u ¯ s - u ) + ] ⁡ d ⁢ x ⁢ d ⁢ t

+ ∫ Q ( η ¯ s - η ) ⁢ ( u ¯ s - u ) + ⁢ 𝑑 x ⁢ 𝑑 t + ∫ Q b ⁢ ( x , t , u ) ⁢ ( u ¯ s - u ) + ⁢ 𝑑 x ⁢ 𝑑 t

(4.16) - ∑ i = 1 k ∫ Q T i ⁢ ( x , t , u ) ⁢ ( u ¯ s - u ) + ⁢ 𝑑 x ⁢ 𝑑 t + ∑ j = 1 m ∫ Q T j ⁢ ( x , t , u ) ⁢ ( u ¯ s - u ) + ⁢ 𝑑 x ⁢ 𝑑 t ≥ 0 .

We have u¯s-u∈W. From (4.12) and condition (iii) in Definition 4.2 (a),

( u ¯ s - u ) + ⁢ ( ⋅ , 0 ) = 0 ,

and thus

〈 ( u - u ¯ s ) t , ( u ¯ s - u ) + 〉 = - 1 2 ⁢ ∥ ( u ¯ s - u ) + ⁢ ( ⋅ , T ) ∥ L 2 ⁢ ( Ω ) 2 ≤ 0 .

On the other hand, it follows from the monotonicity of a⁢(x,t,⋅) in (2.4) that

∫ Q [ a ⁢ ( x , t , ∇ ⁡ u ) - a ⁢ ( x , t , ∇ ⁡ u ¯ s ) ] ⁢ ∇ ⁡ [ ( u ¯ s - u ) + ] ⁡ d ⁢ x ⁢ d ⁢ t

= - ∫ { ( x , t ) ∈ Q : u ¯ s ⁢ ( x , t ) > u ⁢ ( x , t ) } [ a ⁢ ( x , t , ∇ ⁡ u ) - a ⁢ ( x , t , ∇ ⁡ u ¯ s ) ] ⁢ ( ∇ ⁡ u - ∇ ⁡ u ¯ s ) ⁢ 𝑑 x ⁢ 𝑑 t ≤ 0 .

At (x,t)∈Q such that u¯s⁢(x,t)>u⁢(x,t), since u¯s⁢(x,t)≤u¯0⁢(x,t)≤u¯0⁢(x,t), we have from the definition of Tj that Tj⁢(x,t,u⁢(x,t))=0. Therefore,

∫ Q T j ⁢ ( x , t , u ) ⁢ ( u ¯ s - u ) + ⁢ 𝑑 x ⁢ 𝑑 t = ∫ { ( x , t ) ∈ Q : u ¯ s ⁢ ( x , t ) > u ⁢ ( x , t ) } T j ⁢ ( x , t , u ⁢ ( x , t ) ) ⁢ [ u ¯ s ⁢ ( x , t ) - u ⁢ ( x , t ) ] ⁢ 𝑑 x ⁢ 𝑑 t = 0

for all j∈{1,…,m}. Furthermore, for (x,t)∈Q such that u¯s⁢(x,t)>u⁢(x,t), we have u⁢(x,t)<u¯0⁢(x,t). Hence, in view of (4.11) and (4.6), we get η⁢(x,t)∈{η¯⁢(x,t)}, i.e., η⁢(x,t)=η¯⁢(x,t). Also, for such a point (x,t),

T s ⁢ ( x , t , u ⁢ ( x , t ) ) = | η ¯ s ⁢ ( x , t ) - η ¯ ⁢ ( x , t ) | .

The definition of Ti implies that

∫ Q T i ⁢ ( x , t , u ) ⁢ ( u ¯ s - u ) + ⁢ 𝑑 x ⁢ 𝑑 t ≥ 0 for all ⁢ i ∈ { 1 , … , k } .

Consequently,

- ∫ Q ( η - η ¯ s ) ⁢ ( u ¯ s - u ) + ⁢ 𝑑 x ⁢ 𝑑 t - ∑ i = 1 k ∫ Q T i ⁢ ( x , t , u ) ⁢ ( u ¯ s - u ) + ⁢ 𝑑 x ⁢ 𝑑 t

≤ - ∫ Q ( η - η ¯ s ) ⁢ ( u ¯ s - u ) + ⁢ 𝑑 x ⁢ 𝑑 t - ∫ Q T s ⁢ ( x , t , u ) ⁢ ( u ¯ s - u ) + ⁢ 𝑑 x ⁢ 𝑑 t

(4.17) = ∫ { ( x , t ) ∈ Q : u ¯ s ⁢ ( x , t ) > u ⁢ ( x , t ) } { [ η ¯ s ⁢ ( x , t ) - η ¯ ⁢ ( x , t ) ] - | η ¯ ⁢ ( x , t ) - η ¯ s ⁢ ( x , t ) | } ⁢ ( u ¯ s - u ) ⁢ 𝑑 x ⁢ 𝑑 t ≤ 0 .

Combining the estimates from (4.16) to (4.17) yields

0 ≤ ∫ Q b ⁢ ( x , t , u ) ⁢ ( u ¯ s - u ) + ⁢ 𝑑 x ⁢ 𝑑 t = ∫ { ( x , t ) ∈ Q : u ¯ s ⁢ ( x , t ) > u ⁢ ( x , t ) } b ⁢ ( x , t , u ) ⁢ ( u ¯ s - u ) ⁢ 𝑑 x ⁢ 𝑑 t .

By (4.8), if u¯s⁢(x,t)>u⁢(x,t), then u¯>u⁢(x,t) and

b ⁢ ( x , t , u ⁢ ( x , t ) ) = - [ u ¯ 0 ⁢ ( x , t ) - u ⁢ ( x , t ) ] p / 2 .

Hence,

0 ≤ - ∫ { ( x , t ) ∈ Q : u ¯ s ⁢ ( x , t ) > u ⁢ ( x , t ) } [ u ¯ 0 ⁢ ( x , t ) - u ⁢ ( x , t ) ] p / 2 ⁢ [ u ¯ s ⁢ ( x , t ) - u ⁢ ( x , t ) ] ⁢ 𝑑 x ⁢ 𝑑 t .

Since u¯0⁢(x,t)-u⁢(x,t)>0 and u¯s⁢(x,t)-u⁢(x,t)>0 in the set

{ ( x , t ) ∈ Q : u ¯ s ⁢ ( x , t ) > u ⁢ ( x , t ) } ,

this inequality implies that u⁢(x,t)≥u¯s⁢(x,t) for a.e. (x,t)∈Q and thus (4.15) holds true. Since (4.15) is valid for all s∈{1,…,k}, the first inequality of (4.14) follows immediately.

From (4.14) and the definitions of b,Ti, and Tj, we have

b ⁢ ( ⋅ , ⋅ , u ) = T i ⁢ ( ⋅ , ⋅ , u ) = T j ⁢ ( ⋅ , ⋅ , u ) = 0 a.e. on ⁢ Q ,

for all i∈{1,…,k}, j∈{1,…,m}. Furthermore, (4.14), (4.6), and (4.11) imply that

η ⁢ ( x , t ) ∈ F 0 ⁢ ( x , t , u ⁢ ( x , t ) ) = F ⁢ ( x , t , u ⁢ ( x , t ) ) for a.e. ⁢ ( x , t ) ∈ Q .

Thus, if u is a solution of (4.11)–(4.13), it also satisfies (3.8). Our proof of Theorem 4.4 is complete. ∎

4.2 Compactness of Solution Sets – Existence of Extremal Solutions

We can derive from this main existence and enclosure result some other qualitative properties of the solution set of (3.8) between sub- and supersolutions. Assume the conditions in Theorem 4.4 are fulfilled. Let 𝒮 be the set of all solutions of (3.8) between u¯0 and u¯0:

𝒮 := { u ∈ W 0 : u ⁢ is a solution of (3.8) and ⁢ u ¯ 0 ≤ u ≤ u ¯ 0 ⁢ a.e. on ⁢ Q } .

The set 𝒮 is nonempty according to Theorem 4.4. Some properties of 𝒮 are given in the following theorem.

Theorem 4.5.

𝒮 is weakly compact in W 0 and compact in X 0 .
Assume Ψ ¯ ⪯ Ψ ¯ . Then:
1. If u ∈ W 0 is a solution of ( 3.8 ), then u is both a subsolution and a supersolution in the sense of Definition 4.2.
2. 𝒮 is directed both upward and downward, that is, if u 1 , u 2 ∈ 𝒮 then there are u 3 , u 4 ∈ 𝒮 such that u 3 ≤ u 1 , u 2 ≤ u 4 . Moreover, 𝒮 has a least and a greatest element, that is, there are elements u * and u * of 𝒮 such that u * ≤ u ≤ u * for all u ∈ 𝒮 .

Proof.

(a) Since ∥u∥Lp⁢(Q)≤∥u¯0∥Lp⁢(Q)+∥u¯0∥Lp⁢(Q) for all u∈𝒮, the set {∥u∥Lp⁢(Q):u∈𝒮} is bounded. We see from (4.4) that the set {∥η∥L2⁢(Q):η∈F~⁢(u),u∈𝒮} is bounded. Hence, according to Theorem 2.1, the set

{ ∥ u ∥ L ∞ ⁢ ( 0 , T ; W 0 1 , p ⁢ ( Ω ) ) + ∥ u t ∥ L 2 ⁢ ( Q ) : u ∈ 𝒮 }

is also bounded. In particular, 𝒮 is a bounded subset of W0. From the arguments in the proof of Theorem 3.2, we see that 𝒮 is also weakly closed in W0, and thus 𝒮 is weakly compact in W0.

Next, to prove the compactness of 𝒮 in X0, let {un} be a sequence in 𝒮. Since 𝒮 is weakly compact in W0, the sequence {un⁢t} is bounded in L2⁢(Q), and the imbedding W0↪X0 is continuous, by passing to a subsequence, if necessary, we may assume that for some u∈𝒮,

(4.18) u n ⇀ u in ⁢ X 0 and u n ⁢ t ⇀ u t in ⁢ L 2 ⁢ ( Q ) .

We obtain from the theorem of Aubin–Lions (see [22]) that

(4.19) u n → u in ⁢ L p ⁢ ( Q ) .

Together with (4.7) and [8, Theorem 7.26], this limit implies that

(4.20) h L 2 ⁢ ( Q ) * ⁢ ( F 0 ~ ⁢ ( u n ) , F 0 ~ ⁢ ( u ) ) → 0 as ⁢ n → ∞ .

Since u¯0≤un≤u¯0 a.e. on Q for all n∈ℕ, the limit in (4.19) implies that u¯0≤u≤u¯0 a.e. on Q. Hence, by (4.6), we have F0~⁢(un)=F~⁢(un) and F0~⁢(u)=F~⁢(u), and both are closed, bounded, and convex subsets of L2⁢(Q). In particular, they are both weakly compact in L2⁢(Q). For each n∈ℕ, since un∈𝒮, there exists ηn∈F~⁢(un) such that

(4.21) ( u n ⁢ t , v - u n ) + 〈 A ⁢ u n , v - u n 〉 + Ψ ⁢ ( v ) - Ψ ⁢ ( u n ) ≥ ( η n , v - u n ) for all ⁢ v ∈ X 0 .

The limit in (4.20) implies that there exists a sequence {η~n} in F~⁢(u) such that

(4.22) ∥ η n - η ~ n ∥ L 2 ⁢ ( Q ) → 0 .

Since F~⁢(u) is weakly compact in L2⁢(Q), by passing once more to a subsequence, we can assume that η~n⇀η in L2⁢(Q) for some η∈F~⁢(u). This limit and (4.22) imply that

(4.23) η n ⇀ η in ⁢ L 2 ⁢ ( Q ) .

As a consequence of (4.19)–(4.23), we have

(4.24) ( η n , v - u n ) → ( η , v - u ) .

On the other hand, from (4.18) and the lower semicontinuity of Ψ,

(4.25) Ψ ⁢ ( u ) ≤ lim inf ⁡ Ψ ⁢ ( u n ) .

Also, it follows from (4.18) and (4.19) that

(4.26) ( u n ⁢ t , v - u n ) → ( u t , v - u ) .

Letting v=u in (4.21), passing to the limit in the inequality thus obtained, and taking into account the limits in (4.24)–(4.26), we get

(4.27) lim sup n → ∞ ⁡ 〈 A ⁢ u n , u n - u 〉 ≤ 0 .

In view of the evolution triple X0⊂L2⁢(Q)⊂X0*, by applying [7, Lemma 4], we conclude from (4.19) and (4.27) that un→u in X0. As noted above, u∈𝒮, proving the compactness of 𝒮 in X0.

(b) Based on the results in (a), we can now adapt, without significant modifications, the arguments in e.g. [4, 13] for elliptic variational inequalities to the proofs for the results in (b) in our case of parabolic variational inequalities. ∎

Remark 4.6.

Compared to the approach in [3], we see from Theorem 4.5 that with the estimates on the time derivative ut, we are able, using the approach here, to study additional qualitative properties of solutions of parabolic variational inequalities that were not generally obtainable with the method used in [3]. On the other hand, the principal operator here is assumed to have a variational structure, i.e., to have a potential functional, a condition which is not assumed in [3].

Communicated by Patrizia Pucci

References

[1] J. W. Bebernes and K. Schmitt, On the existence of maximal and minimal solutions for parabolic partial differential equations, Proc. Amer. Math. Soc. 73 (1979), no. 2, 211–218. 10.1090/S0002-9939-1979-0516467-3Search in Google Scholar

[2] S. Carl, On the existence of extremal weak solutions for a class of quasilinear parabolic problems, Differential Integral Equations 6 (1993), no. 6, 1493–1505. 10.57262/die/1370019770Search in Google Scholar

[3] S. Carl and V. K. Le, Quasilinear parabolic variational inequalities with multi-valued lower-order terms, Z. Angew. Math. Phys. 65 (2014), no. 5, 845–864. 10.1007/s00033-013-0357-6Search in Google Scholar

[4] S. Carl and V. K. Le, Elliptic inequalities with multi-valued operators: Existence, comparison and related variational-hemivariational type inequalities, Nonlinear Anal. 121 (2015), 130–152. 10.1016/j.na.2014.10.033Search in Google Scholar

[5] S. Carl, V. K. Le and D. Motreanu, The sub-supersolution method and extremal solutions for quasilinear hemivariational inequalities, Differential Integral Equations 17 (2004), no. 1–2, 165–178. 10.57262/die/1356060478Search in Google Scholar

[6] S. Carl, V. K. Le and D. Motreanu, Evolutionary variational-hemivariational inequalities: Existence and comparison results, J. Math. Anal. Appl. 345 (2008), no. 1, 545–558. 10.1016/j.jmaa.2008.04.005Search in Google Scholar

[7] J. Deuel and P. Hess, Nonlinear parabolic boundary value problems with upper and lower solutions, Israel J. Math. 29 (1978), no. 1, 92–104. 10.1007/BF02760403Search in Google Scholar

[8] S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. I, Math. Appl. 419, Kluwer Academic Publishers, Dordrecht, 1997. 10.1007/978-1-4615-6359-4Search in Google Scholar

[9] J. Kačur, Method of Rothe in Evolution Equations, Teubner Text. Math. 80, B. G. Teubner, Leipzig, 1985. 10.1007/BFb0076049Search in Google Scholar

[10] N. Kenmochi, Some nonlinear parabolic variational inequalities, Israel J. Math. 22 (1975), no. 3–4, 304–331. 10.1007/BF02761596Search in Google Scholar

[11] V. K. Le, Sub-supersolutions and the existence of extremal solutions in noncoercive variational inequalities, JIPAM. J. Inequal. Pure Appl. Math. 2 (2001), no. 2, Paper No. 20. Search in Google Scholar

[12] V. K. Le, Subsolution-supersolution method in variational inequalities, Nonlinear Anal. 45 (2001), no. 6, 775–800. 10.1016/S0362-546X(99)00440-XSearch in Google Scholar

[13] V. K. Le, Variational inequalities with general multivalued lower order terms given by integrals, Adv. Nonlinear Stud. 11 (2011), no. 1, 1–24. 10.1515/ans-2011-0101Search in Google Scholar

[14] V. K. Le, On variational and quasi-variational inequalities with multivalued lower order terms and convex functionals, Nonlinear Anal. 94 (2014), 12–31. 10.1016/j.na.2013.07.034Search in Google Scholar

[15] V. K. Le and K. Schmitt, Some general concepts of sub- and supersolutions for nonlinear elliptic problems, Topol. Methods Nonlinear Anal. 28 (2006), no. 1, 87–103. Search in Google Scholar

[16] V. K. Le and K. Schmitt, On a parabolic variational inequality related to a sandpile problem, J. Dynam. Differential Equations 27 (2015), no. 3–4, 879–896. 10.1007/s10884-013-9320-7Search in Google Scholar

[17] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris, 1969. Search in Google Scholar

[18] H. Nagase, On an application of Rothe’s method to nonlinear parabolic variational inequalities, Funkcial. Ekvac. 32 (1989), no. 2, 273–299. Search in Google Scholar

[19] N. S. Papageorgiou, F. Papalini and S. Vercillo, Minimal solutions of nonlinear parabolic problems with unilateral constraints, Houston J. Math. 23 (1997), no. 1, 189–201. Search in Google Scholar

[20] M. Rudd, Weak and strong solvability of parabolic variational inequalities in Banach spaces, J. Evol. Equ. 4 (2004), no. 4, 497–517. 10.1007/s00028-004-0153-zSearch in Google Scholar

[21] M. Rudd and K. Schmitt, Variational inequalities of elliptic and parabolic type, Taiwanese J. Math. 6 (2002), no. 3, 287–322. 10.11650/twjm/1500558298Search in Google Scholar

[22] R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Math. Surveys Monogr. 49, American Mathematical Society, Providence, 1997. Search in Google Scholar

[23] E. Zeidler, Nonlinear Functional Analysis and its Applications. Vol. 3: Variational Methods and Optimization, Springer, New York, 1985. 10.1007/978-1-4612-5020-3Search in Google Scholar

Received: 2017-12-24

Accepted: 2018-02-06

Published Online: 2018-03-08

Published in Print: 2018-04-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/ans-2018-0004

Keywords for this article

Variational Inequalities; Multivalued Term; Convex Functional; Sub-supersolutions; Extremal Solutions

Creative Commons

BY 4.0