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Periodic solutions of semilinear wave equations with discontinuous nonlinearities
Boundary Value Problems volumeÂ 2015, ArticleÂ number:Â 197 (2015)
Abstract
We study the periodic Dirichlet problem for a semilinear wave equation with discontinuous nonlinearity. First we establish a continuation theorem for a semilinear operator equation in a Hilbert space, where a key tool is the BerkovitsTienari degree theory for a class of perturbations of monotone type of a densely defined closed linear operator. Applying the continuation theorem, we prove the main results on the solvability of the given semilinear wave equation, with the aid of spectral theory for densely defined closed linear operators.
1 Introduction
The study of nonlinear wave equations has been developed in various ways of approach by many researchers; for instance, by BrÃ©zis and Nirenberg [1, 2], Rabinowitz [3], Berkovits and Mustonen [4â€“6]. To find periodic solutions of a nonlinear wave equation, Mawhin and Willem [7] established a LeraySchauder type continuation theorem for abstract equations involving some perturbations of monotone type of a linear operator in a Hilbert space, where the Galerkin approximation method was used; see also [8]. In this aspect, Berkovits and Mustonen [9] showed the existence of weak solutions of the periodic Dirichlet problem for a semilinear wave equation under various conditions on the nonlinearity. Moreover, Berkovits and Tienari [10] constructed a topological degree theory for some classes of multivalued operators of monotone type with elliptic superregularization method, with applications to hyperbolic problems with discontinuous nonlinearity.
In this paper, we study a semilinear wave equation of the form
where \(g:[0,\pi]\times\mathbb{R}\times\mathbb{R} \rightarrow\mathbb{R}\) is a possibly discontinuous function in the third variable to be specified later. To seek a weak solution of the problem (1.1), we consider the corresponding semilinear equation
in a Hilbert space \(L^{2}(\Omega)\), where L is a densely defined closed linear operator with an infinite dimensional kernel and N is a bounded upper semicontinuous multivalued operator of monotone type.
Using the BerkovitsTienari degree theory for operators of the type \(L+N\), we first establish a continuation theorem for the semilinear equation (1.2), connecting to a reference map having nonzero degree. Actually, the linear operator L modeled by the differential operator \(\partial^{2}/\partial{t^{2}}\partial^{2}/\partial{x^{2}}\) and the Nemytskii operator N generated by certain nonlinearity g, have the desired properties for the use of the topological degree; see [8, 11, 12].
Next, we prove our main results on the solvability of the problem (1.1), by applying the continuation theorem to the reference maps \(L+P\) and \(L+cI\), where P denotes the orthogonal projection onto the kernel of L, I denotes the identity operator, and âˆ’c is a positive regular value of L. Thus, spectral theory for densely defined closed linear operators plays a decisive role in the study of semilinear equations. Analogous and related results were given in [9, 10]; see also [7, 8].
2 Continuation theorem
Let H be a real Hilbert space with an inner product \((\cdot,\cdot)\). Given a nonempty subset Î© of H, let Î©Ì… and âˆ‚Î© denote the closure and the boundary of Î© in H, respectively. Let \(B_{r}(u)\) denote the open ball in H of radius \(r>0\) centered at u. The symbol â†’ (â‡€) stands for strong (weak) convergence.
Definition 2.1
A multivalued operator \(F: \Omega\subset H \rightarrow2^{H}\) is said to be:

(1)
upper semicontinuous if the set \(F^{1}(A)=\{u\in \Omega \mid Fu\cap A\ne\emptyset\} \) is closed for all closed sets A in H;

(2)
weakly upper semicontinuous if \(F^{1}(A)\) is closed for all weakly closed sets A in H;

(3)
bounded if it maps bounded sets into bounded sets;

(4)
compact if it is upper semicontinuous and the image of any bounded set is relatively compact.
Definition 2.2
A multivalued operator \(F:\Omega\subset H\rightarrow2^{H} \backslash \emptyset\) is said to be:

(1)
of class \((S_{+})\) if for any sequence \((u_{n})\) in Î© and for any sequence \((w_{n})\) in H with \(w_{n}\in Fu_{n}\) such that \(u_{n}\rightharpoonup u\) in H and
$$\limsup_{n\rightarrow\infty}{(w_{n}, u_{n}u)}\leq0, $$we have \(u_{n}\rightarrow u\) in H;

(2)
pseudomonotone if for any sequence \((u_{n})\) in Î© and for any sequence \((w_{n})\) in H with \(w_{n}\in Fu_{n}\) such that \(u_{n}\rightharpoonup u\) in H and
$$\limsup_{n\rightarrow\infty}{( w_{n}, u_{n}u )}\leq0, $$we have \(\lim_{n\rightarrow\infty}{(w_{n}, u_{n}u)}=0\) and if \(u\in \overline{\Omega}\) and \(w_{j}\rightharpoonup w\) in H for some subsequence \((w_{j})\) of \((w_{n})\), then \(w\in Fu\);

(3)
quasimonotone if for any sequence \((u_{n})\) in Î© and for any sequence \((w_{n})\) in H with \(w_{n}\in Fu_{n}\) such that \(u_{n}\rightharpoonup u\) in H, we have
$$\liminf_{n\rightarrow\infty}{(w_{n}, u_{n}u )}\geq0; $$ 
(4)
monotone if
$$(wy, uv)\geq0\quad \text{for all } u,v\in\Omega, w\in Fu \text{ and } y\in Fv. $$
Note that the class of \((S_{+})\)operators is invariant under quasimonotone perturbations and each pseudomonotone bounded operator is quasimonotone. Moreover, if \(F:H\to2^{H}\) is monotone, then it is pseudomonotone.
Let \((H,(\cdot,\cdot))\) be a real separable Hilbert space. Suppose that \(L: D(L)\subset H \to H\) is a densely defined closed linear operator with
and \(K: \operatorname{Im} L \to\operatorname{Im} L\), the inverse of the restriction of L to \(\operatorname{Im} L\cap D(L)\), is compact. Let P and Q be the orthogonal projections onto KerL and ImL, respectively.
In what follows, we always assume that all multivalued operators considered have nonempty closed convex values.
For any open set G in H, let
Following Berkovits and Tienari [10], one can define a topological degree function for the class \(\mathcal{F}_{G}\).
Definition 2.3
Let G be a bounded open set in H and \(L+N\in\mathcal{F}_{G}\). If \(h\notin(L+N)(\partial G \cap D(L))\), then the degree d is defined in terms of the LeraySchauder degree \({d}_{\mathrm{LS}}\) as follows:
where \(F_{\lambda}=I+(KQ+\lambda\Psi^{2}P)N\) and \(h_{\lambda }=(KQ+\lambda\Psi^{2}P)h\). Here \(\Psi: \operatorname{Ker} L \to\operatorname{Ker} L\) is a compact selfadjoint linear injection.
The degree function d defined above satisfies the usual basic properties, as in [10].
Lemma 2.4
Suppose that G is any bounded open subset of X, \(L+N\in\mathcal{F}_{G}\), and \(h\notin(L+N)(\partial G \cap D(L))\). Then the degree d has the following properties:

(a)
(Existence) If \(d (L+N,G,h)\neq0\), then \(h\in (L+N)(G\cap D(L))\).

(b)
(Additivity) If \(G_{1}\) and \(G_{2}\) are disjoint open subsets of G such that \(h\notin(L+N)(\overline{G}\backslash(G_{1}\cup G_{2}))\), then we have
$$d (L+N,G,h)= d (L+N,G_{1},h)+ d (L+N,G_{2},h). $$ 
(c)
(Homotopy invariance) Suppose that \(N, \hat {N}:[0,1]\times\overline{G}\rightarrow2^{H}\) are bounded weakly upper semicontinuous homotopies with nonempty closed convex values such that NÌ‚ is of class \((S_{+})\) and \(PN=P\hat{N}\). If \(h:[0,1]\rightarrow H\) is a continuous path in H such that
$$h(t)\notin\bigl(L+N(t,\cdot)\bigr) \bigl(\partial{G}\cap D(L)\bigr)\quad \textit{for all } t\in[0,1], $$then \(d (L+N(t,\cdot), G, h(t))\) is constant for all \(t\in[0,1]\).

(d)
(Normalization) If \(h\in(L+P)(G\cap D(L))\), then we have \(d (L+P,G,h)=1\).
Here, the homotopy NÌ‚ is of class \((S_{+})\) in the sense that for any sequence \((t_{n},u_{n})\) in \([0,1]\times\overline{G}\) and for any sequence \((a_{n})\) in H with \(a_{n}\in\hat{N}(t_{n},u_{n})\) such that \(u_{n}\rightharpoonup u\) in H and
we have \(u_{n}\rightarrow u\) in H.
Definition 2.5
A singlevalued operator \(L+N_{0}\in\mathcal{F}_{H}\) is called a reference map if it is injective and \(d (L+N_{0},G,h) \ne0\) for any bounded open set \(G\subset H\) with \(h\in(L+N_{0})(G\cap D(L))\).
The typical examples of a reference map are \(L+P\), \(L+cI\) with \(c\notin\sigma(L)\), and \(L+cI+P_{c}\) with \(c\in\sigma(L)\), where \(c>0\) and \(P_{c}\) denotes the orthogonal projection onto \(\operatorname{Ker} (L+cI)\); see [9].
As a key tool, we need the following continuation result on pseudomonotone operators, which follows from the homotopy invariance of the above degree. For the singlevalued case, we refer to Theorem 7 of [9].
Theorem 2.6
Let G be a bounded open set in H with \(0\in G\). Suppose that \(L+N_{0}\in\mathcal{F}_{H}\) is a reference map with \(0\in (L+N_{0})(G\cap D(L))\) and \(N:\overline{G}\to2^{H}\) is a pseudomonotone bounded weakly upper semicontinuous operator. If the following boundary condition holds for a given \(h\in H\):
then the equation \(h\in(L+N)u\) has a solution in \(\overline{G}\cap D(L)\).
Proof
If \(h\in(L+N)(\partial G\cap D(L))\), there is nothing to prove. We now suppose that \(h\notin(L+N)(\partial G\cap D(L))\). If the condition (2.1) holds for the case \(L+N\in\mathcal{F}_{G}\), Lemma 2.4 and Definition 2.5 imply that
To prove our pseudomonotone case, we set
It is clear that \(L+N_{\varepsilon}\in\mathcal{F}_{G}\). Notice that \((L+N)(\overline{G}\cap D(L))\) is closed in H since the graph of L is weakly closed and N is pseudomonotone. Indeed, let \((u_{n})\) be a sequence in \(\overline{G}\cap D(L)\) such that
where \(a_{n}\in Nu_{n}\). Without loss of generality, we may suppose that \(u_{n}\rightharpoonup u\) in H and \(a_{n}\rightharpoonup a \) in H. Since the graph of L is weakly closed, we have \(u\in D(L)\) and \(Lu=ya\). Since the operator K is compact, the sequence \((u_{n})\) has a convergent subsequence \((u_{n_{k}})\), say \(u_{n_{k}}\to v\) for some \(v\in \overline{G}\). By the uniqueness of weak limit, we have \(u=v\in\overline{G}\). Since the operator N is pseudomonotone, \(u_{n_{k}}\to u\) and \(a_{n_{k}}\rightharpoonup a \) imply \(a\in Nu\). Therefore, we get
We have just shown that \((L+N)(\overline{G}\cap D(L))\) is closed in H.
Now there are two cases to consider. First, we assume that there exists a positive number \(\varepsilon_{0}\) such that the following condition holds for every \(\varepsilon\in (0,\varepsilon_{0})\):
Applying (2.2) with \(N_{\varepsilon}\) instead of N, we get
Let \((\varepsilon_{n})\) be a sequence in \((0,\varepsilon_{0})\) with \(\varepsilon_{n}\to0\). In view of Lemma 2.4, let \((u_{n})\) be a sequence in \(G\cap D(L)\) such that
where \(a_{n}\in Nu_{n}\). Then we have \(Lu_{n}+a_{n}\to h\). Since \((L+N)(\overline{G}\cap D(L))\) is closed in H, this implies that
Next, we assume that there are sequences \((\varepsilon_{n})\) in \((0,\infty)\) with \(\varepsilon_{n}\to0\), \((t_{n})\) in \([0,1)\), \((u_{n})\) in \(\partial G\cap D(L)\), and \((a_{n})\) with \(a_{n}\in Nu_{n}\) such that
Passing to subsequences if necessary, we may suppose that
Taking the transformation \(QKQ\) at both sides of (2.3), we get \(Qu_{n}\to Qu\), by the compactness of the operator K. Actually, we have \(t=1\). In fact, assume that \(t\ne1\). Since \(PN_{0}=P\hat{N}_{0}\) for some bounded \((S_{+})\)operator \(\hat{N}_{0}\) and N is bounded and quasimonotone, we have by (2.3)
Since \(\hat{N}_{0}\) is of class \((S_{+})\) and N is pseudomonotone, we obtain \(u_{n}\to u\in\partial G\) and \(a\in Nu\). Moreover, since the graph of L is weakly closed, we obtain from (2.3) that
which contradicts the hypothesis (2.1). Thus, we have just shown that \(t=1\). Therefore, it follows from (2.3) that \(Lu_{n}+a_{n}\to h\), which implies as above
In all possible cases, we conclude that \(h\in(L+N)(\overline{G}\cap D(L))\). This completes the proof.â€ƒâ–¡
Remark 2.7
We point out that in Theorem 2.6 the convexity assumption on the open set G can be relaxed, in comparison to Theorem 7 of [9]. In fact, we did not need the weak closedness of the set G in the whole proof.
3 Main results
This section is devoted to the solvability of a semilinear wave equation with discontinuous nonlinearity, based on the continuation theorem on pseudomonotone operators stated in the previous section.
We consider the semilinear wave equation of the form
where
Let \(\Omega=(0,\pi)\times(0,2\pi)\) and let \(H=L^{2}(\Omega)\) be the real Hilbert space with usual inner product \((\cdot,\cdot)\) and norm \(\ \cdot\\). Let \(\varphi_{nm}(x,t)=\pi^{1} \sin(nx) \exp(imt)\) for \(n\in\mathbb{N}\) and \(m\in\mathbb{Z}\). Each \(u\in H\) has a representation
where \(u_{nm}=(u,\varphi_{nm})\).
We define a linear operator \(L: D(L)\subset H\to H\) by
where
Then \(D(L)\) is a dense subspace of H and L is a selfadjoint operator. Moreover, we have \(\operatorname{Ker} L = \operatorname{span} \{\varphi_{nn}, \varphi_{n,n} : n\in\mathbb{N} \} \), \(\operatorname{Im} L= (\operatorname{Ker} L)^{\perp}\) and L has a pure point spectrum \(\sigma(L)\) of isolated eigenvalues:
Let \(L_{0}\) be the restriction of L to the subspace ImL. Then the operator \(L_{0}: \operatorname{Im} L\cap D(L)\to\operatorname{Im} L\) is a bijection with the inverse \(L_{0}^{1} : \operatorname{Im} L\to\operatorname{Im} L\) given by
where \(\Gamma=\{(n,m)\in\mathbb{N}\times\mathbb{Z} : n^{2}\ne m^{2}\}\). Note that the spectrum of the bounded inverse operator \(L_{0}^{1}\) has no limit point except 0 and \(\operatorname{dim} \operatorname{Ker} (L_{0}^{1}\lambda I)\) is finite for every \(\lambda\in\sigma(L_{0}^{1})\) with \(\lambda\ne0\). This implies that \(L_{0}^{1}\) is compact. See e.g., [7, 12].
Suppose that \(g:\Omega\times\mathbb{R}\times\mathbb{R}\rightarrow \mathbb {R}\) is 2Ï€periodic in the second variable t such that

(g1)
gÌ… and \(\underline{g}\) are superpositionally measurable, that is, \(\overline{g}(\cdot,\cdot,u(\cdot,\cdot))\) and \(\underline {g}(\cdot ,\cdot,u(\cdot,\cdot))\) are measurable on Î© for any measurable function \(u:\Omega \rightarrow\mathbb{R}\);

(g2)
g satisfies the growth condition:
$$ \bigl\vert g(x,t,s)\bigr\vert \leq\thetas + h_{0}(x,t) \quad \text{for almost all } (x,t)\in\Omega \text{ and all } s\in\mathbb{R}, $$where Î¸ is a positive constant and \(h_{0}\in H\) is nonnegative;

(g3)
g is nondecreasing in the third variable s, that is,
$$ \bigl(g(x,t,s) g(x,t,\eta)\bigr) (s\eta) \ge0\quad \text{for almost all } (x,t)\in\Omega \text{ and all } s,\eta\in\mathbb{R}. $$
We define a multivalued operator \(N: H\rightarrow2^{H}\) by setting
Under assumptions (g1) and (g2), the multivalued operator N is bounded, upper semicontinuous, and Nu is nonempty, closed, and convex for every \(u\in H\); see Theorem 1.1 of [11]. Under additional assumption (g3), the operator N is monotone and hence pseudomonotone.
Definition 3.1
A point \(u\in H\) is said to be a weak solution of the problem (3.1) if there exists a point \(w\in Nu\) such that
where \(C^{2}\) denotes the space of twice continuously differentiable functions \(v: \overline{\Omega}\to\mathbb{R}\) such that
In view of (3.2) and (3.3), \(u\in H\) is a weak solution of the problem (3.1) if and only if \(u\in D(L)\) is a solution of the operator equation
Notice that \(\lambda_{0}=3\) is the first positive eigenvalue of âˆ’L and
For our aim, we first present a multivalued version of Proposition A.6 of [1]. For this, we need the Cauchy inequality for every positive number Îµ:
Lemma 3.2
Suppose that \(g:\Omega\times\mathbb{R}\times\mathbb{R}\rightarrow \mathbb{R}\) satisfies the conditions (g1), (g3) and the growth condition (g2) with \(\theta<3\). Then there exists a positive constant Î³ with \(\theta<\gamma<3\) such that the following relation holds for any given \(h,v\in H\) with \(h\in Nv\):
where \(C(h,v)\) is a positive constant that depends on h and v.
Proof
Let \(h,v\in H\) with \(h\in Nv\) be given. Note by (g3) that N is monotone. For all \(u\in H\) and \(w\in Nu\), using the growth condition (g2) and the HÃ¶lder inequality gives
Applying the Cauchy inequality (3.6) with \(\varepsilon=1/2(1/\theta1/3)\) and \(a=\w\\), we obtain
Setting \(\gamma:=[1/2(1/\theta+1/3)]^{1}\), it follows from (3.7) that \(\theta<\gamma<3\) and
where \(C(h,v)\) denotes a positive constant depending on h and v. This completes the proof.â€ƒâ–¡
Now we prove that (3.1) has a weak solution, based on the use of reference map \(L+P\). We give a substantially simpler and more direct proof in a precise manner, in comparison to Corollary 5.2 of [10].
Theorem 3.3
Suppose that \(g:\Omega\times\mathbb{R}\times\mathbb{R}\rightarrow \mathbb {R}\) satisfies the conditions (g1), (g3) and the growth condition (g2) with \(\theta<3\). Further, suppose that there is a positive constant Î· and a measurable function \(h_{1}\in H\) such that
Then (3.1) has a weak solution for each \(h\in H\).
Proof
Let \(h\in H\) be arbitrary but fixed. It follows from condition (3.8) that
Then the monotone operator N is surjective and so there exists a point \(v\in H\) such that \(h\in Nv\). We first show that there is a positive constant R such that
In fact, assume to the contrary that there are sequences \((u_{n})\) in \(D(L)\) with \(\u_{n}\\to\infty\), \((t_{n})\) in the open interval \((0,1)\) with \(t_{n}\to t\), and \((w_{n})\) in H with \(w_{n}\in Nu_{n}\) such that
Taking the orthogonal projection Q of (3.11), we get the estimate
According to Lemma 3.2, there is a positive constant Î³ with \(\theta<\gamma<3\) such that
where \(C(h,v)\) is a positive constant depending on h and v. For all \(n\in\mathbb{N}\), we obtain from (3.5), (3.11), and (3.12)
and hence
and therefore
In view of \(\gamma<3\), the sets \(\{t_{n}\w_{n}\\}\) and \(\{(1t_{n}) \Pu_{n}\^{2} \}\) are bounded. Since \(\{Lu_{n}\}\) is bounded by (3.12), we see that \(\{Qu_{n}\}\) is also bounded. From \(\Pu_{n}\\to\infty\) and \((1t_{n})\Pu_{n}\\to0\), it follows that \(t_{n}\to1\). This implies that \(\{w_{n}\}\) is bounded. On the other hand, it follows from (3.9) that \(\{w_{n}\}\) is unbounded, which is a contradiction. Thus, the assertion (3.10) holds for all \(t\in[0,1)\) because the reference map \(L+P\) is injective with \(0\in(L+P)(B_{R}(0)\cap D(L))\). Applying Theorem 2.6 with \(G=B_{R}(0)\) and \(N_{0}=P\), we conclude that the equation \(h\in Lu+Nu\) has a solution u in \(D(L)\). This completes the proof.â€ƒâ–¡
Next, we are concerned with the solvability of the given equation (3.4) with nonresonance condition. The singlevalued case was discussed in [9].
Theorem 3.4
Suppose that \(g:\Omega\times\mathbb{R}\times\mathbb{R}\rightarrow \mathbb {R}\) satisfies the conditions (g1), (g2), and (g3). Let Ï„ be a positive number such that \(\tau\notin\sigma(L)\) and
where \(\lambda_{N}\) and \(\lambda_{N+1}\) are two consecutive eigenvalues of L and a, b are real numbers. Suppose that there are a positive constant R and a nonnegative function \(h_{2}\in H\) such that the following inequalities hold for almost all \((x,t)\in \Omega\):
Then (3.4) has a solution u in \(D(L)\) for any \(h\in H\).
Proof
Let Ï„ be a positive number such that \(\tau\notin\sigma(L)\) and \(a<\tau<b\). For almost all \((x,t)\in\Omega\) and all \(s\in\mathbb{R}\), we have by (3.14)
where \(\underline{g}(x,t,s) \le w(x,t) \le\overline{g}(x,t,s)\). If we set \(\eta_{\tau}=\max \{ a+\tau, b+\tau\}\), then we have by (3.13)
According to (3.15), we can choose a positive constant Î² such that
Now, let h be any element of H. We first prove that there is a positive constant \(R_{1} \ge R\) such that
We argue by contradiction. Assume that there are sequences \((u_{n})\) in \(D(L)\) with \(\u_{n}\\to \infty\), \((t_{n})\) in \((0,1)\) with \(t_{n}\to t\), and \((w_{n})\) in H with \(w_{n}\in Nu_{n}\) such that
Due to \(\tau\notin\sigma(L)\), it is well known that
For all \(n\in\mathbb{N}\), we have by (3.16) and (3.18)
and hence
which is impossible because the lefthand side of (3.19) tends to infinity. Moreover, (3.17) holds for \(t=0\) since the linear operator \(L+\tau I\) is a reference map. Thus, the boundary condition (3.17) holds for all \(t\in[0,1)\). Applying Theorem 2.6 with \(G=B_{R_{1}}(0)\) and \(N_{0}=\tau I\), the equation \(h\in Lu+Nu\) has a solution u in \(D(L)\). This completes the proof.â€ƒâ–¡
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Acknowledgements
This work was supported by Sungkyun Research Fund, Sungkyunkwan University, 2014.
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ISK conceived of the study and drafted the manuscript. JHB and SJH participated in the coordination. All authors approved the final manuscript.
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Kim, IS., Bae, JH. & Hong, SJ. Periodic solutions of semilinear wave equations with discontinuous nonlinearities. Bound Value Probl 2015, 197 (2015). https://doi.org/10.1186/s136610150462x
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DOI: https://doi.org/10.1186/s136610150462x