Periodic solutions of two-dimensional wave equations with x-dependent coefficients and Sturm–Liouville boundary conditions

Abstract

Abstract This paper considers the periodic solutions of a two-dimensional nonlinear wave equation with x-dependent coefficients v ( x ) y t t − ( v ( x ) y x 1 ) x 1 − ( v ( x ) y x 2 ) x 2 + β y + g ( y ) = f ( x , t ) , subject to the Sturm–Liouville boundary conditions a 11 y ( 0 , x 2 , t ) − b 11 y x 1 ( 0 , x 2 , t ) = 0 , a 12 y ( π , x 2 , t ) + b 12 y x 1 ( π , x 2 , t ) = 0 , a 21 y ( x 1 , 0 , t ) − b 21 y x 2 ( x 1 , 0 , t ) = 0 , a 22 y ( x 1 , π , t ) + b 22 y x 2 ( x 1 , π , t ) = 0 , where x = (x 1, x 2) ∈ (0, π) × (0, π), t ∈ R and a i j 2 + b i j 2 ≠ 0 for i, j = 1, 2. Such a model arises from the forced vibrations of a nonhomogeneous membrane and the propagation of waves in nonhomogeneous media. By using the duality principles of Brézis (1983 Bull. Am. Math. Soc. 8 409–26), we first set up the theoretical framework and give an abstract theorem for the existence of periodic solution. Then, by analysing the spectral asymptotic behaviours of the weighted Sturm–Liouville problem and classifying the boundary conditions, we obtain the fundamental properties of the variable coefficients wave operator under appropriate assumptions for different class of boundary conditions. Finally, based on these properties and with the help of the abstract theorem, we establish some results on the existence and regularity of periodic solutions. To the best of our knowledge, the results are entirely new, and this is the first time to obtain such results for the two-dimensional nonlinear wave equation with x-dependent coefficients.