Existence of $L^p$-solutions for a semilinear wave equation with non-monotone nonlinearity

For Dirichlet-periodic and double periodic boundary conditions, we prove the existence of solutions to a 
forced semilinear wave equation with large forcing terms not flat on 
characteristics. The nonlinearity is assumed to be non-monotone, asymptotically linear, and not resonanant. 
We prove that the solutions are in $L^{p}$, $(p\geq 2)$, when the forcing term is in $L^{p}$. This is optimal; 
even in the linear case there are $L^p$ forcing terms for which the solutions are only in $L^p$. Our results 
extend those in [9] where the forcing term is assumed to be in $L_{\infty}$, and 
are in contrast with 
those in [6] where the non-existence of continuous solutions is established for $C^{\infty}$ forcing terms flat on characteristics. 
 200 words.

Hence, there exists M > 0 such that Also, without loss of generality, we may assume that −1 < β < 0. The spectrum of subject to the boundary condition (2) is given by In both cases all the eigenvalues have finite multiplicity except for 0 which has infinite multiplicity. We denote Ω 1 = (0, π) × (0, 2π) and Ω 2 = (0, 2π) × (0, 2π). Also we denote by p the norm in L p (Ω 1 ) or L p (Ω 2 ), depending on the context. By the arguments in [3] one sees that if g is monotone then for every f ∈ L 2 (Ω 1 ), the equation (1), (2) has a solution. Morover, if g and f are smooth and |g (u)| > for some > 0 and all u ∈ R then such a solution is of class C ∞ . In the non-monotone case, [16] and [12] proved that for f in a dense subset of L 2 (Ω 1 ) the equation (1), (2) has a weak solution; however, no mechanism is provided for determining the values of f for which (1) (2) has a solution. In the double-periodic case without resonance (−τ ∈ σ 2 ( )), [6] gives a class of smooth forcing terms for which the problem has no continuous solution. In [9], a class of forcing terms in L ∞ for which the double-periodic and the Dirichlet periodic problems have solutions is found.
In this paper we give a sufficient condition on the forcing term, f ∈ L p (Ω i ), i = 1, 2, for (1)-(2) and (1)-(3) to have solutions in L p .
In order to state our main result we introduce the concept of flatness on characteristics.
We say that φ is not flat on characteristics if given > 0 there exists δ > 0 such that where m(A) is the Lebesgue measure of the set A.
Our main result is the following.

(8)
If ϕ is not flat on characteristics then there exist c 0 such that for |c| ≥ c 0 the equation (1),(2) has a weak solution u ∈ L p (Ω 1 ) (see (11)). Remark 1. In section 5 we extend this result to the the double periodic case (1)-(3). Theorem 1.2 is optimal; even in the linear case all we can expect is to have L p solutions when f is in L p . Theorem 1.2 generalizes Theorem A in [10] where additional smoothness was assumed on the forcing term q. Also Theorem 1.2 generalizes Theorem 1.2 in [9] where the q was assumed to be in L ∞ .
The central idea for the proof of Theorem 1.2 is the estimation, in the L 2 sense, of the projection into the kernel of of approximate solutions to (1), (2). We achieve this using relation (25) below; similar arguments were used in [9]. Examples of functions satisfying (7) are plentiful, for instance q(x, t) = sin(x + t) + sin(t − x) satisfies (7). For studies on (1),(2) with g superlinear and monotone we refer the reader to [15]. For other recent results on wave equations with non-monotone nonlinearities the reader is referred to [2]. Extensions of the results in [2] using techniques introduced in [6] are found in [7].
3. Proof of Theorem 1.2 for p = 2. Let |c| ≥ c 1 , see Lemma 2.1. From [12] (see also [16]), there exist sequences { n } and {u n } in L 2 (Ω 1 ) such that n → 0 in L 2 (Ω 1 ) and u n is a weak solution to By (8) and (20), we have Let us denote z n = u n − cϕ, Π N (z n ) = v n and Π N ⊥ (z n ) = w n . Hence z n = v n + w n and Since, by Lemma 2.1, { w n C 1/2 } is bounded we may assume that {w n } converges uniformly in Ω 1 .
Following the arguments leading to (54) in [9], we see that v n is solution of (23) if and only if v n is solution to a.e. r ∈ [0, 2π].

5.
The double-periodic case. Now we turn our attention to the equation (1) subject to the condition (3). In this case the kernel N of is the closure of the linear space generated by the functions {sin(kx) sin(kt), cos(kx) cos(kt), sin(kx) cos(kt), cos(kx) sin(kt), k = 0, 1, 2, . . . } Here, Y and the weak solutions are defined as in (11) with Ω 1 replaced by Ω 2 . Imitating the proof of Theorem 1.2 one proves the following result.
As in the proof of Theorem 1.2 we project approximate solutions to the equation (1)-(3) onto N and Y . The only technical difference is that projection onto N of the approximating solutions are now of the form v n (x, t) =v n + v 1,n (x + t) + v 2n (t − x), and (24) becomes v n = 1 4π 2 τ Ω2 ( n − h(u n )) (52) 2πτ (v 1,n (r) +v n ) +