On the Existence of Solutions for Some Nondegenerate Nonlinear Wave Equations of Kirchhoff Type

摘要

Let Ω be a bounded domain in #x211D;n with a smooth boundary Γ. In this work we study the existence of solutions for the following boundary value problem: $$begin{gathered}frac{{partial ^2 y}}{{partial t^2 }} - Mleft( {int_Omega {left| {nabla y} right|^2 dx} } right)Delta y - frac{partial }{{partial t}}Delta y = fleft( y right)inQ = Omega times left( {0,infty } right), hfill y = 0insum _0 = Gamma _0 times left( {0,infty} right), hfill Mleft( {int_Omega {left| {nabla y} right|^2 dx}} right)frac{{partial y}}{{partial nu }} + frac{partial }{{partial t}}left( {frac{{partial y}}{{partial nu }}} right) = ginsum _0 = Gamma _0 times left( {0,infty } right), hfill yleft( 0 right) = y_0 frac{partial }{{partial t}}left( 0 right) = y1,inOmega , hfill end{gathered} $$ where M is a C 1-function such that M(λ) ⩾ λ0 > 0 for every λ ⩾ 0 and f(y) = |y|α y for α ⩾ 0.