Maria do Rosário GROSSINHO †

We prove the solvability of the parabolic problem 8 > i=1 bi(x; t)@xi u=f(x; t; u) in IR; u(x; t) = 0 on @ IR; u(x; t) = u(x; t + T ) in IR; assuming certain conditions on the asymptotic behaviour of the ratio 2 R s 0 f(x; t; )d =s 2 with respect to the principal eigenvalue of the as- sociated linear problem. The method of proof, which is based on the construction of upper and lower solutions, also yields information on the localization and the stability of the solution.


Introduction and statements
Let Ω (⊂ IR N ) be a bounded domain, with a boundary ∂Ω of class C 2 , and let T > 0 be a fixed number.Set Q = Ω×]0, T [ and Σ = ∂Ω × [0, T ].Let us consider the parabolic problem in Ω. (1.1) We assume throughout that where Q) for i, j, k = 1, . . ., N .We also suppose that the operator ∂ t + A is uniformly parabolic, i.e. there exists a constant η > 0 such that, for all (x, t) ∈ Q and ξ ∈ IR N , N i,j=1 a ij (x, t)ξ i ξ j ≥ η|ξ| 2 .
We further assume that f : Ω×]0, T [×IR → IR satisfies the L p −Carathéodory conditions, for some p > N + 2, and there exist continuous functions g ± : IR → IR such that, for a.e.(x, t) ∈ Q f (x, t, s) ≤ g + (s) for s ≥ 0 and f (x, t, s) ≥ g − (s) for s ≤ 0. (1. 2) It is convenient, for the sequel, to suppose that all functions, which are defined on Ω×]0, T [, have been extended by T -periodicity on Ω × IR.
In this paper we are concerned with the solvability of (1.1) when the nonlinearity f lies in some sense to the left of the principal eigenvalue λ 1 of the linear problem EJQTDE, 1999 No. 9, p. 2 It was proven in [2] that the Dolph-type condition lim sup s→±∞ g ± (s) s < λ 1 (1.3) guarantees the existence of a solution of (1.1).On the other hand, it does not seem yet known whether the same conclusion holds under the more general Hammerstein-type condition lim sup s→±∞ 2G ± (s) where G ± (s) = s 0 g ± (σ)dσ for s ∈ IR.Our purpose here is to provide some partial answers to this question.Of course, the main difficulty, in order to use in this context conditions on the potential like (1.4), is due to the lack of variational structure of problem (1.1); whereas the only known proof of Hammerstein's result, for a selfadjoint elliptic problem in dimension N ≥ 2, relies on the use of variational methods.Accordingly, we will employ a technique based on the construction of upper and lower solutions, which will be obtained as solutions of some related, possibly one-dimensional, problems.We stress that an important feature of the upper and lower solution method is that it also provides information about the localization and, to a certain extent, about the stability of the solutions.Yet, since we impose here rather weak regularity conditions on the coefficients of the operator A and on the domain Ω and we require no regularity at all on the function f , the classical results in [11], [1], [3], [10] do not apply.Therefore, we will use the following theorem recently proved in [4,Theorem 4.5].Before stating it, we recall that a lower solution α of (1.1) is a function α Similarly, an upper solution β of (1.1) is defined by reversing all the above inequalities.A solution of (1.1) is a function u which is simultaneously a lower and an upper solution.
Lemma 1.1 Assume that α is a lower solution and β is an upper solution of (1.1), satisfying α ≤ β in Q.Then, there exist a minimum solution v EJQTDE, 1999 No. 9, p. 3 and a maximum solution w of (1.1), with in Ω (1.5) Remark 1.1 We will say in the sequel that v (resp.w) is relatively attractive from below (resp.from above).Of course, this weak form of stability can be considerably strenghthened provided that more regularity is assumed in (1.1) (cf.[3]).

Remark 1.2
The condition α(•, 0) = 0 on ∂Ω is not restrictive.Indeed, if it is not satisfied, we can replace α by the unique solution α, where k ρ is the function associated to f by Lemma 3.3 in [4] and corresponding to ρ = max{|α| ∞ , |β| ∞ }.A similar observation holds for β.
We start noting that Hammerstein's result can be easily extended to a special class of parabolic equations, which includes the heat equation.
Theorem 1.1 Assume that b i = 0, for i = 1, . . ., N , and suppose that there exist constants c and q, with c > 0 and Moreover, assume that condition (1.4) holds.Then, problem (1.1) has a solution v and a solution w, satisfying v ≤ w, such that v is relatively attractive from below and w is relatively attractive from above.
EJQTDE, 1999 No. 9, p. 4 We stress that this theorem completes, for what concerns the stability information, the classical result of Hammerstein for the selfadjoint elliptic problem As already pointed out, we do not know whether a statement similar to Theorem 1.1 holds for a general parabolic operator as that considered in (1.1).The next two results provide some contributions in this direction, although they do not give a complete answer to the posed question.In order to state the former, we need to settle some notation.For each i = 1, . . ., N , denote by ]A i , B i [ the projection of Ω onto the x i -axis and set Then, define Then, the same conclusions of Theorem 1.1 hold.
The constant λ1 depends only on the coefficients of the operator A and on the domain Ω.It is strictly positive and generally smaller than the principal eigenvalue λ 1 ; therefore, it provides an explicitly computable lower estimate for λ 1 .Moreover, λ1 coincides with λ 1 when N = 1, a 11 = 1 and b 1 = 0, so that the equation in (1.1) is the one-dimensional heat equation.On the other hand, it must be stressed that the restriction from above on a limit superior required by (1.4) is replaced in (1.7) by a restriction from above on a limit inferior.Furthermore, in Theorem 1.2 the growth condition (1.6) is not needed anymore.We recall that conditions similar to (1.7) were first EJQTDE, 1999 No. 9, p. 5 introduced in [6] for solving the one-dimensional two-point boundary value problem and were later used in [8] for studying the higher dimensional elliptic problem It is worth noticing at this point that, if the coefficients of the operator A and the function f do not depend on t, then the same proof of Theorem 1.2 yields the solvability, under (1.7), of the, possibly non-selfadjoint, elliptic problem This observation provides an extension of the result in [8] to the more general problem (1.9), which could not be directly handled by the approach introduced in that paper.A preliminary version of Theorem 1.2 was announced in [9].
In our last result we show that the constant λ1 considered in Theorem 1.2 can be replaced by the principal eigenvalue λ 1 , provided that a further control on the functions g ± is assumed.
Then, the same conclusions of Theorem 1.1 hold.
We point out that the sole condition (1.10), which is a weakened form of (1.3), is not sufficient to yield the solvability of (1.1) (cf.[2]).Theorem 1.3 extends to the parabolic setting a previous result obtained in [5] for the selfadjoint elliptic problem (1.8).By the same proof one also obtains the solvability, under (1.10) and (1.11), of the, possibly non-selfadjoint, elliptic problem (1.9).We stress that, although the proof of Theorem 1.3 exploits some ideas borrowed from [5], nevertheless from the technical point of view it is much more delicate, due to the different regularity that solutions of (1.1) exhibit with respect to the space and the time variables.

Preliminaries
In this subsection we state some results concerning the linear problem associated to (1.1), which apparently are not well-settled in the literature, when low regularity conditions are assumed on the coefficients of the operator A and on the domain Ω.
where η is the constant of uniform parabolicity of the operator ∂ t + A. Then, Proposition 2.1 in [4] guarantees that, for every f ∈ L p (Q), the problem in Ω. ( has a unique solution v ∈ W 2,1 p (Q) and, therefore, v ∈ C 1+µ,µ (Q), for some µ > 0. Let f ∈ L p (Q) be given and let v be the corresponding solution of (2.3).Set β = v + sϕ 1 , where s > 0 is such that β ≥ 0 and s(λ 1 − ess sup Q q)ϕ 1 ≥ kv.We have that is β is an upper solution of (2.1).In a quite similar way we define a lower solution α of (2.1), with α ≤ 0. Therefore Lemma 1.1 yields the existence of a solution u ∈ W 2,1 p (Q) of problem (2.1), with α ≤ u ≤ β.The uniqueness of the solution is a direct consequence of the parabolic maximum principle (see e.g.[4, Proposition 2.2]) and its asymptotic stability follows from [4,Theorem 4.6].Accordingly, the operator and the open mapping theorem implies that its inverse is continuous, that is, (2.2) holds.Finally, the last statement follows from the parabolic strong maximum principle, as soon as one observes that if f ≥ 0 a.e. in Q, then α = 0 is a lower solution of (2.1).EJQTDE, 1999 No. 9, p. 9 Proposition 2.3 For i = 1, 2, let q i ∈ L ∞ (Q) be such that q 1 ≤ q 2 a.e. in Q and let u i be nontrivial solutions of in Ω, respectively.If u 2 ≥ 0, then q 1 = q 2 a.e. in Q and there exists a constant c ∈ IR such that u 1 = c u 2 .
Proof.We can assume, without loss of generality, that u + 1 = 0. Since and hence either v > > 0, or v = 0.The minimality of c actually yields v = 0 and therefore u 1 = cu 2 .This finally implies and therefore q 1 = q 2 a.e. in Q.

Proof of Theorem 1.2
Again we show how to construct an upper solution β of (1.1), with β ≥ 0; a lower solution α, with α ≤ 0, being obtained similarly.Exactly as in the proof of Theorem 1.1, we can reduce ourselves to the case where g + (s) > 0 for s ≥ 0.Then, we define a function h as in (2.4).The remainder of the proof is divided in two steps: in the former, we study some simple properties of the solutions of a second order ordinary differential equation related to problem (1.1); in the latter, we use the facts established in the previous step for constructing an upper solution of the original parabolic problem.
Step 1.Let A < B be given constants and let p, q : [A, B] → IR be functions, with p absolutely continuous and q continuous, satisfying 0 < p 0 := min and 0 < q 0 := min

.7)
Let also h : IR → IR be a continuous function and set H(s) = s 0 h(σ) dσ for s ∈ IR.Consider the initial value problem (2.10) Then, there exists a solution u of (2.8), which is defined on [A, B] and satisfies Proof of the Claim.Let u be a maximal solution of (2.8).Note that, by (2.6), (2.7) and (2.9), u has a local maximum at the point A+B We want to prove that ω − = A and ω + = B. Assume, by contradiction, that EJQTDE, 1999 No. 9, p. 12 By (2.6) and (2.11), we have, for each and hence −u (x) Integrating this relation between A+B and ω + and changing variable, we get by (2.12) Then, condition (2.10) yields a contradiction and the conclusions of the Claim follow.

Proof of Theorem 1.3
Again we describe how to build an upper solution β of (1.1), with β ≥ 0; a lower solution α, with α ≤ 0, being constructed in a similar way.As in the proof of Theorem 1.1, we can reduce ourselves to the case where g + (s) > 0 for s ≥ 0.Then, we define a function h as in (2.4) Let us consider the problem in Ω (2.21) EJQTDE, 1999 No. 9, p. 15 and let us prove that it admits at least one solution.Observe that, since h(s) > 0 for every s, any solution u of (2.21) is such that u > > 0 and then, by condition (1.2), is an upper solution of (1.1).Fix p > N + 2 and associate to (2.21) the solution operator S : It follows from Proposition 2.2 that S is completely continuous and its fixed points are precisely the solutions of (2.21).Let us consider the equation with µ ∈ [0, 1], which corresponds to in Ω. Claim 1.Let (u n ) n be a sequence of solutions of Then, possibly passing to subsequences, where v = c ϕ 1 , for some c > 0, and EJQTDE, 1999 No. 9, p. 16 Proof of Claim 1.Let us write, for s ∈ IR, h(s) = q(s) s + r(s), with q, r continuous functions such that in Ω. ( The sequence (v n ) n is bounded in W 2,1 p (Q) and therefore, possibly passing to a subsequence, it converges weakly in W 2,1 p (Q) and strongly in C 1+α,α (Q), for some α > 0, to a function v ∈ W 2,1 p (Q), with |v| ∞ = 1.We can also suppose that µ n → µ 0 ∈ [0, 1] and q(u n ) converges in L ∞ (Q), with respect to the weak* topology, to a function q 0 ∈ L ∞ (Q), satisfying by (2.25) a.e. in Q.Moreover, by (2.26), we have uniformly a.e. in Q.The weak continuity of the operator in Ω.
We also have Claim 2. There exists a sequence (S n ) n , with S n → +∞, such that, if u is a solution of (2.23), for some µ ∈ [0, 1], then max Q u = S n , for every n.
Proof of Claim 2. By (2.20), we can find a sequence (s n ) n , with s n → +∞, and a constant ε > 0, such that for every n.Assume, by contradiction, that there exist a subsequence of (s n ) n , which we still denote by (s n ) n , and a sequence (u n ) n of solutions of (2.24) such that max , and therefore in C 1,0 (Q), with v = cϕ 1 , for some c > 0, and There is also a constant K > 0 such that for every n.Moreover, we have EJQTDE, 1999 No. 9, p. 18 and, possibly for a subsequence, x n → x 0 and t n → t 0 , with (x 0 , t 0 ) ∈ Ω × [0, T ], because (x 0 , t 0 ) is a maximum point of v. Using Fubini's theorem and possibly passing to subsequences, we also obtain from (2.32) and (2.35), respectively, in L p (Ω), for a.e.t ∈ [0, T ], and in L p (0, T ), for a.e.x ∈ Ω.Moreover, we have that for a.e.x ∈ Ω.Let us write where the choices of points t ∈ [0, T ], such that (2.36) and (2.37) hold, x ∈ Ω, such that (2.38), (2.39) and (2.40) hold, and x * ∈ ∂Ω will be specified later.Let us observe that, for each n, we can find a sequence (w Hence, using Fubini's theorem and possibly passing to a subsequence, we get EJQTDE, 1999 No. 9, p. 19 ) in L p (0, T ) for a.e.x ∈ Ω.Hence, it follows that, for each n, in L p (0, T ), for a.e.x ∈ Ω, and therefore, for a.e.t ∈ [0, T ], Moreover, for each n, we have H(u n (•, t)) ∈ C 1 (Ω) for every t ∈ [0, T ] and hence, by (2.33) and (2.34), we obtain, for every x ∈ Ω, where σ n is a path, joining x n to x and having range contained in Ω, and (σ n ) denotes its length.Because x n → x 0 , with x n , x 0 ∈ Ω, and x can be chosen in a dense subset of Ω, we can suppose that EJQTDE, 1999 No. 9, p. 20 (2.45) for all large n.Since t n → t 0 and t can be chosen in a dense subset of [0, T ], we can pick t such that for all large n.Notice that, at this point, x ∈ Ω and t ∈ [0, T ] have been fixed.Next, let B be a ball of radius R, centered at x and containing Ω, and set, for each n From (2.38), it follows that γ n → 0 in L 1 (B).We now assume N ≥ 2; the case where N = 1 can be dealt with in a similar (and even simpler) way.We introduce spherical coordinates in IR N centered at for all large n.Combining the above estimates (from (2.41) to (2.47)), we get a contradiction with (2.31).Accordingly, we take as (S n ) n a tail-end of (s n ) n .
We are now ready to prove the existence of a solution of (2.21).Let us define the following open bounded set in C 0 (Q), with 0 ∈ O, O = u ∈ C 0 (Q) | − S n < u(x, t) < S n for every (x, t) ∈ Q , EJQTDE, 1999 No. 9, p. 22 where S n , for any fixed n, comes from Claim 2. Let u be a solution of (2.23), for some µ ∈ [0, 1], such that u ∈ O. Observing that any solution u of (2.23), for any µ ∈]0, 1], satisfies u > > 0 and using Claim 2, we conclude that u ∈ O.Then, the homotopy invariance of the degree yields the existence of a solution in O of (2.23) for µ = 1, that is a solution of problem (2.21).

2 u
.15) u (x) > 0 for x ∈ [A, A+B 2 [ and u (x) < 0 for x ∈ ] A+B 2 , B]. (2.16) From the definition of p it follows that u is of class C 2 on [A, B] \ A+B 2 and satisfies the equation −a u + b sign x − A+B = h(u), (2.17) everywhere on [A, B] \ A+B 2

( 2 .
23) By the Leray-Schauder degree theory, equation (2.22), with µ = 1, and therefore problem (2.21), is solvable, if there exists an open bounded set O in C 0 (Q), with 0 ∈ O, such that no solution of (2.22), or equivalently of (2.23), for any µ ∈ [0, 1], belongs to the boundary of O.The remainder of this proof basically consists of building such a set O.