Existence of stable periodic solutions of a semilinear parabolic problem under Hammerstein–type conditions ^{∗}

Maria do Rosário GROSSINHO ^{†}

Departamento de Matemática, ISEG, Universidade Técnica de Lisboa, Rua do Quelhas 6, 1200 Lisboa, Portugal

and

CMAF, Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal

            E-mail: mrg@ptmat.lmc.fc.ul.pt

Pierpaolo OMARI ^{‡}

Dipartimento di Scienze Matematiche, Università di Trieste, Piazzale Europa 1, I–34127 Trieste, Italia

             E-mail: omari@univ.trieste.it

Abstract

We prove the solvability of the parabolic problem

∂_{t}u − ∑_{i,j=1}^{N}∂_{x_{i}}(a_{ij}(x, t)∂_{x_{j}}u)+∑_{i=1}^{N} b_{i}(x, t)∂_{x_{i}}u=f(x, t, u) in Ω×IR,
u(x, t) = 0  on ∂Ω×IR,
u(x, t) = u(x, t + T)  in Ω×IR,

assuming certain conditions on the asymptotic behaviour of the ratio 2 ∫_{0}^{s} f(x, t, σ)dσ/s^{2} with respect to the principal eigenvalue of the associated 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.

1991 Mathematics subject classification : 35K20, 35B10, 35B35. Keywords : parabolic equation, periodic solution, Hammerstein’s condition, upper and lower solutions, existence, localization, stability.

∗Research supported by CNR–JNICT.

†Supported also by FCT, PRAXIS XXI, FEDER, under projects PRAXIS/PCEX/P/ MAT/36/96 and PRAXIS/2/2.1/MAT/125/94.

‡Supported also by MURST 40% and 60% research funds.

1 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

\{ \begin{matrix}∂_{t}u + A(x, t, ∂_{x})u = f(x, t, u) & in Q, \\ u(x, t) = 0 & on Σ, \\ u(x, 0) = u(x, T) & in Ω. \end{matrix}  (1.1)

We assume throughout that

A(x, t, ∂_{x}) = − ∑_{i,j=1}^{N} ∂_{x_{i}}(a_{ij}(x, t)∂_{x_{j}}) + ∑_{i=1}^{N} b_{i}(x, t)∂_{x_{i}},

where a_{ij} ∈ C^{0}(\overline{Q}), a_{ij} = a_{ji}, a_{ij}(x, 0) = a_{ij}(x, T) in Ω, ∂_{x_{k}}a_{ij} ∈ L^{∞}(Q), b_{i} ∈ L^{∞}(Q) and ∂_{x_{k}}b_{i} ∈ L^{∞}(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) ∈ \overline{Q} and ξ ∈ IR^{N},

∑_{i,j=1}^{N} 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

∂_{t}u + A(x, t, ∂_{x})u = λu in Q,
u(x, t) = 0  on Σ,
u(x, 0) = u(x, T)  in Ω.

It was proven in [2] that the Dolph–type condition

lim sup_{s→±∞} \frac{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→±∞} \frac{2G_{±}(s)}{s^{2}} < λ_{1},  (1.4)

where G_{±}(s) = ∫_{0}^{s} 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 ∈ W_{p}^{2,1}(Q) (p > N + 2) such that

∂_{t}α + A(x, t, ∂_{x})α ≤ f(x, t, α) a.e. in Q,
α(x, t) ≤ 0  on Σ,
α(x, 0) ≤ α(x, T)  in Ω.

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 and a maximum solution w of (1.1), with α ≤ v ≤ w ≤ β in Q. Moreover, if α(·, 0) = 0 = β(·, 0) on ∂Ω, then the following holds: for every u_{0} ∈ W_{p}^{2−2/p}(Ω) ∩ H_{0}^{1}(Ω), with α(·, 0) ≤ u_{0} ≤ v(·, 0) (resp. w(·, 0) ≤ u_{0} ≤ β(·, 0)) in Ω, the set S_{u_{0}} of all functions u : \overline{Ω}×[0, +∞[→ IR, with u ∈ W_{p}^{2,1}(Ω×]0, σ[) for every σ > 0, satisfying

\{ \begin{matrix}∂_{t}u + A(x, t, ∂_{x})u = f(x, t, u) & a.e. in Ω ×]0, +∞[, \\ u(x, t) = 0 & on ∂Ω ×]0, +∞[, \\ u(x, 0) = u_{0}(x) & in Ω \end{matrix}  (1.5)

and α ≤ u ≤ v (resp. w ≤ u ≤ β) in Ω×]0, +∞[, is non–empty and every u ∈ S_{u_{0}} is such that lim_{t→+∞} |u(·, t) − v(·, t)|_{∞} = 0 (resp. lim_{t→+∞} |u(·, t) − w(·, t)|_{∞} = 0).

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 \overline{α}, with α ≤ \overline{α} ≤ v in Q, of

∂_{t}\overline{α} + A(x, t, ∂_{x})\overline{α} = f(x, t, α) + k_{ρ}(x, t, α, \overline{α}) in Q,
\overline{α}(x, t) = 0  on Σ,
\overline{α}(x, 0) = \overline{α}(x, T)  in Ω,

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 q ∈]1, \frac{2N}{N−2}[ if N ≥ 3, or q ∈]1, +∞[ if N = 2, such that

|g_{±}(s)| ≤ |s|^{q−1} + c  for s ∈ IR.  (1.6)

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.

We stress that this theorem completes, for what concerns the stability information, the classical result of Hammerstein for the selfadjoint elliptic problem

\{ \begin{matrix}− ∑_{i,j=1}^{N} ∂_{x_{i}}(a_{ij}(x)∂_{x_{j}}u) = f(x, u) & in Ω, \\ u = 0 & on ∂Ω. \end{matrix}

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

\overline{a}_{i} = min_{\overline{Q}} a_{ii} and \overline{b}_{i} = |b_{i} − ∑_{j=1}^{N} ∂_{x_{j}}a_{ji}|_{∞}.

Then, define

λ̂_{1} = max_{i=1,...,N} \{(\frac{π}{B_{i} − A_{i}} )^{2} \overline{a}_{i} exp (−\begin{matrix}\overline{b} & i \\ \overline{a} & i \end{matrix} (B_{i} − A_{i}))\} .

Theorem 1.2 Assume

lim inf_{s→±∞} \frac{2G_{±}(s)}{s^{2}} < λ̂_{1}.  (1.7)

Then, the same conclusions of Theorem 1.1 hold.

The constant λ̂_{1} 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, λ̂_{1} 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 introduced in [6] for solving the one–dimensional two–point boundary value problem

\{ \begin{matrix}−u′′ = f(x, u) & in ]A, B[, \\ u(A) = u(B) = 0 \end{matrix}

and were later used in [8] for studying the higher dimensional elliptic problem

\{ \begin{matrix}−Δu = f(x, u) & in Ω, \\ u = 0 & on ∂Ω. \end{matrix}  (1.8)

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

\{ \begin{matrix}− ∑_{i,j=1}^{N} ∂_{x_{i}}(a_{ij}(x)∂_{x_{j}}u) + ∑_{i=1}^{N} b_{i}(x)∂_{x_{i}}u = f(x, u) & in Ω, \\ u = 0 & on ∂Ω. \end{matrix}  (1.9)

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 λ̂_{1} 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.

Theorem 1.3 Assume

lim sup_{s→±∞} \frac{g_{±}(s)}{s} ≤ λ_{1}  (1.10)

and

lim inf_{s→±∞} \frac{2G_{±}(s)}{s^{2}} < λ_{1}.  (1.11)

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.

2 Proofs

2.1 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.

We start with some notation. Fixed t_{1}, t_{2}, with t_{1} ≤ t_{2}, and given u, v ∈ C^{1,0}(\overline{Ω} × [t_{1}, t_{2}]), we write:

• u ≥ v if, for every (x, t) ∈ \overline{Ω} × [t_{1}, t_{2}], u(x, t) ≥ v(x, t);

• u >> v if, for every (x, t) ∈ Ω×[t_{1}, t_{2}], u(x, t) > v(x, t) and, for every (x, t) ∈ ∂Ω × [t_{1}, t_{2}], either u(x, t) > v(x, t), or u(x, t) = v(x, t) and ∂_{ν}u(x, t) < ∂_{ν}v(x, t), where ν = (ν_{0}, 0) ∈ IR^{N+1}, ν_{0} ∈ IR^{N} being the outer normal to Ω at x ∈ ∂Ω.

Proposition 2.1 There exist a number λ_{1} > 0 and functions φ_{1}, φ_{1}^{∗} ∈ W_{p}^{2,1}(Q), for every p, satisfying, respectively,

∂_{t}φ_{1} − ∑_{i,j=1}^{N} ∂_{x_{i}}(a_{ij}(x, t)∂_{x_{j}}φ_{1}) + ∑_{i=1}^{N} b_{i}(x, t)∂_{x_{i}}φ_{1} = λ_{1}φ_{1} in Q,
φ_{1}(x, t) = 0  on Σ,
φ_{1}(x, 0) = φ_{1}(x, T)  in Ω

and

−∂_{t}φ_{1}^{∗} − ∑_{i,j=1}^{N} ∂_{x_{j}}(a_{ij}(x, t)∂_{x_{i}}φ_{1}^{∗}) − ∑_{i=1}^{N} ∂_{x_{i}}(b_{i}(x, t)φ_{1}^{∗}) = λ_{1}φ_{1}^{∗} in Q,
φ_{1}^{∗}(x, t) = 0  on Σ,
φ_{1}^{∗}(x, 0) = φ_{1}^{∗}(x, T)  in Ω.

Moreover, the following statements hold:

(i) φ_{1} >> 0 and φ_{1}^{∗} >> 0; (ii) if ψ is a solution of

∂_{t}ψ − ∑_{i,j=1}^{N} ∂_{x_{i}}(a_{ij}(x, t)∂_{x_{j}}ψ) + ∑_{i=1}^{N} b_{i}(x, t)∂_{x_{i}}ψ = λ_{1}ψ in Q,
ψ(x, t) = 0  on Σ,
ψ(x, 0) = ψ(x, T)  in Ω,

or, respectively, of

−∂_{t}ψ − ∑_{i,j=1}^{N} ∂_{x_{j}}(a_{ij}(x, t)∂_{x_{i}}ψ) − ∑_{i=1}^{N} ∂_{x_{i}}(b_{i}(x, t)ψ) = λ_{1}ψ in Q,
ψ(x, t) = 0  on Σ,
ψ(x, 0) = ψ(x, T)  in Ω,

then ψ = cφ_{1}, or, respectively, ψ = cφ_{1}^{∗}, for some c ∈ IR; (iii) λ_{1} is the smallest number λ for which the problems

∂_{t}u − ∑_{i,j=1}^{N} ∂_{x_{i}}(a_{ij}(x, t)∂_{x_{j}}u) + ∑_{i=1}^{N} b_{i}(x, t)∂_{x_{i}}u = λu in Q,
u(x, t) = 0  on Σ,
u(x, 0) = u(x, T)  in Ω

and

−∂_{t}u − ∑_{i,j=1}^{N} ∂_{x_{j}}(a_{ij}(x, t)∂_{x_{i}}u) − ∑_{i=1}^{N} ∂_{x_{i}}(b_{i}(x, t)u) = λu in Q,
u(x, t) = 0  on Σ,
u(x, 0) = u(x, T)  in Ω

have nontrivial solutions.

Proposition 2.1 is a immediate consequence of [4, Proposition 2.3].

Proposition 2.2 Fix p > N + 2. Let q ∈ L_{∞}(Q) satisfy ess sup_{Q}q < λ_{1}. Then, for every f ∈ L_{p}(Q) the problem

\{ \begin{matrix}∂_{t}u + A(x, t, ∂_{x})u = qu + f(x, t) & in Q, \\ u(x, t) = 0 & on Σ, \\ u(x, 0) = u(x, T) & in Ω \end{matrix}  (2.1)

has a unique solution u ∈ W_{p}^{2,1}(Q) (which is asymptotically stable). Moreover, there exists a constant C > 0, independent of f, such that

|u|_{W_{p}^{2,1}} ≤ C |f|_{p}.  (2.2)

Finally, if f ≥ 0 a.e. in Q, with strict inequality on a set of positive measure, then u >> 0.

Proof. Fix a constant k ≥ 0 such that

ess inf_{Q}(k − q) > (2η)^{−1}(max_{i=1,...,N} |b_{i}|_{∞}),

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

\{ \begin{matrix}∂_{t}v + A(x, t, ∂_{x})v + (k − q)v = f(x, t) & in Q, \\ v(x, t) = 0 & on Σ, \\ v(x, 0) = v(x, T) & in Ω. \end{matrix}  (2.3)

has a unique solution v ∈ W_{p}^{2,1}(Q) and, therefore, v ∈ C^{1+μ,μ}(\overline{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

∂_{t}β + A(x, t, ∂_{x})β = qβ + f + s(λ_{1} − q)φ_{1} − kv ≥ qβ + f a.e. in Q,

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_{p}^{2,1}(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 ∂_{t} + A : W_{p}^{2,1}(Q) → L_{p}(Q) is invertible 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).

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

∂_{t}u_{i} + A(x, t, ∂_{x})u_{i} = q_{i}u_{i} in Q,
u_{i}(x, t) = 0  on Σ,
u_{i}(x, 0) = u_{i}(x, T),  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

∂_{t}u_{2} + A(x, t, ∂_{x})u_{2} + q_{2}^{−}u_{2} = q_{2}^{+}u_{2} ≥ 0 a.e. in Q,

we have u_{2} >> 0. If we set c = min{d ∈ IR | d u_{2} ≥ u_{1}} and v = c u_{2} − u_{1}, we get, as c > 0 and v ≥ 0,

∂_{t}v + A(x, t, ∂_{x})v + q_{1}^{−}v = q_{1}^{+}v + c (q_{2} − q_{1}) u_{2} ≥ 0 a.e. in Q

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

0 = ∂_{t}v + A(x, t, ∂_{x})v = (q_{2} − q_{1}) c u_{2} a.e. in Q

and therefore q_{1} = q_{2} a.e. in Q.

2.2 Proof of Theorem 1.1

We indicate how to build an upper solution β of (1.1), with β ≥ 0; a lower solution α, with α ≤ 0, can be constructed in a similar way. If there exists a constant β ≥ 0 such that g_{+}(β) ≤ 0, β is an upper solution of (1.1). Therefore, suppose that g_{+}(s) > 0 for s ≥ 0, and set

h(s) = \{ \begin{matrix}g_{+}(s) & if s ≥ 0, \\ g_{+}(0) & if s < 0. \end{matrix}  (2.4)

Let us consider the elliptic problem

\{ \begin{matrix}− ∑_{i,j=1}^{N} ∂_{x_{i}}(a_{ij}(x)∂_{x_{j}}u) = h(u) & in Ω, \\ u = 0 & on ∂Ω. \end{matrix}  (2.5)

From conditions (1.4) and (1.6), it follows that (2.5) admits a solution u ∈ H_{0}^{1}(Ω). A bootstrap argument, like in [7], shows that u ∈ W_{p}^{2}(Ω), for all finite p, and the strong maximum principle implies that u >> 0. The function β, defined by setting β(x, t) = u(x) for (x, t) ∈ \overline{Q}, is by (1.2) an upper solution of (1.1).

2.3 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_{[A,B]} p(x) ≤ max_{[A,B]} p(x) =: p_{∞}  (2.6)

and

0 < q_{0} := min_{[A,B]} q(x) ≤ max_{[A,B]} q(x) =: q_{∞}.  (2.7)

Let also h : IR → IR be a continuous function and set H(s) = ∫_{0}^{s} h(σ) dσ for s ∈ IR. Consider the initial value problem

\{ \begin{matrix}−(pu′)′ = qh(u), \\ u(\frac{A+B}{2} ) = d, \\ u′(\frac{A+B}{2} ) = 0, \end{matrix}  (2.8)

where d is a real parameter. By a solution of (2.8) we mean a function u of class C^{1}, with pu′ of class C^{1}, defined on some interval I ⊂ [A, B], with \frac{A+B}{2} ∈I^{∘}, which satisfies the equation on I and the initial conditions.

Claim. Assume that there are constants c, d, with 0 ≤ c < d, such that

h(s) > 0 for s ∈ [c, d]  (2.9)

and

∫_{c}^{d} \frac{dσ}{\sqrt{H(d) − H(σ)}} ≥ (\frac{\sqrt{2 p_{∞}q_{∞}}}{p_{0}} ) \frac{B − A}{2}.  (2.10)

Then, there exists a solution u of (2.8), which is defined on [A, B] and satisfies

c ≤ u(x) ≤ d for x ∈ [A, B],

u′(x) > 0 for x ∈ [A, \frac{A + B}{2} [ and u′(x) < 0 for x ∈]\frac{A + B}{2} , B].

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 \frac{A+B}{2} and, if ]ω_{−}, ω_{+}[ denotes the maximal interval included in ]A, B[ where u(x) ∈]c, d], we have

u′(x) > 0 for x ∈]ω_{−}, \frac{A+B}{2} [ and u′(x) < 0 for x ∈]\frac{A+B}{2} , ω_{+}[. (2.11)

We want to prove that ω_{−} = A and ω_{+} = B. Assume, by contradiction, that

ω_{+} < B.  (2.12)

Similarly one should argue if ω_{−} > A. From (2.11) we derive that u is decreasing on [\frac{A+B}{2} , ω_{+}[ and, by the definition of ω_{+}, we have

lim_{x→ω_{+}} u(x) = c =: u(ω_{+}).

Now, pick x ∈ [\frac{A+B}{2} , ω_{+}[, multiply the equation in (2.8) by −pu′ and integrate between \frac{A+B}{2} and x. Taking into account that, by (2.9) and (2.11), h(u)u′ < 0 on ]\frac{A+B}{2} , ω_{+}[, we obtain, using (2.6) and (2.7) as well,

\frac{1}{2}(p(x)u′(x))^{2} = − ∫ x_{\frac{A+B}{2}} pqh(u)u′dt
≤ p_{∞}q_{∞} (H (u (\frac{A+B}{2} )) − H(u(x)))
= p_{∞}q_{∞} (H(d) − H(u(x))) .

By (2.6) and (2.11), we have, for each x ∈]\frac{A+B}{2} , ω_{+}[,

(0 <) |u′(x)|^{2} ≤ 2 (\frac{p_{∞}q_{∞}}{p_{0}^{2}} ) (H(d) − H(u(x)))

and hence

\frac{−u′(x)}{\sqrt{H(d) − H(u(x))}} ≤ \frac{\sqrt{2 p_{∞}q_{∞}}}{p_{0}} .

Integrating this relation between \frac{A+B}{2} and ω_{+} and changing variable, we get by (2.12)

∫_{c}^{d} \frac{dσ}{\sqrt{H(d) − H(σ)}} < (\frac{\sqrt{2p_{∞}q_{∞}}}{p_{0}} ) \frac{B − A}{2}.

Then, condition (2.10) yields a contradiction and the conclusions of the Claim follow.

Step 2. We prove now that problem (1.1) has an upper solution β ∈ C^{2,1}(\overline{Q}), with β >> 0. Assume, without loss of generality, that

max_{i=1,...,N} \{(\frac{π}{B_{i} − A_{i}} )^{2} \overline{a}_{i} exp (−\begin{matrix}\overline{b} & i \\ \overline{a} & i \end{matrix} (B_{i} − A_{i}))\}

is attained at i = 1 and set, for the sake of simplicity,

]A, B[ := ]A_{1}, B_{1}[,

\overline{a} := \overline{a}_{1} = min_{\overline{Q}} a_{11}

and

\overline{b} := \overline{b}_{1} = |b_{1} − ∑_{j=1}^{N} ∂_{x_{j}}a_{j1}|_{∞}.

Note that

\overline{a} > 0 and \overline{b} ≥ 0.  (2.13)

Let us set, for x ∈ [A, B],

p(x) := exp (−\frac{\overline{b}}{\overline{a}} |x − \frac{A+B}{2} |) and q(x) := \frac{1}{\overline{a}}p(x)

and consider the ordinary differential equation

−(pu′)′ = q h(u),  (2.14)

where h is defined in (2.4). It is clear that p, q and h satisfy, respectively, (2.6), (2.7) and (2.9), for any fixed c, d, with 0 ≤ c < d. Observe that (2.10) is also fulfilled, for c = 0 and for some d > 0. Indeed, since (1.7) implies that

lim inf_{s→+∞} (2 H(s) − λ̂_{1}s^{2}) = −∞,

we can find a sequence (d_{n})_{n}, with d_{n} → +∞, such that, for each n,

(0 <) H(d_{n}) − H(s) < \frac{λ̂_{1}}{2} (d_{n}^{2} − s^{2}) for s ∈ [0, d_{n}[

and hence

∫_{0}^{d_{n}}\frac{dσ}{\sqrt{H(d_{n}) − H(σ)}} > \sqrt{\frac{2}{λ̂_{1}}} ∫_{0}^{d_{n}}\frac{dσ}{\sqrt{d_{n}^{2} − σ^{2}}} = \sqrt{\frac{2}{λ̂_{1}}} \frac{π}{2} = (\frac{\sqrt{2p_{∞}q_{∞}}}{p_{0}} ) \frac{B − A}{2}.

Therefore, taking d := d_{n}, for some fixed n, we conclude that (2.10) holds. Accordingly, by the Claim, there exists a solution u of (2.14), which is defined on [A, B] and satisfies

u(x) > 0 for x ∈]A, B[,  (2.15)
u′(x) > 0 for x ∈ [A, \frac{A+B}{2} [ and u′(x) < 0 for x ∈]\frac{A+B}{2} , B].

(2.16)

From the definition of p it follows that u is of class C^{2} on [A, B] \ \{\frac{A+B}{2} \} and satisfies the equation

−\overline{a} u′′ + \overline{b} sign (x − \frac{A+B}{2} ) u′ = h(u),  (2.17)

everywhere on [A, B] \ \{\frac{A+B}{2} \}. Actually, since u′(\frac{A+B}{2} ) = 0, a direct inspection of (2.17) shows that u is of class C^{2} and satisfies equation (2.17) everywhere on [A, B]. Moreover, using (2.13), (2.15), (2.16) and (2.9), with c = 0 and any d > 0, we derive from (2.17)

u′′(x) = \frac{1}{\overline{a}} (\overline{b} sign (x − \frac{A+B}{2} ) u′(x) − h(u(x))) < 0 on [A, B]. (2.18)

Now, we set

β(x_{1}, …, x_{N}, t) = u(x_{1}) for (x_{1} … x_{N}, t) ∈ \overline{Q}.

We have that β ∈ C^{2,1}(\overline{Q}) and β >> 0. Let us check that β is an upper solution of problem (1.1). Indeed, using (2.13), (2.17), (2.18) and (1.2), as well as the definitions of \overline{a} and \overline{b}, we have, for each (x_{1}, …, x_{N}, t) ∈ Q,

∂_{t}β(x_{1}, ..., x_{N}, t) − ∑_{i,j=1}^{N} a_{ij}(x_{1}, ..., x_{N}, t) ∂_{x_{i}x_{j}}β(x_{1}, ..., x_{N}, t) +

+ ∑_{i=1}^{N} (b_{i}(x_{1}, ..., x_{N}, t) − ∑_{j=1}^{N} ∂_{x_{j}}a_{ji}(x_{1}, ..., x_{N}, t)) ∂_{x_{i}}β(x_{1}, ..., x_{N}, t)

= −a_{11}(x_{1}, ..., x_{N}, t)u′′(x_{1}) + (b_{1}(x_{1}, ..., x_{N}, t) − ∑_{j=1}^{N} ∂_{x_{j}}a_{j1}(x_{1}, ..., x_{N}, t))u′(x_{1})

≥ −\overline{a}u′′(x_{1}) + \overline{b} sign(x_{1} − \frac{A+B}{2} )u′(x_{1}) = h(u(x_{1})) = g_{+}(u(x_{1})) ≥ f(x_{1}, ..., x_{N}, t, β(x_{1}, ..., x_{N}, t)).

This concludes the proof of the Theorem 1.2.

2.4 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), which by (1.10) and (1.11) satisfies

lim sup_{s→+∞} \frac{h(s)}{s} ≤ λ_{1}  (2.19)

and

lim inf_{s→+∞} \frac{2H(s)}{s^{2}} < λ_{1}.  (2.20)

Let us consider the problem

\{ \begin{matrix}∂_{t}u + A(x, t, ∂_{x})u = h(u) & in Q, \\ u(x, t) = 0 & on Σ, \\ u(x, 0) = u(x, T) & in Ω \end{matrix}  (2.21)

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 : C^{0}(\overline{Q}) → C^{0}(\overline{Q}) which sends any function u ∈ C^{0}(\overline{Q}) onto the unique solution v ∈ W_{p}^{2,1}(Q) of

∂_{t}v + A(x, t, ∂_{x})v = h(u) in Q,
v(x, t) = 0  on Σ,
v(x, 0) = v(x, T)  in Ω.

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

u = μ Su,  (2.22)

with μ ∈ [0, 1], which corresponds to

\{ \begin{matrix}∂_{t}u + A(x, t, ∂_{x})u = μh(u) & in Q, \\ u(x, t) = 0 & on Σ, \\ u(x, 0) = u(x, T) & in Ω. \end{matrix}  (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}(\overline{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.

Claim 1. Let (u_{n})_{n} be a sequence of solutions of

\{ \begin{matrix}∂_{t}u_{n} + A(x, t, ∂_{x})u_{n} = μ_{n}h(u_{n}) & in Q, \\ u_{n}(x, t) = 0 & on Σ, \\ u_{n}(x, 0) = u_{n}(x, T) & in Ω, \end{matrix}  (2.24)

with μ_{n} ∈ [0, 1], such that |u_{n}|_{∞} → +∞. Then, possibly passing to subsequences,

\frac{u_{n}}{|u_{n}|_{∞}} → v in W_{p}^{2,1}(Q),

where v = c φ_{1}, for some c > 0, and

\frac{h(u_{n})}{|u_{n}|_{∞}} → λ_{1}v in L_{p}(Q).

Proof of Claim 1. Let us write, for s ∈ IR,

h(s) = q(s) s + r(s),

with q, r continuous functions such that

0 ≤ q(s) ≤ λ_{1}  (2.25)

and

\frac{r(s)}{s} → 0, as |s| → +∞.  (2.26)

Let us set, for each n,

v_{n} = \frac{u_{n}}{|u_{n}|_{∞}} ,

where v_{n} satisfies

\{ \begin{matrix}∂_{t}v_{n} + A(x, t, ∂_{x})v_{n} = μ_{n}q(u_{n})v_{n} + μ_{n}r(u_{n})/|u_{n}|_{∞} & in Q, \\ v_{n}(x, t) = 0 & on Σ, \\ v_{n}(x, 0) = v_{n}(x, T) & in Ω. \end{matrix}  (2.27)

The sequence (v_{n})_{n} is bounded in W_{p}^{2,1}(Q) and therefore, possibly passing to a subsequence, it converges weakly in W_{p}^{2,1}(Q) and strongly in C^{1+α,α}(\overline{Q}), for some α > 0, to a function v ∈ W_{p}^{2,1}(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)

0 ≤ q_{0}(x, t) ≤ λ_{1}  (2.28)

a.e. in Q. Moreover, by (2.26), we have

\frac{r(u_{n}(x, t))}{|u_{n}|_{∞}} → 0  (2.29)

uniformly a.e. in Q. The weak continuity of the operator ∂_{t} +A : W_{p}^{2,1}(Q) → L_{p}(Q) implies that v satisfies

\{ \begin{matrix}∂_{t}v + A(x, t, ∂_{x})v = μ_{0}q_{0}v & in Q, \\ v(x, t) = 0 & on Σ, \\ v(x, 0) = v(x, T) & in Ω. \end{matrix}  (2.30)

Now, if we set q_{1} = μ_{0}q_{0}, q_{2} = λ_{1}, v_{1} = v and v_{2} = φ_{1}, Proposition 2.3 yields μ_{0}q_{0} = λ_{1} a.e. in Q, so that, by (2.28), μ_{0} = 1, and v = cφ_{1}, for some c > 0. We also have

\begin{aligned}∫_{Q} |λ_{1} − q(u_{n})|^{p} & ≤ & |λ_{1} − q(u_{n})|_{∞}^{p−1} ∫_{Q} |λ_{1} − q(u_{n})| \\ & ≤ & λ_{1}^{p−1} ∫_{Q}(λ_{1} − q(u_{n})) → 0, \end{aligned}

i.e. q(u_{n}) → λ_{1} in L_{p}(Q), and therefore, by (2.29), h(u_{n})/|u_{n}|_{∞} → λ_{1}v in L_{p}(Q). Finally, Proposition 2.2 implies that v_{n} → v in W_{p}^{2,1}(Q).

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\overline{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

λ_{1} − \frac{2H(s_{n})}{s_{n}^{2}} > ε  (2.31)

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_{\overline{Q}} u_{n} = u_{n}(x_{n}, t_{n}) = s_{n},

where (x_{n}, t_{n}) ∈ Ω × [0, T]. Since |u_{n}|_{∞} → +∞, we can suppose by Claim 1 that v_{n} = u_{n}/|u_{n}|_{∞} → v in W_{p}^{2,1}(Q), and therefore in C^{1,0}(\overline{Q}), with v = cφ_{1}, for some c > 0, and

|u_{n}|_{∞}^{−1}|λ_{1}u_{n} − h(u_{n})|_{p} → 0.  (2.32)

There is also a constant K > 0 such that

|u_{n}|_{∞}^{−1}|λ_{1}u_{n} − h(u_{n})|_{∞} ≤ K  (2.33)

and

|∇_{x}v_{n}|_{∞} ≤ K,  (2.34)

for every n. Moreover, we have

|∂_{t}v_{n} − ∂_{t}v|_{p} → 0  (2.35)

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,

|u_{n}|_{∞}^{−1}(λ_{1}u_{n}(·, \overline{t}) − h(u_{n}(·, \overline{t}))) → 0,  (2.36)
∂_{t}v_{n}(·, \overline{t}) − ∂_{t}v(·, \overline{t}) → 0  (2.37)

in L_{p}(Ω), for a.e. \overline{t} ∈ [0, T], and

|u_{n}|_{∞}^{−1}(λ_{1}u_{n}(\overline{x}, ·) − h(u_{n}(\overline{x}, ·))) → 0,  (2.38)
∂_{t}v_{n}(\overline{x}, ·) − ∂_{t}v(\overline{x}, ·) → 0  (2.39)

in L_{p}(0, T), for a.e. \overline{x} ∈ Ω. Moreover, we have that

∫_{0}^{T} |∂_{t}v(\overline{x}, τ)|^{2} dτ is finite  (2.40)

for a.e. \overline{x} ∈ Ω. Let us write

\frac{λ_{1}}{2} s_{n}^{2} − H(s_{n}) = \frac{λ_{1}}{2} u_{n}^{2}(x_{n}, t_{n}) − H(u_{n}(x_{n}, t_{n}))

= [\frac{λ_{1}}{2} (u_{n}^{2}(x_{n}, t_{n}) − u_{n}^{2}(\overline{x}, t_{n})) − H(u_{n}(x_{n}, t_{n})) + H(u_{n}(\overline{x}, t_{n}))]
+ [\frac{λ_{1}}{2} (u_{n}^{2}(\overline{x}, t_{n}) − u_{n}^{2}(\overline{x}, \overline{t})) − H(u_{n}(\overline{x}, t_{n})) + H(u_{n}(\overline{x}, \overline{t}))]
+ [\frac{λ_{1}}{2} (u_{n}^{2}(\overline{x}, \overline{t}) − u_{n}^{2}(x^{∗}, \overline{t})) − H(u_{n}(\overline{x}, \overline{t})) + H(u_{n}(x^{∗}, \overline{t}))] , (2.41)

where the choices of points \overline{t} ∈ [0, T], such that (2.36) and (2.37) hold, \overline{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_{k}^{(n)})_{k} in C^{1}(\overline{Q}) such that w_{k}^{(n)} → u_{n} in W_{p}^{1}(Q) and therefore in C^{0}(\overline{Q}), since p > N +2. This implies in particular that w_{k}^{(n)} → u_{n} in L_{∞}(Q) and ∂_{t}w_{k}^{(n)} → ∂_{t}u_{n} in L_{p}(Q). Hence, using Fubini’s theorem and possibly passing to a subsequence, we get ∂_{t}w_{k}^{(n)}(x, ·) → ∂_{t}u_{n}(x, .) in L_{p}(0, T) for a.e. x ∈ Ω. Hence, it follows that, for each n,

(λ_{1}w_{k}^{(n)}(\overline{x}, ·) − h(w_{k}^{(n)}(\overline{x}, ·))) ∂_{t}w_{k}^{(n)}(\overline{x}, ·)
→ (λ_{1}u_{n}(\overline{x}, ·) − h(u_{n}(\overline{x}, ·))) ∂_{t}u_{n}(\overline{x}, ·)

in L_{p}(0, T), for a.e. \overline{x} ∈ Ω, and therefore, for a.e. \overline{t} ∈ [0, T],

\frac{λ_{1}}{2} (u_{n}^{2}(\overline{x}, t_{n}) − u_{n}^{2}(\overline{x}, t)) − (H(u_{n}(\overline{x}, t_{n})) − H(u_{n}(\overline{x}, \overline{t})))

= lim_{k→+∞} [\frac{λ_{1}}{2} (w_{k}^{(n)}^{2}(\overline{x}, t_{n}) − w_{k}^{(n)}^{2}(\overline{x}, \overline{t}))
− (H(w_{k}^{(n)}(\overline{x}, t_{n})) − H(w_{k}^{(n)}(\overline{x}, \overline{t})))]
= lim_{k→+∞} ∫ t_{n} ; \overline{t} (λ_{1}w_{k}^{(n)}(\overline{x}, τ) − h(w_{k}^{(n)}(\overline{x}, τ))) ∂_{t}w_{k}^{(n)}(\overline{x}, τ) dτ
= ∫ t_{n} ; \overline{t} (λ_{1}u_{n}(\overline{x}, τ) − h(u_{n}(\overline{x}, τ))) ∂_{t}u_{n}(\overline{x}, τ) dτ.  (2.42)

Moreover, for each n, we have H(u_{n}(·, t)) ∈ C^{1}(\overline{Ω}) for every t ∈ [0, T] and hence, by (2.33) and (2.34), we obtain, for every \overline{x} ∈ Ω,

|u_{n}|_{∞}^{−2} |\frac{λ_{1}}{2} (u_{n}^{2}(x_{n}, t_{n}) − u_{n}^{2}(\overline{x}, t_{n})) − (H(u_{n}(x_{n}, t_{n})) − H(u_{n}(\overline{x}, t_{n})))|

≤ ∫_{0}^{1} |λ_{1}v_{n}(σ_{n}(τ), t_{n}) − |u_{n}|_{∞}^{−1}h(u_{n}(σ_{n}(τ), t_{n}))| ×
×|∇_{x}v_{n}(σ_{n}(τ), t_{n})| |σ′_{n}(τ)| dτ
≤ K^{2} ℓ(σ_{n}), (2.43)

where σ_{n} is a path, joining x_{n} to \overline{x} and having range contained in Ω, and ℓ(σ_{n}) denotes its length. Because x_{n} → x_{0}, with x_{n}, x_{0} ∈ Ω, and \overline{x} can be chosen in a dense subset of Ω, we can suppose that

K^{2}ℓ(σ_{n}) < \frac{ε}{4},  (2.44)

for all large n. Fix \overline{x} ∈ Ω such that (2.38), (2.39), (2.40), (2.42) and (2.44) hold. For every \overline{t} ∈ [0, T], we derive from (2.33), (2.39) and (2.42)

|u_{n}|_{∞}^{−2} |\frac{λ_{1}}{2} (u_{n}^{2}(\overline{x}, t_{n}) − u_{n}^{2}(\overline{x}, \overline{t})) − (H(u_{n}(\overline{x}, t_{n})) − H(u_{n}(\overline{x}, \overline{t})))|

≤ |∫ t_{n} ; \overline{t} |λ_{1}v_{n}(\overline{x}, τ) − |u_{n}|_{∞}^{−1}h(u_{n}(\overline{x}, τ))| |∂_{t}v_{n}(\overline{x}, τ)| dτ|

≤ K ∫ t_{n} ; \overline{t} |∂_{t}v_{n}(\overline{x}, τ)| dτ

≤ K ∫_{0}^{T} |∂_{t}v_{n}(\overline{x}, τ) − ∂_{t}v(\overline{x}, τ)| dτ + K ∫ t_{n} ; \overline{t} |∂_{t}v(\overline{x}, τ)| dτ

≤ \frac{ε}{4} + K |t_{n} − \overline{t}|^{1/2} (∫_{0}^{T} |∂_{t}v(\overline{x}, τ)|^{2} dτ )^{1/2}  (2.45)

for all large n. Since t_{n} → t_{0} and \overline{t} can be chosen in a dense subset of [0, T], we can pick \overline{t} such that

K (∫_{0}^{T} |∂_{t}v(\overline{x}, τ)|^{2} dτ )^{1/2} |t_{n} − \overline{t}|^{1/2} < \frac{ε}{4},  (2.46)

for all large n. Notice that, at this point, \overline{x} ∈ Ω and \overline{t} ∈ [0, T] have been fixed. Next, let B be a ball of radius R, centered at \overline{x} and containing \overline{Ω}, and set, for each n

γ_{n}(x) = \{ \begin{matrix}|u_{n}|_{∞}^{−1} |λ_{1}u_{n}(x, \overline{t}) − h(u_{n}(x, \overline{t}))| & if x ∈ Ω, \\ |u_{n}|_{∞}^{−1}h(0) & if x ∈ B \ Ω . \end{matrix}

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 \overline{x}. Denoting by (ρ, ϕ_{1}, …, ϕ_{N−2}, ψ) with ρ ∈ [0, R], (ϕ_{1}, …, ϕ_{N−2}) ∈ [0, π]^{N−2}, ψ ∈ [0, 2π] the spherical coordinates of a point x ∈ B and by Φ this change of coordinates, we get

\begin{matrix}∫_{[0,π]^{N−2}×[0,2π]} (∫_{0}^{R} γ_{n}(Φ)| det Φ′|dρ) dϕ_{1} … dϕ_{N−2}dψ \\ = ∫_{B} γ_{n}dx → 0 , \end{matrix}

where | det Φ′| = ρ^{N−1}(sin ϕ_{1})^{N−2}(sin ϕ_{2})^{N−3} · . . . · sin ϕ_{N−2}. Hence, possibly passing to a subsequence, we have

∫_{0}^{R} |γ_{n}(Φ)| ρ^{N−1}dρ → 0

for a.e. (ϕ_{1}, …, ϕ_{N−2}, ψ) ∈ [0, π]^{N−2} × [0, 2π]. Passing to a further subsequence, we also have that, for a.e. fixed (ϕ_{1}, …, ϕ_{N−2}, ψ),

γ_{n}(Φ) → 0

for a.e. ρ ∈ [0, R]. On the other hand, the functions γ_{n} are continuous and uniformly bounded, by the linear growth of h, and therefore, by Lebesgue’s theorem,

∫_{0}^{R} γ_{n}(Φ)dρ → 0

for a.e. (ϕ_{1}, …, ϕ_{N−2}, ψ). This means that

∫_{[\overline{x},y]} γ_{n} → 0

for a.e. y ∈ ∂B. Denoting by x^{∗} the first intersection point of [\overline{x}, y] with ∂Ω, we obtain

∫_{0}^{1} |u_{n}|_{∞}^{−1} |λ_{1}u_{n}(x^{∗} + τ(\overline{x} − x^{∗}), \overline{t}) − h(u_{n}(x^{∗} + τ(\overline{x} − x^{∗}), \overline{t}))| dτ → 0 .

Hence, using (2.34) and (2.36), we have

|u_{n}|_{∞}^{−2} |\frac{λ_{1}}{2} (u_{n}^{2}(\overline{x}, \overline{t}) − u_{n}^{2}(x^{∗}, \overline{t})) − (H(u_{n}(\overline{x}, \overline{t})) − H(u_{n}(x^{∗}, \overline{t})))|

= ∫_{0}^{1} |λ_{1}v_{n}(x^{∗} + τ(\overline{x} − x^{∗}), \overline{t}) − |u_{n}|_{∞}^{−1}h(u_{n}(x^{∗} + τ(\overline{x} − x^{∗}), \overline{t}))| ×
× |∇_{x}v_{n}(x^{∗} + τ(\overline{x} − x^{∗}), \overline{t}) · (\overline{x} − x^{∗})| dτ
≤ K| \overline{x} − x^{∗}| ∫_{0}^{1} |λ_{1}v_{n}(x^{∗} + τ(\overline{x} − x^{∗}), \overline{t}) +
−|u_{n}|_{∞}^{−1}h(u_{n}(x^{∗} + τ(\overline{x} − x^{∗}), \overline{t}))| dτ < \frac{ε}{4}  (2.47)

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}(\overline{Q}), with 0 ∈ O,

O = \{u ∈ C^{0}(\overline{Q}) | − S_{n} < u(x, t) < S_{n} for every (x, t) ∈ \overline{Q}\} ,

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 ∈ \overline{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 \overline{O} of (2.23) for μ = 1, that is a solution of problem (2.21).

References

[1] H. Amann, Periodic solutions of semilinear parabolic equations, in “Nonlinear Analysis: A collection of papers in honor of E. H. Rothe”, Cesari, Kannan, Weinberger Eds., Academic Press, New York (1987), 1–29.

[2] A. Castro, A. C. Lazer, Results on periodic solutions of parabolic equations suggested by elliptic theory, Boll. U.M.I. 1-B (1982), 1089– 1104.

[3] E. N. Dancer, P. Hess, On stable solutions of quasilinear periodic– parabolic problems, Annali Sc. Norm. Sup. Pisa 14 (1987), 123–141.

[4] C. De Coster, P. Omari, Unstable periodic solutions of a parabolic problem in presence of non–well–ordered lower and upper solutions, Université du Littoral, Cahiers du LMPA N^{o} 101, June 1999, pp. 1–54 (J. Functional Analysis, to appear).

[5] D. Del Santo, P. Omari, Nonresonance conditions on the potential for a semilinear elliptic problem, J. Differential Equations, 108 (1994), 120–138.

[6] M. L. C. Fernandes, P. Omari, F. Zanolin, On the solvability of a semilinear two–point BVP around the first eigenvalue, Differential and Integral Equations 2 (1989), 63–79.

[7] D.G. de Figueiredo, Positive solutions of semilinear elliptic equations, Lecture Notes in Math. vol. 957, Springer–Verlag, Berlin, 1982; pp. 34–87.

[8] A. Fonda, J. P. Gossez, F. Zanolin, On a nonresonance condition for a semilinear elliptic problem, Differential and Integral Equations 4 (1991), 945–952.

[9] M. R. Grossinho, P. Omari, A Hammerstein–type result for a semilinear parabolic problem, in “Proceedings Equadiff 95” (L. Magalhães, C. Rocha and L. Sanchez, eds) World Scientific, Singapore (1997), pp. 403-408.

[10] P. Hess, “Periodic–parabolic Boundary Value Problems and Positivity”, Wiley, New York, 1991.

[11] J. S. Kolesov, A test for the existence of periodic solutions to parabolic equations, Soviet Math. Dokl. 7 (1966), 1318–1320.