- Research
- Open Access
Periodic solutions of semilinear wave equations with discontinuous nonlinearities
- In-Sook Kim^{1}Email author,
- Jung-Hyun Bae^{1} and
- Suk-Joon Hong^{1}
- Received: 4 September 2015
- Accepted: 21 October 2015
- Published: 29 October 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 Berkovits-Tienari 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.
Keywords
- semilinear wave equation
- multi-valued operator
- pseudomonotone operator
- degree theory
MSC
- 35L05
- 35R05
- 47H04
- 47H05
- 47H11
1 Introduction
The study of nonlinear wave equations has been developed in various ways of approach by many researchers; for instance, by Brézis and Nirenberg [1, 2], Rabinowitz [3], Berkovits and Mustonen [4–6]. To find periodic solutions of a nonlinear wave equation, Mawhin and Willem [7] established a Leray-Schauder 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 multi-valued operators of monotone type with elliptic super-regularization method, with applications to hyperbolic problems with discontinuous nonlinearity.
Using the Berkovits-Tienari 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 −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 Ω of H, let Ω̅ and ∂Ω denote the closure and the boundary of Ω in H, respectively. Let \(B_{r}(u)\) denote the open ball in H of radius \(r>0\) centered at u. The symbol → (⇀) stands for strong (weak) convergence.
Definition 2.1
- (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
- (1)of class \((S_{+})\) if for any sequence \((u_{n})\) in Ω and for any sequence \((w_{n})\) in H with \(w_{n}\in Fu_{n}\) such that \(u_{n}\rightharpoonup u\) in H andwe have \(u_{n}\rightarrow u\) in H;$$\limsup_{n\rightarrow\infty}{(w_{n}, u_{n}-u)}\leq0, $$
- (2)pseudomonotone if for any sequence \((u_{n})\) in Ω and for any sequence \((w_{n})\) in H with \(w_{n}\in Fu_{n}\) such that \(u_{n}\rightharpoonup u\) in H andwe 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 \(w\in Fu\);$$\limsup_{n\rightarrow\infty}{( w_{n}, u_{n}-u )}\leq0, $$
- (3)quasimonotone if for any sequence \((u_{n})\) in Ω 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$$(w-y, u-v)\geq0\quad \text{for all } u,v\in\Omega, w\in Fu \text{ and } y\in Fv. $$
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.
In what follows, we always assume that all multi-valued operators considered have nonempty closed convex values.
Definition 2.3
The degree function d defined above satisfies the usual basic properties, as in [10].
Lemma 2.4
- (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 N̂ is of class \((S_{+})\) and \(PN=P\hat{N}\). If \(h:[0,1]\rightarrow H\) is a continuous path in H such thatthen \(d (L+N(t,\cdot), G, h(t))\) is constant for all \(t\in[0,1]\).$$h(t)\notin\bigl(L+N(t,\cdot)\bigr) \bigl(\partial{G}\cap D(L)\bigr)\quad \textit{for all } t\in[0,1], $$
- (d)
(Normalization) If \(h\in(L+P)(G\cap D(L))\), then we have \(d (L+P,G,h)=1\).
Definition 2.5
A single-valued 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 single-valued case, we refer to Theorem 7 of [9].
Theorem 2.6
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.
- (g1)
g̅ 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 Ω for any measurable function \(u:\Omega \rightarrow\mathbb{R}\);
- (g2)g satisfies the growth condition:where θ is a positive constant and \(h_{0}\in H\) is nonnegative;$$ \bigl\vert g(x,t,s)\bigr\vert \leq\theta|s| + h_{0}(x,t) \quad \text{for almost all } (x,t)\in\Omega \text{ and all } s\in\mathbb{R}, $$
- (g3)g is nondecreasing in the third variable s, that is,$$ \bigl(g(x,t,s)- g(x,t,\eta)\bigr) (s-\eta) \ge0\quad \text{for almost all } (x,t)\in\Omega \text{ and all } s,\eta\in\mathbb{R}. $$
Under assumptions (g1) and (g2), the multi-valued 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
Lemma 3.2
Proof
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
Proof
Next, we are concerned with the solvability of the given equation (3.4) with nonresonance condition. The single-valued case was discussed in [9].
Theorem 3.4
Proof
Declarations
Acknowledgements
This work was supported by Sungkyun Research Fund, Sungkyunkwan University, 2014.
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Authors’ Affiliations
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