On the nonlinear wave equation Utt−B(t,‖Ux‖2)Uxx=f(x,t,U,Ux,Ut) associated with the mixed homogeneous conditions

AbstractIn this paper we consider the following nonlinear wave equation (1)utt−Bt,‖ux‖2uxx=f(x,t,u,ux,ut),x∈Ω=(0,1),0<t<T, (2)ux(0,t)−h0u(0,t)=0,ux(1,t)+h1u(1,t)=0, (3)u(x,0)=ũ0(x),ut(x,0)=ũ1(x), where h0, h1 are given nonnegative constants and B, f, ũ0, ũ1 are given functions. In Eq. (1) the coefficient B(t,‖ux‖2) containing an integral ‖ux‖2=∫01|ux(x,t)|2dx. 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 B∈C2(R+2), B⩾b0>0, B1∈C1(R+2), B1⩾0, f∈C2(Ω×[0,∞)×R3) and f1∈C1(Ω×[0,∞)×R3) we obtain from the following equation utt−Bt,‖ux‖2+εB1t,‖ux‖2uxx=f(x,t,u,ux,ut)+εf1(x,t,u,ux,ut) associated to (2), (3) a weak solution uε(x,t) having an asymptotic expansion of order 2 in ε, for ε sufficiently small