ENCLOSURE OF SOLUTIONS FOR QUASILINEAR

DYNAMIC HEMIVARIATIONAL INEQUALITIES

Siegfried Carl

Martin-Luther Universität Halle-Wittenberg, Fachbereich Mathematik und Informatik, Institut für Analysis, 06099 Halle, Germany

Abstract. In this paper we provide a criterion for the existence and enclosure of weak solutions for a class of quasilinear parabolic hemivariational inequalities in terms of appropriately defined upper and lower solutions. Regularization, truncation and comparison techniques combined with the theory of pseudomonotone operators are the main tools used in the proof.

1. Introduction

Hemivariational inequalities have been introduced by Panagiotopoulos in [15] (see also [16,19]) to describe various mechanical problems involving nonmonotone, possibly multivalued constitutive laws derived by nonconvex so called superpotentials. Because of the multivalued character and the lack of differentiability of superpotentials one also speaks of multivalued or nonsmooth mechanics. For applications we refer to [15,16,19].

Let Ω ⊂ ℝ^{N} be a bounded domain with Lipschitz boundary ∂Ω, Q = Ω × (0, τ) and Γ = ∂Ω × (0, τ), τ > 0. In this paper we consider the multivalued version of a dynamic hemivariational inequality in form of the following quasilinear nonmonotone parabolic inclusion problem

(P)  \{ \begin{matrix}\frac{∂u}{∂t} + Au ∈ f̂(u) + h  in Q , \\ u = ψ in Ω , u = 0 on Γ , \end{matrix}

where A is a second order quasilinear differential operator in divergence form of Leray-Lions type. The nonlinear multivalued term f̂ on the right-hand side of (P) is assumed to be generated by a locally bounded Baire-measurable function f : ℝ ↦ ℝ and defined by

1991 Mathematics Subject Classification. 35K60, 35R45, 35R70, 49J40.

Key words and phrases. Quasilinear parabolic inclusions, discontinuous nonlinearities, regularization, upper and lower bounds, truncation technique.

The work on this paper was supported by a grant from the Deutsche Akademie der Naturforscher LEOPOLDINA

(1.1)  f̂(s) = [\underline{f}(s), f̄(s)] , s ∈ ℝ ,

where the endpoints \underline{f}(s) and f̄(s) of the interval in (1.1) are given by

(1.2)  \underline{f}(s) = lim_{ε↓0} \underline{f}_{ε}(s) , f̄(s) = lim_{ε↓0} f̄_{ε}(s) ,

with

(1.3)  \underline{f}_{ε}(s) = ess inf_{|ξ−s|<ε} f(ξ) , f̄_{ε}(s) = ess sup_{|ξ−s|<ε} f(ξ) .

Let j : ℝ ↦ ℝ be defined as

j(s) = ∫_{0}^{s} f(ξ) dξ

with f as given above, i.e. f ∈ L_{loc}^{∞}(ℝ). Then j is locally Lipschitz continuous and the generalized gradient ∂j at s (in the sense of Clarke) satisfies

∂j(s) ⊆ f̂(s) ,

and if the one-sided limits f(s ± 0) exit at s ∈ ℝ then ∂j(s) = f̂(s), cf.[6]. This means in a physicist’s language that the multivalued law is characterized by the generalized gradient of a nonsmooth potential j.

The corresponding stationary problems to (P) have been studied for example in [4, 6, 7] using critical point theory for nondifferentiable functionals, in [17, 20] using the Galerkin method (see also [15,16]), in [12] applying finite element technique, and in [2, 14] employing the theory of pseudomonotone operators.

Only recently discontinuous nonlinear nonmonotone dynamic problems have been studied in [5, 11], see also [18]. In [11] existence results have been obtained for parabolic inclusions of the form (P) with a linear operator A by regularization of the discontinuous nonlinearity and by applying Galerkin’s method, cf. also [18]. This approach, however, requires certain growth and sign conditions imposed on the discontinuous nonlinearity involved. In [5] existence of extremal solutions for a nonmonotone discontinuous quasilinear parabolic equation (not an inclusion) under initial and Dirichlet boundary conditions could be proved by means of upper and lower solution method combined with some generalized monotone iteration technique. The much more precise results obtained in [5] compared with [11] was possible by imposing some additional condition on the discontinuous nonlinearity.

The aim of this paper is to provide criteria that ensure not only the existence but also bounds for solutions of the quasilinear parabolic inclusion (P). For continuous nonlinear parabolic initial boundary value problems (IBVP for short) the method of upper and lower solutions has been proved to be a powerful tool for obtaining existence and enclosure results, cf. [3, 9]. Here we extend this method to multivalued problems (P). Assuming the existence of appropriately defined upper and lower solutions ordered in a natural way we prove the existence of weak solutions for problem (P) lying between them. Our proof is based on the application of truncation, regularization and comparison techniques combined with the theory of pseudomonotone operators.

Applying the results of this paper to a problem recently treated in [11] we are able to obtain more precise results under less restrictive assumptions on the discontinuous nonlinearity involved.

2. Hypotheses, definitions and the main result

Let W^{1,p}(Ω) denote the usual Sobolev space and (W^{1,p}(Ω))^{∗} its dual space. For the sake of simplicity we shall assume p ≥ 2, and q ∈ ℝ being the dual real satisfying 1/p + 1/q = 1 . Then W^{1,p}(Ω) ⊂ L^{2}(Ω) ⊂ (W^{1,p}(Ω))^{∗} forms an evolution triple with all the embeddings being continuous, dense and compact, cf. [21].

We set V = L^{p}(0, τ; W^{1,p}(Ω)) , denote its dual space by V^{∗} = L^{q}(0, τ; (W^{1,p}(Ω))^{∗}) , and define a function space W by

W = \{w ∈ V | \frac{∂w}{∂t} ∈ V^{∗}\} ,

where the derivative ∂/∂t is understood in the sense of vector-valued distributions, cf.[10,21]. The space W endowed with the norm

∥w∥_{W} = ∥w∥_{V} + ∥∂w/∂t∥_{V^{∗}}

is a Banach space which is separable and reflexive due to the separability and reflexivity of V and V^{∗}, respectively. Furthermore it is well known that the embedding W ⊂ C([0, τ], L^{2}(Ω)) is continuous, cf. [10,21]. Finally, because W^{1,p}(Ω) ⊂ L^{p}(Ω) is compactly embedded, we have a compact embedding of W ⊂ L^{p}(Q) , cf. [10].

By W_{0}^{1,p}(Ω) we denote the subspace of W^{1,p}(Ω) whose elements have generalized homogeneous boundary values. Let W^{−1,q}(Ω) denote the dual space of W_{0}^{1,p}(Ω). Then obviously W_{0}^{1,p}(Ω) ⊂ L^{2}(Ω) ⊂ W^{−1,q}(Ω) forms an evolution triple and all statements made above remain true also in this situation when setting V_{0} = L^{p}(0, τ; W_{0}^{1,p}(Ω)) , V_{0}^{∗} = L^{q}(0, τ; W^{−1,q}(Ω)) and W_{0} = \{w ∈ V_{0} | \frac{∂w}{∂t} ∈ V_{0}^{∗}\} . The differential operator A is given by

Au(x, t) = − ∑_{i=1}^{N} \frac{∂}{∂x_{i}} a_{i}(x, t, ∇u(x, t)) ,

where we impose the following standard conditions of Leray-Lions type on the coefficient functions a_{i}: Q × ℝ^{N} ↦ ℝ, i = 1, …, N, e.g. cf. [9]. (A1) Each a_{i}(x, t, ξ) is a Carathéodory function, i.e. measurable in (x, t) ∈ Q for all

ξ ∈ ℝ^{N} and continuous in ξ ∈ ℝ^{N} for a.a. (x, t) ∈ Q. (A2) There exists a constant c_{0} ≥ 0 and a function k_{0} ∈ L_{+}^{q}(Q) such that

|a_{i}(x, t, ξ)| ≤ k_{0}(x, t) + c_{0} |ξ|^{p−1}, i = 1, …, N for a.a. (x, t) ∈ Q and for all ξ ∈ ℝ^{N}.

(A3) ∑_{i=1}^{N}(a_{i}(x, t, ξ)−a_{i}(x, t, ξ′))(ξ_{i}−ξ′_{i}) > 0 for a.a. (x, t) ∈ Q and for all ξ, ξ′ ∈ ℝ^{N}

with ξ ≠ ξ′. (A4) There exists a positive constant c_{1} and a function k_{1} ∈ L_{+}^{1}(Q) such that

∑_{i=1}^{N} a_{i}(x, t, ξ) ξ_{i} ≥ c_{1}|ξ|^{p} − k_{1}(x, t), for a.a. (x, t) ∈ Q and for all ξ ∈ ℝ^{N}.

Denote by ⟨·, ·⟩ the duality pairing between the elements of V^{∗} and V (respectively V_{0}^{∗} and V_{0}). Then as a consequence of (A1) and (A2) the semilinear form a associated with the operator A by

⟨Au, φ⟩ = a(u, φ) = ∑_{i=1}^{N} ∫_{Q} a_{i}(x, t, ∇u)\frac{∂φ}{∂x_{i}} dxdt

is well-defined on V × V, and the operator A : V ↦ V^{∗} ⊂ V_{0}^{∗} is continuous and bounded. The norm (strong) convergence is denoted by →, and the weak convergence by ⇀.

A partial ordering in L^{p}(Q) is defined by u ≤ w if and only if w − u belongs to the set L_{+}^{p}(Q) of all nonnegative elements of L^{p}(Q). This induces a corresponding partial ordering also in the subset W of L^{p}(Q), and if \underline{u}, ū ∈ W with \underline{u} ≤ ū then

[\underline{u}, ū] = \{u ∈ W | \underline{u} ≤ u ≤ ū\}

denotes the order interval formed by \underline{u} and ū. We assume ψ ∈ L^{2}(Ω) and h ∈ V_{0}^{∗}. Let us introduce the notion of a (weak) solution of the inclusion (P).

Definition 2.1. A function u ∈ W_{0} is called a solution of problem (P) if there is a function v ∈ L^{q}(Q) such that

(i) u(·, 0) = ψ in Ω ,

(ii) v(x, t) ∈ f̂(u(x, t)) a.e. (x, t) ∈ Q ,

(iii) ⟨\frac{∂u}{∂t}, φ⟩ + a(u, φ) = ∫_{Q} vφ dxdt + ⟨h, φ⟩ for all φ ∈ V_{0}.

The notion of upper (resp. lower) solution for problem (P) will be defined by means of the function f̄ (resp. \underline{f}) : ℝ ↦ ℝ given by (1.2) which is upper (resp. lower) semicontinuous (cf. [6]) and thus is sup-measurable, i.e., the superposition f̄(u(x, t)) (resp. \underline{f}(u(x, t))): Q ↦ ℝ is measurable whenever u : Q ↦ ℝ is measurable.

Definition 2.2. A function ū ∈ W is called an upper solution to problem (P) if (i) ū ≥ 0 on Γ , ū(·, 0) ≥ ψ(·) in Ω , (ii) ⟨\frac{∂ū}{∂t}, φ⟩ + a(ū, φ) ≥ ∫_{Q} f̄(ū)φ dxdt + ⟨h, φ⟩ for all φ ∈ V_{0} ∩ L_{+}^{p}(Q).

Similarly a function \underline{u} ∈ W is a lower solution to (P) if the reversed inequalities hold in (i) and (ii) of Definition 2.2 where f̄ is replaced by \underline{f}.

Further we shall make the following hypotheses. (H1) Suppose problem (P) has an upper solution ū and a lower solution \underline{u} such that

\underline{u} ≤ ū. (H2) Let f : ℝ ↦ ℝ be a locally bounded and Baire-measurable function. Suppose

that there exist a constant α > 0 and a function k_{2} ∈ L_{+}^{q}(Q) such that

|f(s)| ≤ k_{2}(x, t)

for a.a. (x, t) ∈ Q and for all s ∈ [\underline{u}(x, t) − α, ū(x, t) + α].

Remark 2.1. By hypotheses (H2) the nonlinearity f ∈ L_{loc}^{∞}(ℝ) is supposed to satisfy a L^{q}(Q) - boundedness condition with respect to the order interval [\underline{u} − α, ū + α]. In particular, there is no growth restriction needed.

Let J : ℝ ↦ ℝ be a mollifier, that is J ∈ C_{0}^{∞}((−1, +1)), J ≥ 0 with

∫_{−∞}^{∞} J(s) ds = 1 .

For ε > 0 the regularization of f, denoted by f^{ε}, is then defined by the convolution

(2.1)  f^{ε}(s) = \frac{1}{ε} ∫_{−∞}^{∞} f(s − ξ)J(\frac{ξ}{ε}) dξ .

From hypothesis (H2) it readily follows that for any u ∈ [\underline{u}, ū] and for any ε : 0 < ε < α we get the estimate

(2.2)  |f^{ε}(u(x, t))| ≤ k_{2}(x, t) .

The main result of the present paper is given by the following theorem.

Theorem 2.1. Let hypotheses (A1)-(A4) and (H1), (H2) be satisfied. Then problem (P) has at least one solution within the order interval [\underline{u}, ū].

Remark 2.2. Theorem 2.1 provides simultaneously an existence and enclosure result for problem (P). In the special case that A is linear an existence result for problem (P) has been proved recently in [11] by using the Galerkin method. For this purpose the nonlinearity f had been assumed to satisfy a sign condition of the form

(2.3)  ∃ λ > 0 : ess inf_{s∈ℝ} f(s)s ≥ −λ

and the following growth condition

(2.4)  ∃ C > 0 : |f(s)| ≤ C(1 + |s|) , s ∈ ℝ .

(Actually in [11] the growth condition has been assumed to hold a.e. in ℝ which is only slightly more general.) In the next subsection we shall show that in this special case the growth condition (2.4) already implies all the hypotheses of Theorem 2.1. This means that in this situation our assumptions are weaker than those of [11] and at the same time provide stronger results.

2.1. Special case. Let A be a linear operator of the form

Au = − ∑_{i,j=1}^{N} \frac{∂}{∂x_{i}} (a_{ij} \frac{∂u}{∂x_{j}}) ,

with a_{ij} ∈ L^{∞}(Q) , a_{ij}(x, t)ξ_{i}ξ_{j} ≥ μ|ξ|^{2} for a.a. (x, t) ∈ Q, for all ξ ∈ ℝ^{N} and μ being some positive constant. Then all hypotheses (A1)-(A4) are trivially satisfied for p = q = 2. It should be noted that an extended linear operator A of the form

Au = − ∑_{i,j=1}^{N} \frac{∂}{∂x_{i}} (a_{ij} \frac{∂u}{∂x_{j}}) + ∑_{i=1}^{N} b_{i} \frac{∂u}{∂x_{i}} + cu

with b_{i} , c ∈ L^{∞}(Q) could be taken into account as well without any difficulties (which has been omitted only for simplicity).

Now the nonlinearity f : ℝ ↦ ℝ, f ∈ L_{loc}^{∞}(ℝ) is only supposed to satisfy the growth condition (2.4). Then it can easily be seen that the upper (resp. lower) semicontinuous function f̄ (resp. \underline{f}) : ℝ ↦ ℝ satisfies the same growth condition (2.4) with a possibly different constant C, i.e., we have

(2.5)  −C(1 + |s|) ≤ f̄(s) , \underline{f}(s) ≤ C(1 + |s|)

for all s ∈ ℝ.

Consider the following IBVP for functions ū and \underline{u}, respectively, of the form

(2.6)  \{ \begin{matrix}\frac{∂ū}{∂t} + Aū = C(1 + |ū|) + h in Q , \\ ū = ψ in Ω , ū = 0 on Γ , \end{matrix}

(2.7)  \{ \begin{matrix}\frac{∂\underline{u}}{∂t} + A\underline{u} = −C(1 + |\underline{u}|) + h in Q , \\ \underline{u} = ψ in Ω , \underline{u} = 0 on Γ . \end{matrix}

Since the nonlinearities on the right-hand sides of (2.6) and (2.7), respectively, are uniformly Lipschitz continuous, the existence of a uniquely defined solution for the semilinear IBVP (2.6) and (2.7), respectively, follows by applying standard theory for evolutionary equations involving strongly monotone, continuous and coercive operators. (If necessary one has to pass to a new variable ũ by the tranformation u = ũ exp(κt) with some sufficiently large positive constant κ.)

By (2.5) the solution ū of the IBVP (2.6) is an upper solution of problem (P) according to Definition 2.2 and the solution \underline{u} of (2.7) is a lower solution. Further, the maximum principle implies ū ≥ \underline{u}, since the difference w = ū − \underline{u} satisfies the inequality

\frac{∂w}{∂t} + Aw = C(1 + |ū|) + C(1 + |\underline{u}|) ≥ 0

and w = 0 in Ω, w = 0 on Γ, which results in w ≥ 0 in Q. Thus hypothesis (H1) is satisfied. In order to show that also hypothesis (H2) is satisfied let α > 0 be given. Then the growth condition (2.4) yields

|f(u(x, t))| ≤ C(1 + |u(x, t)|)
≤ C(1 + |\underline{u}(x, t)| + |ū(x, t)| + α) ∈ L^{2}(Q)

for all u(x, t) ∈ [\underline{u}(x, t) − α, ū(x, t) + α], and a.e. in Q (notice p = q = 2). Hence the assertion of Remark 2.2 is proved, that is, we are able to give an existence and enclosure result by means of Theorem 2.1 without assuming the sign condition 2.3.

3. A regularized auxiliary IBVP

Throughout this section we shall assume that the hypotheses (A1)-(A4) and (H1), (H2) are satisfied.

We introduce the truncation operator T defined by

T u(x, t) = \{ \begin{matrix}ū(x, t) & if  u(x, t) > ū(x, t) , \\ u(x, t) & if  \underline{u}(x, t) ≤ u(x, t) ≤ ū(x, t) , \\ \underline{u}(x, t) & if  u(x, t) < \underline{u}(x, t) , \end{matrix}

where \underline{u} and ū are the assumed lower and upper solution, respectively. The operator T : L^{p}(Q) ↦ L^{p}(Q) is bounded and continuous, cf. [8]. Further, let b : Q × ℝ ↦ ℝ be the following cut off function

b(x, t, s) = \{ \begin{matrix}(s − ū(x, t))^{p−1} & if  s > ū(x, t) , \\ 0 & if  \underline{u}(x, t) ≤ s ≤ ū(x, t) , \\ −(\underline{u}(x, t) − s)^{p−1} & if  s < \underline{u}(x, t) . \end{matrix}

Then one readily verifies that b is a Carathéodory function satisfying a growth condition of the form

(3.1)  |b(x, t, s)| ≤ k_{3}(x, t) + c_{2}|s|^{p−1}

for some positive constant c_{2} and some function k_{3} ∈ L^{q}(Q), and it can be shown that an estimate of the form

(3.2)  ∫_{Q} b(x, t, u(x, t))u(x, t) dxdt ≥ c_{3}∥u∥_{L^{p}(Q)}^{p} − c_{4}

is valid for some positive constant c_{3} , c_{4}.

By (3.1) it follows that the Nemytskij operator B associated with the function b is bounded and continuous from L^{p}(Q) into L^{q}(Q). Let F^{ε} denote the Nemytskij operator related with the regularization f^{ε} given by (2.1). Then we define the superposition operator F^{ε} ∘ T by

F^{ε} ∘ T u(x, t) = f^{ε}(T u(x, t)) .

The aim of this section is to prove existence and a priori estimates of solutions of the following regularized truncated IBVP:

Find u_{ε} ∈ W_{0} such that

(P_{ε})  \{ \begin{matrix}\frac{∂u_{ε}}{∂t} + Au_{ε} + Bu_{ε} & = F^{ε} ∘ T u_{ε} + h in Q , \\ u_{ε} & = ψ in Ω . \end{matrix}

Lemma 3.1. For any ε : 0 < ε < α the IBVP (P_{ε}) admits a solution u_{ε} ∈ W_{0}.

Proof. By hypothesis (H2) and the definition of the regularization f^{ε} given in (2.1) it follows that for any ε : 0 < ε < α the Nemytskij operator F^{ε} : [\underline{u}, ū] ⊂ L^{p}(Q) ↦ L^{q}(Q) is continuous and uniformly bounded. Thus F^{ε} ∘ T : L^{p}(Q) ↦ L^{q}(Q) is continuous and uniformly bounded as well. Let w ∈ W_{0} denote an extension of the initial values ψ and the homogeneous boundary values into the domain Q. Such an extension always exists. Then performing the simple change of variable

(3.3)  z_{ε} = u_{ε} − w

problem (P_{ε}) can be transformed into an equivalent IBVP with homogeneous initial and boundary values of the form

(P̃_{ε})  \{ \begin{matrix}\frac{∂z_{ε}}{∂t} + Ãz_{ε} + B̃z_{ε} & = F^{ε} ∘ T̃z_{ε} + h̃ in Q , \\ z_{ε} & = 0 in Ω , \end{matrix}

where the functions ã_{i} and b̃ generating the operator Ã and B̃, respectively, are given by ã_{i}(x, t, ξ) = a_{i}(x, t, ξ + ∇w(x, t)) , b̃(x, t, s) = b(x, t, s + w(x, t)), and h̃ = h − ∂w/∂t ∈ V_{0}^{∗} as well as T̃z_{ε} = T (z_{ε} + w). One readily observes that the coefficients ã_{i} and b̃ satisfy analogous structure conditions (A1) - (A4) and (3.1), (3.2), respectively, and that F^{ε} ∘ T̃ : L^{p}(Q) ↦ L^{q}(Q) is continuous and uniformly bounded for ε : 0 < ε < α.

Let L = ∂/∂t and its domain D(L) ⊂ V_{0} be defined by

D(L) = {u ∈ W_{0} | u(·, 0) = 0 in Ω} .

The linear operator L : D(L) ⊂ V_{0} ↦ V_{0}^{∗} can be shown to be closed, densely defined and maximal monotone, cf. [21]. The assumptions (A1) - (A4), and the properties of the operators B̃ and F^{ε} ∘ T̃ mentioned above imply that the operator

Ã + B̃ − F^{ε} ∘ T̃ : V_{0} ↦ V_{0}^{∗}

is continuous, pseudomonotone and coercive with respect to D(L), cf. [1]. Hence due to [1, Theorem 5] (see also [10,13]) the IBVP (P̃_{ε}) admits a solution z_{ε} ∈ D(L) for any h̃ ∈ V_{0}^{∗}. This in turn implies the existence of solutions u_{ε} for the IBVP (P_{ε}). ⊓ ⊔ Lemma 3.2. For any ε ∈ (0, α) the solutions u_{ε} of the IBVP (P_{ε}) are uniformly bounded in W_{0}, i.e.

(3.4)  ∥u_{ε}∥_{W_{0}} ≤ C .

Proof. Equivalently to (3.4) we shall prove that

∥z_{ε}∥_{W_{0}} ≤ C

holds. Let z_{ε} be any solution of (P̃_{ε}), that is, z_{ε} ∈ W_{0}, z_{ε}(·, 0) = 0 in Ω and for all φ ∈ V_{0} we have

(3.5)  ⟨\frac{∂z_{ε}}{∂t} , φ⟩ + ã(z_{ε}, φ) + ∫_{Q}(B̃z_{ε})φ dxdt = ∫_{Q}(F^{ε} ∘ T̃z_{ε})φ dxdt + ⟨h̃, φ⟩ .

Let c > 0, k > 0 be some universal constants which may have different values but which do not depend on the solutions z_{ε}. Since the data of (P̃_{ε}) satisfy analogous structure conditions like those of (P_{ε}) from (A4) and (3.2) we obtain

(3.6)  ã(z_{ε}, z_{ε}) + ∫_{Q}(B̃z_{ε})z_{ε} dxdt ≥ c (∥∇z_{ε}∥_{L^{p}(Q)}^{p} + ∥z_{ε}∥_{L^{p}(Q)}^{p}) − k ,

and the uniform boundedness of F^{ε} ∘ T̃ yields

(3.7)  |∫_{Q}(F^{ε} ∘ T̃z_{ε}) z_{ε} dxdt| + |⟨h̃, z_{ε}⟩| ≤ c∥z_{ε}∥_{V_{0}} .

Since z_{ε}(·, 0) = 0 we have

(3.8)  ⟨∂z_{ε}/∂t, z_{ε}⟩ ≥ 0 .

Thus from (3.5) with the special test function φ = z_{ε} we obtain by means of (3.6), (3.7) and (3.8)

∥z_{ε}∥_{V_{0}}^{p} ≤ c (1 + ∥z_{ε}∥_{V_{0}})

which implies

(3.9)  ∥z_{ε}∥_{V_{0}}^{p} ≤ c .

To complete the proof we have to show that ∥∂z_{ε}/∂t∥_{V_{0}^{∗}} ≤ c holds.

From (3.5) and by means of (A2) and (3.1) we get

\begin{aligned}|⟨\frac{∂z_{ε}}{∂t} , φ⟩| & ≤ |ã(z_{ε}, φ)| + |∫_{Q}(B̃z_{ε}) φ dxdt| \\ & + |∫_{Q}(F^{ε} ∘ T̃z_{ε}) φ dxdt| + |⟨h̃, φ⟩| \\ & ≤ c (1 + ∥∇z_{ε}∥_{L^{p}(Q)}^{p−1})∥∇φ∥_{L^{p}(Q)} \\ & + c (1 + ∥z_{ε}∥_{L^{p}(Q)}^{p−1})∥φ∥_{L^{p}(Q)} + c ∥φ∥_{V_{0}} \\ & ≤ c (1 + ∥z_{ε}∥_{V_{0}}^{p−1})∥φ∥_{V_{0}} . \end{aligned} (3.10)

From (3.10) and (3.9) it follows

∥\frac{∂z_{ε}}{∂t} ∥_{V_{0}^{∗}} = sup\{|⟨\frac{∂z_{ε}}{∂t} , φ⟩| , ∥φ∥_{V_{0}} ≤ 1\} ≤ c ,

and hence the assertion of the lemma. ⊓ ⊔

Lemma 3.3. There exists a sequence (u_{ε_{n}}) of solutions of (P_{ε_{n}}) such that the following convergence properties hold as ε_{n} → 0 for n → ∞:

(i) u_{ε_{n}} ⇀ u in W_{0}, i.e., u_{ε_{n}} ⇀ u in V_{0}, \frac{∂u_{εn}}{∂t} ⇀ \frac{∂u}{∂t} in V_{0}^{∗}, (ii) u_{ε_{n}} → u in L^{p}(Q), (iii) F^{ε_{n}} ∘ T u_{ε_{n}} ⇀ v in L^{q}(Q).

Proof. Let ε_{n} : 0 < ε_{n} < α be such that ε_{n} → 0 as n → ∞ . To each ε_{n} we associate a solution u_{ε_{n}} of the IBVP (P_{ε_{n}}). Then by Lemma 3.2 and due to the weak compactness of a ball in the reflexive Banach space W_{0} there is a subsequence of (u_{ε_{n}}) again denoted by (u_{ε_{n}}) which is weakly convergent to some u ∈ W_{0}, that is, property (i). The compact embedding W_{0} ⊂ L^{p}(Q) implies property (ii). Finally the uniform boundedness of (F^{ε_{n}} ∘ T u_{ε_{n}}) in L^{q}(Q) yields the existence of a weakly convergent subsequence again denoted by (F^{ε_{n}} ∘ T u_{ε_{n}}) in L^{q}(Q), i.e. property (iii). ⊓ ⊔

Lemma 3.4. The sequence (u_{ε_{n}}) of Lemma 3.3 is strongly convergent to u in V_{0}, i.e., u_{ε_{n}} → u in V_{0} as n → ∞ .

Proof. By a result of [9, Lemma 4] we have u_{ε_{n}} → u in V_{0} provided that u_{ε_{n}} ⇀ u in W_{0} and lim sup_{n→∞} a(u_{ε_{n}}, u_{ε_{n}} − u) ≤ 0 holds.

By definition u_{ε_{n}} solve the IBVP (P_{ε_{n}}), that is u_{ε_{n}}(x, 0) = ψ(x) in Ω and

(3.11) ⟨\frac{∂u_{ε_{n}}}{∂t} , φ⟩ + a(u_{ε_{n}}, φ) + ∫_{Q}(Bu_{ε_{n}}) φ dxdt = ∫_{Q}(F^{ε_{n}} ∘ T u_{ε_{n}}) φ dxdt + ⟨h, φ⟩

for all φ ∈ V_{0}. Since u_{ε_{n}} ⇀ u in W_{0} according to Lemma 3.3 and u_{ε_{n}}(x, 0) = ψ(x) one can show that u(x, 0) = ψ(x). As special test function in (3.11) we take φ = u_{ε_{n}} − u. Then due to

⟨\frac{∂(u_{ε_{n}} − u)}{∂t}, u_{ε_{n}} − u⟩ ≥ 0

we get

(3.12)  ⟨\frac{∂u_{ε_{n}}}{∂t} , u_{ε_{n}} − u⟩ ≥ ⟨\frac{∂u}{∂t}, u_{ε_{n}} − u⟩ .

By (3.12) we get from (3.11)

(3.13) a(u_{ε_{n}}, u_{ε_{n}} − u) ≤ ⟨h − \frac{∂u}{∂t}, u_{ε_{n}} − u⟩ + ∫_{Q}(F^{ε_{n}} ∘ T u_{ε_{n}} − Bu_{ε_{n}})(u_{ε_{n}} − u) dxdt .

The right-hand side of (3.13) tends to zero as n → ∞ due to u_{ε_{n}} ⇀ u in V_{0}, u_{ε_{n}} → u in L^{p}(Q), the boundedness of (F^{ε_{n}} ∘ T u_{ε_{n}}) in L^{q}(Q) and the continuity of B : L^{p}(Q) ↦ L^{q}(Q). Hence it follows

lim sup_{n→∞} a(u_{ε_{n}}, u_{ε_{n}} − u) ≤ 0 ,

which implies u_{ε_{n}} → u in V_{0}. ⊓ ⊔

Next we shall show that the weak limit v according to Lemma 3.3 satisfies

v ∈ f̂(T u) .

Lemma 3.5. Let the sequence (u_{ε_{n}}) be given by Lemma 3.3. Then its limit u satisfies

(3.14)  v ∈ f̂(T u) ,

where F^{ε_{n}} ∘ T u_{ε_{n}} ⇀ v in L^{q}(Q) .

Proof. Under the convergence properties of Lemma 3.3 we have to show that the weak limit v satisfies the inclusion v(x, t) ∈ f̂(T u(x, t)) a.e. in Q. To this end we appropriately modify an approach used in [16].

The strong convergence of u_{ε_{n}} → u in L^{p}(Q) and the continuity of the truncation mapping T : L^{p}(Q) ↦ L^{p}(Q) yields

(3.15)  T u_{ε_{n}} → T u in L^{p}(Q) .

By the definition of the regularization f^{ε} according to (2.1) one readily verifies the following inequality

ess inf_{|ξ|≤ε_{n}} f(s − ξ) ≤ f^{ε_{n}}(s) ≤ ess sup_{|ξ|≤ε_{n}} f(s − ξ)

which yields

(3.16)  ess inf_{|s−T u_{εn} |≤ε_{n}} f(s) ≤ f^{ε_{n}}(T u_{ε_{n}}) ≤ ess sup_{|s−T u_{εn} |≤ε_{n}} f(s) .

The strong convergence of (T u_{ε_{n}}) in L^{p}(Q) implies the almost everywhere convergence of some subsequence (again denoted by (T u_{ε_{n}})), i.e., T u_{ε_{n}}(x, t) → T u(x, t), a.e. in Q. By applying Egoroff’s theorem for any δ > 0 there is a measurable set ω ⊂ Q with meas ω < δ such that

T u_{ε_{n}} → T u uniformly in Q \ ω .

Let ϱ > 0 be arbitrarily given. Then there is an ε_{n_{0}} ∈ (0, α) such that for all ε_{n} : 0 < ε_{n} < min(ε_{n_{0}}, ϱ/2) we have

(3.17)  |T u_{ε_{n}}(x, t) − T u(x, t)| < ϱ/2 in Q \ ω ,

and in view of (3.16)

(3.18)  ess inf_{|s−T u_{εn} |<ϱ/2} f(s) ≤ f^{ε_{n}}(T u_{ε_{n}}) ≤ ess sup_{|s−T u_{εn} |<ϱ/2} f(s) .

Hence by (3.17) and (3.18) for all ε_{n} < min(ε_{n_{0}}, ϱ/2) we obtain

(3.19)  f^{ε_{n}}(T u_{ε_{n}}(x, t)) ≤ f̄_{\frac{ϱ}{2}} (T u_{ε_{n}}(x, t)) ≤ f̄_{ϱ}(T u(x, t)) in Q \ ω ,

and analogously

(3.20)  \underline{f}_{ϱ}(T u(x, t)) ≤ \underline{f}_{\frac{ϱ}{2}} (T u_{ε_{n}}(x, t)) ≤ f^{ε_{n}}(T u_{ε_{n}}(x, t)) in Q \ ω .

Multiplying (3.19) by φ ∈ L_{+}^{p}(Q \ ω) and taking into account that

F^{ε_{n}} ∘ T u_{ε_{n}} ⇀ v in L^{q}(Q) as ε_{n} → 0 ,

we get

\begin{aligned}∫_{Q\ω} v φ dxdt & = lim_{ε_{n}↓0} ∫_{Q\ω} f^{ε_{n}}(T u_{ε_{n}}) φ dxdt \\ & ≤ lim_{ε_{n}↓0} ∫_{Q\ω} f̄_{ϱ}(T u) φ dxdt = ∫_{Q\ω} f̄_{ϱ}(T u) φ dxdt , \end{aligned} (3.21)

and analogously from (3.20)

(3.22)  ∫_{Q\ω} \underline{f}_{ϱ}(T u) φ dxdt ≤ ∫_{Q\ω} v φ dxdt .

Hence by (3.21) and (3.22) for any ϱ > 0 it follows

∫_{Q\ω} \underline{f}_{ϱ}(T u) φ dxdt ≤ ∫_{Q\ω} v φ dxdt ≤ ∫_{Q\ω} f̄_{ϱ}(T u) φ dxdt

which yields by applying Lebesgue’s theorem as ϱ → 0 the inequality

∫_{Q\ω} \underline{f}(T u) φ dxdt ≤ ∫_{Q\ω} v φ dxdt ≤ ∫_{Q\ω} f̄(T u) φ dxdt

for any φ ∈ L_{+}^{p}(Q \ ω) and thus

v(x, t) ∈ [\underline{f}(T u(x, t)), f̄(T u(x, t))] = f̂(T u(x, t)) a.e. in Q \ ω ,

where meas ω < δ. Since δ > 0 has been chosen arbitrarily we can choose δ as small as we wish such that we finally obtain

v(x, t) ∈ f̂(T u(x, t)) a.e. in Q , i.e. v ∈ f̂(T u) ,

which proves the lemma. ⊓ ⊔

Now we are ready to prove our main result Theorem 2.1.

4. Proof of Theorem 2.1

According to Lemmata 3.3, 3.4 and 3.5 there exists a sequence (u_{ε_{n}}) of solutions of the regularized auxiliary problems (P_{ε_{n}}) satisfying the following convergence properties as ε_{n} → 0 :

u_{ε_{n}} ⇀ u  in W_{0} ,
u_{ε_{n}} → u  in L^{p}(Q) ,
u_{ε_{n}} → u  in V_{0} ,
F^{ε_{n}} ∘ T u_{ε_{n}} ⇀ v  in L^{q}(Q) , with v ∈ f̂(T u) .

This allows the passage to the limit as ε_{n} → 0 in the weak formulation of the problems (P_{ε_{n}}), and we obtain that the limits u ∈ W_{0} and v ∈ L^{q}(Q), respectively, satisfy the following IBVP

(P_{0})  \{ \begin{matrix}\frac{∂u}{∂t} + Au + Bu & = v + h in Q , \\ u & = ψ in Ω , \end{matrix}

or equivalently

(P_{0})  \{ \begin{matrix}\frac{∂u}{∂t} + Au + Bu & ∈ f̂(T u) + h in Q , \\ u & = ψ in Ω . \end{matrix}

We shall show that the limit u which solves problem (P_{0}) is in fact a solution of the original inclusion problem (P) which is enclosed by the upper and lower solution ū and \underline{u}, respectively. To this end we only have to show that the limit u belongs to the interval [\underline{u}, ū], since then we have T u = u, Bu = 0 and thus u satisfies (P), i.e., u ∈ W_{0} and

(P)  \{ \begin{matrix}\frac{∂u}{∂t} + Au & ∈ f̂(u) + h in Q , \\ u & = ψ in Ω . \end{matrix}

We prove that u ≤ ū in Q, the proof of \underline{u} ≤ u can be done similarly. By definition the upper solution ū satisfies: ū ∈ W and ū ≥ 0 on Γ, ū ≥ ψ in Ω , and

(4.1)  ⟨\frac{∂ū}{∂t}, φ⟩ + a(ū, φ) ≥ ∫_{Q} f̄(ū) φ dxdt + ⟨h, φ⟩ ,

for all φ ∈ V_{0} ∩ L_{+}^{p}(Q). The limits u ∈ W_{0} and v ∈ L^{q}(Q) satisfy (P_{0}), i.e., v ∈ f̂(T u) such that

(4.2)  ⟨\frac{∂u}{∂t}, φ⟩ + a(u, φ) + ∫_{Q} Bu φ dxdt = ∫_{Q} v φ dxdt + ⟨h, φ⟩ ,

for all φ ∈ V_{0}. Subtracting (4.1) from (4.2) we get for all φ ∈ V_{0} ∩ L_{+}^{p}(Q) the inequality

(4.3)  ⟨\frac{∂(u − ū)}{∂t}, φ⟩ + a(u, φ) − a(ū, φ) + ∫_{Q} Bu φ dxdt ≤ ∫_{Q}(v − f̄(ū)) φ dxdt .

Let w^{+} = max(w, 0) then (u − ū)^{+} ∈ V_{0} ∩ L_{+}^{p}(Q) is an admissible nonnegative testfunction for (4.3) satisfying (u − ū)^{+}(x, 0) = 0 in Ω, which implies (cf. [10]),

(4.4)  ⟨\frac{∂(u − ū)}{∂t}, (u − ū)^{+}⟩ ≥ 0 .

Thus from (4.3) we obtain with the special test function φ = (u − ū)^{+} and in view of (4.4)

(4.5) a(u, (u−ū)^{+})−a(ū, (u−ū)^{+})+∫_{Q} Bu(u−ū)^{+} dxdt ≤ ∫_{Q}(v−f̄(ū))(u−ū)^{+} dxdt .

Since v ∈ f̂(T u) = [\underline{f}(T u), f̄(T u)] we have for u > ū: T u = ū and v ∈ [\underline{f}(ū), f̄(ū)]. Hence the right-hand side of (4.5) yields

(4.6)  ∫_{Q}(v − f̄(ū)(u − ū)^{+} dxdt = ∫_{{u>ū}} (v − f̄(ū))(u − ū) dxdt ≤ 0 .

Further by hypothesis (A3) it follows

\begin{matrix}a(u, & (u − ū)^{+}) − a(ū, (u − ū)^{+}) \\ = ∫_{Q} ∑_{i=1}^{N}(a_{i}(x, t, ∇u) − a_{i}(x, t, ∇ū))\frac{∂(u − ū)^{+}}{∂x_{i}} dxdt \\ = ∫_{\{u>ū\}} ∑_{i=1}^{N}(a_{i}(x, t, ∇u) − a_{i}(x, t, ∇ū))\frac{∂(u − ū)}{∂x_{i}} dxdt ≥ 0 . \end{matrix} (4.7)

From (4.5) we finally obtain by means of (4.6) and (4.7)

(4.8)  ∫_{Q} Bu(u − ū)^{+} dxdt ≤ 0 .

By definition of the function b generating the Nemytskij operator B we get

\begin{aligned}∫_{Q} Bu (u − ū)^{+} dxdt & = ∫_{\{u>ū\}} b(x, t, u) (u − ū) dxdt \\ & = ∫_{\{u>ū\}} (u − ū)^{p−1}(u − ū) dxdt = ∫_{Q}((u − ū)^{+})^{p} dxdt ≥ 0 \end{aligned}

which in view of (4.8) results in

0 ≤ ∥(u − ū)^{+}∥_{L^{p}(Q)}^{p} ≤ 0 , i.e., (u − ū)^{+} = 0 ,

and hence u ≤ ū. This completes the proof of Theorem 2.1.

Remark 4.1. For problem (P) one can take into account also a more general operator A of the form

Au(x, t) = − ∑_{i=1}^{N} \frac{∂}{∂x_{i}} a_{i}(x, t, u(x, t), ∇u(x, t))

satisfying the usual Leray-Lions conditions. In this case, however, the operator A of the regularized auxiliary problem (P_{ε}) has to be modified appropriately. Furthermore, Neumann and Robin type boundary conditions can be taken into consideration.

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E-mail address: Internet: carl@ mathematik.uni-halle.d400.de