On the nonlinear wave equation utt−B(t,||u||2,||ux||2,||ut||2)uxx=f(x,t,u,ux,ut,||u||2,||ux||2,||ut||2) associated with the mixed homogeneous conditions Original Research Article

In this paper we consider the following nonlinear wave equation: equation(1) View the MathML source equation(2) View the MathML source equation(3) View the MathML source where h0>0,h1⩾0 are given constants and View the MathML source are given functions. In Eq. (1), the nonlinear terms View the MathML source depending on the integrals View the MathML source and View the MathML source. In this paper we associate with problem (1)–(3) a linear recursive scheme for which the existence of a local and unique solution is proved by using standard compactness argument. In case of View the MathML source and f1∈CN([0,1]×R+×R3×R+3) we obtain from the following equation utt−[B(t,||u||2,||ux||2,||ut||2)+εB1(t,||u||2,||ux||2,||ut||2)]uxx=f(x,t,u,ux,ut,||u||2,||ux||2,||ut||2)+εf1(x,t,u,ux,ut,||u||2,||ux||2,||ut||2) associated to (2), (3) a weak solution uε(x,t) having an asymptotic expansion of order N+1 in ε, for ε sufficiently small.