AN ABSTRACT EXISTENCE THEOREM FOR PARABOLIC SYSTEMS

. In this paper we prove an abstract existence theorem which can be applied to solve parabolic problems in a wide range of applications. It also applies to parabolic variational inequalities. The abstract theorem is based on a Gelfand triple ( V,H,V ∗ ), where the standard realization for parabolic systems of second order is ( W 1 , 2 (Ω) ,L 2 (Ω) ,W 1 , 2 (Ω) ∗ ). But also realizations to other problems are possible, for example, to fourth order systems. In all applications to boundary value problems the set M ⊂ V is an aﬃne subspace, whereas for variational inequalities the constraint M is a closed convex set. The proof is purely abstract and new. The corresponding compactness theorem is based on [5]. The present paper is suitable for lectures, since it relays on the corresponding abstract elliptic theory.

1. Introduction. In this paper we give an abstract existence proof for parabolic systems. The abstract theorem has been applied to many boundary value problems. Among other things it includes also cases in which the parabolic part is degenerated, therefore it contains elliptic-parabolic problems. It also includes the case of a convex constraint, therefore it contains variational inequalities. The proof for the combination of both effects is new, and has been presented by me in the lecture about partial differential equations in 2003.
There are two main reasons for this approach. One is mathematical, and consists of degenerate parabolic systems occurring in physical applications. The theory in this paper applies for example to boundary value problems of parabolic systems as shown in 11.1. Other applications one finds in [5], [6], [18], [4]. The proof is based on the estimates given in sections 7 and 8. The theory is more general than the parabolic existence theorems in [1], [12], [15], [19].
The other reason lies in theoretical physics and is the entropy principle as formulated in rational thermodynamics, see e.g. [20]. It implies that the estimate, which is the basic estimate of our approach, is equivalent with this entropy inequality. Thus the equations coming from physics are left for mathematical treatment in the original physical setting.
The method of this paper is worthwhile to make some comments. First of all it is a purely abstract formulation of the underlying variational inequality. In this formulation spaces with respect to the space variable are a Hilbert space H for the A different situation arises for a variational inequality. We assume that the inequality is given by the inequality u ≥ 0. Then the strong version of the Dirichlet problem, with Dirichlet data u 1 ≥ 0 and initial data u 0 ≥ 0, reads Now, in general one does not know that ∂ t u is a function, it is only defined as a distribution. However, then the term u∂ t u in the equation is not defined, but this term formally can be written as 1 2 ∂ t (u 2 ) and then can be integrated. One obtains the following weak version u ∈ L 2 ([0, T ]; M ) and This is of the form of the existence theorem, as we formulate it in 6.2, in the special case that b(u) = u. Therefore in this paper the goal is to prove the general existence theorem 6.2 in an abstract setting for a general closed and convex set M ⊂ V . The usual version for parabolic equations, that is M is a subspace, is a consequence of this general theorem and formulated in 6.3.
There is a large class of problems to which this existence theorem can be applied. Some are presented in section 11. Realistic elliptic-parabolic boundary value problems, which fall under the theorem in this paper, one finds in [11]. We mention, that this paper also contains vector valued versions of such variational inequalities. However, the large class of problems where u → b(u) has a jump, is not contained in this paper. However, the basic estimates in this paper generalize to such jump nonlinearities, so that the existence theorems can be used as an approximating step.
The constraint in this paper is a time independent set M ⊂ V . It often happens that the more general case of a time dependent constraint occurs. The proof is more involved, therefore not contained in this paper (see the argumentation in [13]).
There are different approaches to parabolic existence theory with a constraint. In particular the approach by [15], where a variational formulation is formulated and the elliptic part is of gradient structure. In this situation one can multiply by ∂ t u and obtains an estimate on the time derivative.
Note: The main part of this paper has been presented during my lecture in 2003. I hope that this theory, which builds on the theory of corresponding stationary problems, can therefore be used in future lectures on functional analysis or on partial differential equations.
2. Motivation. In the following we present some formal observations, which show that the type of system we consider is a consequence of general necessities. Consider a system of partial differential equations ∂ t v k + divq k = τ k for k = 1, . . . , N, (2.1) where, in order to complete the system, we have to specify terms and determine the independent variables. Independent of this we want to derive an estimate. Therefore let us multiply the k-th equation by a function λ k . Then summing over k we get This identity should behave like a parabolic one, that is, integrated over a domain ]t 0 , t 1 [×Ω it should be controlled by initial and boundary terms. Thus we require that an identity holds with a quantity w. If this is true, we obtain where q k = (q kj ) j=1,...,n . Writing τ k = r k + g k this gives where usually the q kj -term and the r k -term are dissipative terms and g k denotes an external term. Integrating this identity over ]t 0 , t 1 [×Ω we obtain The first term on the right side contains initial conditions and the second term boundary conditions, whereas the last one is the external term. Therefore the two terms on the left side are the essential ones. The positivity of the second integrand if postulated, means that the dissipative term has a sign. Assuming that our N equations are linearly independent we introduce independent variables u k , k = 1, . . . , N , and denote the vector u = (u k ) k . Let us for a moment assume that further q k = − l a kl (u)∇u, and for simplicity r k = 0. Then, if λ k = λ k (u), the dissipative term reads EXISTENCE THEOREM FOR PARABOLIC SYSTEMS   2083 which has the correct sign D ≥ 0, if the matrix k λ k um (u)a kl (u) ml is positive semidefinite. We have a freedom to choose the functions u k , they are not determined by the equations, and properties determined by the equations are independent of the choice of u k . One particular choice is λ k = u k , in which case Another possibility would be, to choose the functions v k as independent variable, which would not change the procedure of this paper. However, what we do not assume, is that u k and v k are the same, that is a very special case.
Thus what remains is to study the first term on the left side of (2.2), that is (2.5). Let us introduce the vector notation u = (u k ) k as above and Therefore, if the derivatives ∂ t v k are independent from each other, we derive as necessary condition On the other hand, if v = β(u), hence we introduce u as independent set of variables, which one can write as If the derivatives ∂ t u l are independent from each other, we derive as necessary condition By taking the derivative of this equation with respect to u m one obtains from which it follows that (β m u l (u)) m,l is a symmetric matrix, therefore If this is true for all u, one has for a certain function ψ v k = β k (u) = ψ u k (u) for all k. (2.8) We have seen, that (2.5) implies conditions (2.6) and (2.8), that is v k = ψ u k (u) and u k = ϕ v k (v) for all k, (2.9) which means, that u and v are dual variables to each other. As a consequence is a mapping with the same properties, that is, ψ * : H → IR ∪ {+∞} is not identical +∞, convex and lower semicontinuous. The map ψ * is called the conjugate convex function of ψ, or Fenchel transformation of ψ. The main inequality reads which is Young's inequality. In this inequality, for given z * , the equality holds for z, if the supremum of ψ * (z * ) in definition (3.1) is attained for this z. It also follows that (ψ * ) * = ψ. Simple cases are 3.1 Examples. Let H = IR.
(2) If the convex function ψ is given by is the corresponding conjugate function ψ * .
A subgradient z * ∈ H of ψ in z is defined by and the subdifferential ∂ψ(z) := {z * ∈ H ; z * is subgradient of ψ in z}.
and one of these statements implies, that ψ(z) < +∞ and ψ * (z * ) < +∞, and is equivalent to Proof. The statement z * ∈ ∂ψ(z) implies by (3.4), that (3.3) is satisfied, which implies ψ(z) < +∞, since ψ(z) is not infinite for allz ∈ H, that is, is finite for somē z ∈ H. Then (3.3) is equivalent to which is symmetric in (ψ, z) and (ψ * , z * ). We mention, that this inequality is the inverse Young inequality (3.2) and therefore really must be an equality.
The Weierstraß function E ψ is nonnegative, if the function ψ is convex. Then the graph of ψ lies above a plain.

Weierstraß
) is the usual E-function depending on two variables.
For z 0 ∈ H the translated function is ψ z0 (z) := ψ(z 0 + z). Assume that ψ is finite and continuously differentiable on D and ψ = ∞ in H \ D.
With this the Weierstraß E-function is defined for (z 1 , z 2 ) ∈ D × D by Proof. By 3.4(3) , if we take the definition.
Concerning b we will need for the next sections the following definition, which essentially is the conjugate function z * → ψ * z0 (z * ) for z * = b(z).
Proof. Using 3.4(3) and 3.4(1) one obtains the identities for B z0 (z). By the convexity of ψ, or the monotonicity of b, or the nonnegativity of E ψ , the terms in the definition are nonnegative.
Proof. By the previous definition 3.6 using the formula for E ψ in (3.6).
Whereas ψ * z0 for the example in 3.1 with p < ∞ grows at infinity of order p * , in general one has only the following lemma.
3.8 Superlinearity of ψ * z0 . Let ψ : H → H be convex and lower semicontinuous, and bounded on bounded subsets of H. Then for δ > 0 and z 0 ∈ H there exists a constant C δ,z0 so that Proof. We compute This implies, if ψ and D = H as in 3.4, and b as in 3.4(1) and B as in 3.6, that there is a constant C δ,z0 := C δ,z0 − δψ z0 (0) with b(z) H ≤ δB z0 (z) + C δ,z0 for all z ∈ H and δ > 0. (3.7) 4. Elliptic theorem. The parabolic existence proof is based on the following elliptic theorem, which is formulated on a closed, convex set M of a Banach space V , M ⊂ V nonempty, closed, and convex.
is given, where V * is the dual space of V . By (w, w * ) → w , w * V := w * (w) for w ∈ V and w * ∈ V * we denote the dual product of V . The main assumption for the elliptic existence theorem 4.2 is the 4.1 Continuity condition. The following holds: Let u m , u ∈ V and v * ∈ V * with With this the following theorem is satisfied.

Theorem.
Let V be a separable reflexive Banach space and M ⊂ V as in (4.1), and let F : M → V * with the following properties: (1) Boundedness. The map F is bounded on bounded subsets of M .
Under these assumptions there exists u ∈ M , so that Note: The condition, thatū belongs to M , in general is necessary for the theorem.
We call (4.3) the variational inequality for F with respect to M .
There are many examples, which fall under this theorem, the standard ones are monotone operators and compact perturbations of monotone operators. The condition 4.2(2), that is 4.1, is usually connected with the name "pseudomonotone operators", although the definition of pseudomonotone is a little bit different, but it is equivalent under the complete assumptions of 4.2.

5.
Time discrete problem. We consider a Hilbert space H and a Banach space V as in the introduction, that is (1.2), is satisfied. For more general V we refer to 13.2. The approximative problem is given for discrete times t i with t i < t i+1 . For simplicity we consider the case of a constant time step h > 0, that is, The constraint is approximated by where here the set M i may change in time. On M i an "elliptic" operator is given by 4) where this map is defined recursively, that is, it may depend on the solution for smaller i. The "parabolic" part is given by a map b = ∇ψ with ψ : H → IR convex and continuously differentiable. (5.5) We approximate the parabolic problem by a time discrete version, that is we replace the time derivative of b(u) by time differences Under assumptions (5.1)-(5.5) the problem is to find inductively in i a solution with given starting value u 0 ∈ H.
Here u i−1 , for i = 1, is the initial value u 0 := u 0 ∈ H, and for i ≥ 2, is the known vector from previous time step. Thus the solution u i ∈ M i is constructed inductively in i, therefore the operator A i may contain also information from the previous time steps, e.g. it may depend on u i−1 .
For the existence proof we deal with certain assumptions about the operator among them the continuity condition 4.1. We will show that the "parabolic" part b = ∇ψ by assumption gives the necessary property, provided the embedding from V into H is compact. As consequence only the map A i : M i → V * has to satisfy 4.1. The map F i satisfies the following

Lemma. Let the inclusion Id
Proof. We assume that 4.1 is satisfied for A i , and we have to show that condition Since it is assumed that 4.1 for the map A i is satisfied, we conclude Inserting the definition of v * one gets Therefore it has been shown that condition 4.1 for the map F i is fulfilled.
Similar one can show, that property 4.1 for F i implies this property for A i . It is also enough to assume the boundedness condition for A i . This is because the following lemma holds.

Lemma. Let Id
Proof. Assume this is not true. Then there is an We can now formulate the theorem for time discrete solutions.
Under these assumptions it follows, that for given u 0 ∈ H and for h ≤ 1 λ the time discrete problem 5.1 has a solution.
Proof. Let u 0 be as in 5.1 and i ≥ 1. Because of remark 5.2 the problem in 5.1 can be formulated as Property (1) implies 4.2(1) for F i h by using 5.4. Since (2) is satisfied, the statement 5.3 shows that 4.2(2) is valid for the map F i h . Since ψ is convex, the first term on the right side of Consequently there is a solution of the variational inequality (5.9).
Alternatively, the time discrete solution in 5.1 can be formulated as in 5.6. For this we construct for each sequence u j ∈ H with j ∈ IN ∪ {0} a step function in time by (5.10)

Similar one defines the elliptic operator
With these definitions equation 5.1 becomes By a step function we mean a function as in (5.10).
6. The main theorem. In the following we describe the main theorem of this paper. Given a Hilbert space H and a Banach space V such that (V, H, V * ) is a Gelfand triple satisfying V → H → V * . As pointed out in section 13, we can work with a special case and can assume that Besides these spaces we consider a set with a time independent constraint M ⊂ V nonempty, closed, and convex.
We assume that the "parabolic part" of our problem is given by a map b : H → H monotone and continuous, in fact b = ∇ψ, ψ : H → IR convex and continuously differentiable, (6.4) where the functional B 0 in the above definition is given by see the definition in 3.6. On M we denote the "elliptic part" of the problem by We shall present some illustrating examples in section 11, in particular 11.1. According to the continuity condition in 4.1 we assume for the parabolic problem in this section, that a time version of this condition is satisfied.
With this assumption we can prove the following existence theorem, where the structure of the theorem is the same as in 5.5. We mention, that in concrete cases the proof that A maps into L p * ([0, T ]; V * ) usually immediately gives the boundedness condition 6.2(1). (2) Continuity condition. A satisfies the condition 6.1.

Existence theorem. Let
, and c 0 > 0 and C 0 are constants.
Then there exist solutions of the "evolution problem", that is for given Proof. The parabolic terms in the solution property in (6.6) we denote as see definition 8.1. It is used in the proof the theorem in section 10.
In the special case that the set M ⊂ V is an affine space, that is the constraints are defined by equations only, the existence theorem has the following form. Then there exist solutions of the "evolution equation", that is if The proof of this statement uses the general existence theorem.
Proof. In 6.2 we have proved the inequality, using the notation in (6.7), Here M now is an affine subspace contained in V . It follows that this inequality then also holds for all

HANS WILHELM ALT
We also get This is obviously equivalent to the assertion. The left side is a linear form in ξ. Now we can replace ξ by −ξ to obtain that the left side equals zero. If we now , that is with compact support in [0, T [, then ξ vanishes in a neighbourhood of T , and therefore one choosest close to T in order to get For sets, which are not constant in time, additional terms will occur in the following lemmata. To be precise, let us consider solutions These elements can be given by different circumstances such as the time discrete solution with w * h (t) = A h (t, u h (t)). We assume that the quantities in (7.2) are step functions in time, that is in the following computations (5.10) is assumed, which means For the convergence of the time discrete solutions we have to show estimates which are independent of h. The first basic estimate is the 7.1 Energy estimate. Let u h and w * h as in (7.2). Then we conclude, ifū is a step function, and ift is a multiple of h, If the functionū is constant, that isū ∈ M , and if we neglect the last part on the left side, E ψ ≥ 0, we obtain the standard version of the estimate Proof. With (7.3) the inequality (7.2) reads Here we have used the notation u * i := b(u i ). It holds where for the inequality we also can write by taking the identity for ψ * (u * i−1 ) into account (see section 3). Therefore one obtains

HANS WILHELM ALT
Using that the first term is a telescope sum, and using the definition in 3.6, one gets that this is Here in the term Bū0(u 0 ) and in the term 1 h (ū i −ū i−1 ) for i = 1 the functionū 0 occurs. Now rewriting terms as step functions in time, one gets the result.
The second estimate is the following 7.2 Compactness in time. Let (7.1) be satisfied, and let u h and w * h as in (7.2) as well as u * h (t) = b(u h (t)). Then for t an s being a multiple of h, s = jh, we infer Proof. It is assumed that t and s are multiple of h, say, As in the previous proof we write for t i = ih, i ∈ IN, problem (7.2) as for v ∈ M , where again we use the notation u * i := b(u i ). Now choose k ∈ IN and set v = u k , and sum over i = k + 1, . . . , k + j. The result is Now by the identity 3.4(3) and Young's inequality (3.2), see (7.5), we compute for the left side with E ψ * defined in 3.3 and since u k ∈ ∂ψ * (u * k ), a consequence of u * k ∈ ∂ψ(u k ), see 3.2. Thus we have shown We rewrite this as This proof works for a general convex set M . If M is a subspace one obtains a slightly better estimate.
Proof. As in the previous proof we know that (7.7) is satisfied, but now for an affine subspace M , so that for v ∈ M 1 , a subspace for which M =ū 1 + M 1 withū 1 ∈ M . Now again choose k ∈ IN, and sum over i = k + 1, and write the result in terms of functions in time, that is t = kh, see (7.3).

Parabolic identity.
In all theories about parabolic problems there is one equation, which plays an exceptional role, and it has to be proved for the continuous limit problem. For some parabolic problems it is connected to an inequality, which is postulated for the formulation of a solution. In this paper it is connected to the following. The definition can also be written as

Definition. Let
We mention, that only Bū(u) ∈ L ∞ ([0, T ]) is assumed. The fact that u 0 is a "initial value" for u, is only determined by this definition. The term Φū(u, v) is the parabolic term in the differential inequality (6.6). That this coincides with the parabolic term ∂ t b(u) in the differential equation, is shown formally in the following lemma. The argumentation is essentially the same as the proof in the time discrete case in 10.3.

Proof. We write
and if one considers the limit δ → 0 after dividing by δ, one obtains formally Therefore the first term in the above identity is

Add both terms in order to get Φū(u, v)(t).
This statement indicates, that formally Φū(u, u) = 0. In a rigorous way this will be proved in the following lemma.
is the backward differential quotient. Now by definition 8.1

HANS WILHELM ALT
We see that with discrete partial integration, defining u(t) := u 0 for t < 0 and therefore b(u(t)) = b(u 0 ) for t < 0, Here we have used thatt − δ > 0, and that u δ (t) = u(t) for t > 0. Now, let us have a look at each of the three terms in the above identity. In the first term, since t → b(u(t)) is in L ∞ ([0, T ]; H) by assumption and (3.7), the integrand t → (ū − u(t) , b(u(t)) − b(u 0 ) ) H is integrable. Therefore this term converges for almost allt to (ū − u(t) , b(u(t)) − b(u 0 ) ) H , a term which occurs also in the formula for Φū(u, u δ )(t), hence this term cancels.
The last term is, again since t → b(u(t)) is in L ∞ ([0, T ]; H), Concerning the second term we use the inequality 3.7, that is for all t, s (Bū(u(t + δ)) − Bū(u(t))) = ∂ +δ t Bū(u(t)), and the second term becomes Since t → Bū(u(t)) is integrable, this converges for a subsequence δ → 0 (a subsequence of an a-priori given sequence δ → 0, see the remark at the end of this proof) for almost allt to Bū(u(t)) − Bū(u 0 ), a term which occurs in the formula for Φū(u, u δ )(t).
Remark: We mention that for the sequence δ → 0 one has to apply a certain trick. First one chooses a subsequence so that the limit with respect to this subsequence is the limes inferior in the assertion. With this subsequence one has to go into the above proof with the choice of a subsequence.
9. Compactness theorem. The main statement of this section is the compactness result in 9.3, whose proof is based on the corresponding result in [5]. Before we show this we present a useful lemma. The lemma applies to our function ω(s) = s. By the way, the property ω(0) = 0 is assumed in 9.3.
Proof. Let s be arbitrary, 0 < s <t, and choose j ∈ IN with by (9.1), since ω is concave.
The following statement we will apply in the main proof.
Let such numbers R > 0 and ε > 0 be given. Hence for small δ > 0 there are The boundedness in V and the compactness of the embedding V → H imply that there are u 1 , u 2 ∈ H with u 1 δ → u 1 and u 2 δ → u 2 in H for a subsequence δ → 0. Since b is continuous it follows for this subsequence that b(u 1 Similarly since ψ is continuous we obtain ψ(u 1 δ ) → ψ(u 1 ) and ψ(u 2 δ ) → ψ(u 2 ) in IR. Hence by 3.5 Besides this we compute for every v ∈ H by inserting the identity (9.5). We get since ψ is differentiable. We conclude b(u 1 ) = b(u 2 ) and therefore by (9.4) With this we are able to prove the main result.

Compactness result. Let V be a separable reflexive Banach space and H a Hilbert space with compact embedding Id
as in (6.4) (see also (5.5)). For C > 0 let Here ω is continuous with ω(0) = 0. Then Proof part 1. We prove that For this define where we assume that ω(s) = 0. Then for u ∈ K C t1−s t0 Considering the integrand on [t 0 , t 1 − s] \ T s R (u) we see that therefore the Lebesgue measure of the set [t 0 , t 1 − s] \ T s R (u) is estimated by a constant depending on R alone. By (3.7) we compute for u ∈ K C b(u(t)) H ≤ C 1 := C + C 1,0 for almost all t ∈ [t 0 , t 1 ]. (9.8) Then by 9.2 and (9.8) , and integrating this gives The right side is independent of u. First we choose R large enough, so that the second term on the right becomes small, and then s small, so that the first term is small. It follows that the integral is small uniformly in u ∈ K C .
Proof part 2. We prove that both ω R and ω are assumed to be concave and monotone, and of course ω R (0) = 0 and ω(0) = 0. The estimate is true for every time interval [0, δ] with This gives the result.
Proof main part. We choose a time step δ > 0 as in (9.9) and approximate each function by a step function as follows. We define for v ∈ L p ([t 0 , t 1 ]; H) and s ∈ [0, t 0 − t 1 ] and we approximate b(u) by β(s, u δ R ) for u ∈ K C and a suitable s which we choose later. Here where it is assumed thatū ∈ K C . We compute and by (9.8) and (9.7) Integrating over the offset s we obtain if R is large and then δ is small. We then can choose an s = s u such that for u ∈ K C t1 t0 b(u(t)) − β(s u , u δ R )(t) H dt ≤ 2 · ε. (9.11) It follows that the functions b(u) lie in an 2ε-neighbourhood of the step functions β(s u , u δ R ), ; H) ; u ∈ K C , R large, δ small} in the topology with respect to L 1 ([t 0 , t 1 ]; H).
Proof last part. From the previous proof it follows, that the precompactness of follows from the precompactness of ; H), since the first set is contained in an 2ε-neigbourhood of the second set, ε an arbitrary small number. The second set depends on ε, which is allowed. Indeed this is true, since R was chosen large enough and δ small enough, both depending on ε.
Therefore the precompactness of the second set in L 1 ([t 0 , t 1 ]; H) has to be shown. Since these are step functions, we have to show the precompactness of the steps in H, that is the precompactness of {b(u δ R (t 0 + (i − 1)δ + s u )) ; u ∈ K C , i ≤ k, R large, δ small} ⊂ H. We show instead the precompactness of the larger set {b(u) ; u ∈ V, u V ≤ R} ⊂ H for large R. But this follows from the compactness of the embedding V → H. Then bounded sets in V are precompact in H, and the continuous function b transforms this to a precompact set in H.
The compactness of the functions b(u h ) in L 1 ([0, T ]; H) implies, that for a sequence h → 0 these functions have a strong limit b * in L 1 ([0, T ]; H). Then, if the functions u h already have a weak limit u, one can apply the following lemma, whose proof is classical. It shows that the limits satisfy b * = b(u). Note, that this is true, even if b is not strictly increasing, however it must be monotone.
We want to show, that R(u, b * ) = 0, that is b(u) = b * . Now it follows from the assumption, that for a subsequence b(u m (t)) → b * (t) strongly in H for almost all t as m → ∞. Therefore (for this subsequence) To continue, we have to use the standard monotonicity argument for b, which implies and therefore 0 ≤ ( v(t) − u m (t) , R(v, b(u m ))(t) ) H for almost all t. It follows that for v ∈ L p ([0, T ]; H). Now to the convergence of the second term. Since u m → u weakly in L p ([0, T ]; V ), which is continuously embedded into L p ([0, T ]; H), and since Altogether we conclude We apply now a Minty type argument, that is we replace v by u + ε(v − u) and letting ε 0, to obtain , and therefore also which finally implies R(u, b * ) = 0 almost everywhere.
10. Convergence proof. In the following we give a convergence proof of the main theorem of this paper, formulated in 6.2. We consider the case of a time independent constraint M ⊂ V , and the boundedness condition 6.2(1) and the continuity condition in 6.1 are satisfied. Besides this we assume the coerciveness in 6.2(3). First we show that we have approximative solutions of the time discrete problem. We define the time discrete operator to the map (s, z) → A(s, z) from (6.5) by The initial data are u 0 ∈ H and b is given by (6.4). With this we show 10.1. There are solutions u h , which are step function with u h (t) = u 0 for t < 0 and u h (t) ∈ M for t > 0, of the time discrete problem for v ∈ M . This is true for t > 0 which are multiple of h, and then also for all t > 0.
With We want to prove 5.5 (1). Let S be a bounded set in M i . Define Hence on S i also the L ∞ ([0, T ])-norm of B 0 is bounded. It follows from 6.2(1), that A on S i is bounded in L p * ([0, T ]; V * ), say, Therefore A i is bounded on S, which shows 5.5 (1). We want to prove 5.5(2) and we know 6.2 (2), that is the continuity condition 6.1. Let a sequence be given for A i as in 5.5 (2), which is stated in 4.1, that is We have to show, that the conclusions in 4.1 are true. Define and u i to u as u im to u m . Then for m → ∞ it converges u im → u i weakly in Since the embedding V → H is compact, u m converges to u strongly in H.
Plugging in the definitions for u im and u i , the last identity becomes Thus the conclusions of the continuity condition 5.5(2), see 4.1, is satisfied. Therefore the continuity condition 5.5(2) is fulfilled.
It remains to show 5.5 (3). From the coerciveness 6.2(3) for all t > 0 and u ∈ M one gets, sinceū does not depend on time, the same estimate for the operator A h , Since by 3.7 This shows coercivity 5.5(3) withū i =ū. Since all assumptions are fulfilled 5.5 is applicable.

.
We prove: There is u ∈ L p ([0, T ]; V ) with u(t) ∈ M for almost all t, such that for a subsequence h → 0 the following convergence holds: Here the approximative functions u h are defined only in [0, T h ], where T h is the largest multiple of h less or equal T . It is irrelevant, how u h is defined in ]T h , T ], the main thing is that it stays bounded, for example we define it byū. Then the statements hold on the interval [0, T ].
We take a solution of 10.1, set w * h (t) := A h (t, u h (t)) in (7.2), and use the a-priori estimate 7.1. We obtain The coerciveness (10.4) leads to

This implies
Since G 0 ∈ L 1 ([0, T ]) the last term on the right is bounded uniformly in h. Then a Gronwall argumentation on the inequality (10.6) gives the boundedness of the sets Using this one gets from (10.6) the "parabolic" estimate ess sup where C depends only on u 0 ,ū, G 0 , Bū, c 0 , T , and obvious quantities like p, n, V , H. In particular, C is independent of h. Hence for a subsequence h → 0 there exists the weak limit u h → u in L p ([0, T ]; V ). Since u h (t) ∈ M for almost all t, it follows that u(t) ∈ M for almost all t. Equation (10.7) says that (define for example u h (t) :=ū for T h < t < T ) the set {u h ; 0 < h < h 0 } satisfies the required boundedness assumption in 6.2(1). It follows by the boundedness condition 6.2(1) that the set {A(u h ) ; 0 < h < h 0 } is bounded in L p * ([0, T ]; V * ). Therefore there is a subsequence h → 0 so (all subsequent subsequences have to be chosen as subsequence of the previous subsequence), that the weak What is missing is the strong convergence of b(u h ) as h → 0. To derive this we go into the second estimate in section 7, that is 7.2, This gives . By the estimates proved so far, that is the estimates for u h and w * h = A(u h ) the right side is bounded by a constant times s. Thus we obtain This estimate is fulfilled for all s > 0, not only for multiple of h, see lemma 9.1. Then the compactness theorem 9.3 implies that {b(u h ) ; 0 < h < h 0 } is precompact in L 1 ([0, T ]; H). Hence there is a subsequence h → 0 so (all subsequent subsequences have to be chosen as subsequence of the previous subsequence), that b(u h ) strongly in L 1 ([0, T ]; H) to a limit b * . But then by 9. For the sequence u h we have the following time discrete inequality and u h (t) := u 0 for t < 0. We now define for a givent a t =t h as a multiple of h 11. Examples. In the following we present some concrete examples. First there are second order boundary value problems, where Ω ⊂ IR n is a bounded Lipschitz domain and H = L 2 (Ω; IR N ) and V = W 1,2 (Ω; IR N ).
11.1 Second order problem. We let Ω ⊂ IR n as above and take a closed (may be empty) set Γ ⊂ ∂Ω. On the time interval [0, T ] we consider the elliptic-parabolic boundary value problem Here functions u 1 ∈ W 1,2 (Ω) and u 0 ∈ L 2 (Ω) and a right side f ∈ L 2 ([0, T ] × Ω) are given. Moreover β : Ω × IR → IR and a : Ω × IR × IR n → IR n are Carathéodory functions, that is measurable in the first argument and continuous in the other arguments. We assume the following monotonicity and growth conditions (1) Then, with a correct choice of M, A, and b, this example is of the general type. The condition 6.2(1) of theorem 6.3 is satisfied.
The fact, that one proves that A maps into L 2 ([0, T ]; V * ), usually gives the boundedness condition 6.2(1) as a byproduct. We mention, that also the case, that a(x, u(t, x), ∇u(t, x)) has a controlled unbounded term in u(t, x), can be treated. Similar arguments apply to a right side f (t, x, u(t, x)).
Also the case of systems of elliptic equations is covered, that is the case N > 1. Then different components of this system may satisfy different boundary conditions.
The problem where u − = u + and β − < β + is not contained in this theorem, although the theory is capable to treat this case as a limit u + − u − 0, but we do not discuss this here.
The standard case of parabolic equations is, that the solutions are continuous in time. This is not the case here, and it has again to do with the elliptic-parabolic character of the problem, see [5,Introduction]. where Ω = B R (0) is a ball, u 1 < 0 continuous, and β(u) = max(u, 0). Then the following holds: