A priori bounds and stability in the retarded nonlinear wave equation
The object of the authors' work is to give sufficient conditions to ensure that the zero solution of certain generalizations of \(u_{tt}=(a(x)g(u_ x(t-h(t),x)))_ x+(f(u_ x))_{xt}\) is asymptotically stable and to study limit sets. Here, \(0\leq h(t)\leq h\) say, \(a(x)>0\), \(xg(x)>0\) if \(x\neq 0\), \(f'(x)>0\) and \(g'(x)>0\). They impose the boundary conditions \(u(t,0)=u(t,1)=0\). Among other conditions, it is crucial to have \[ \int^ 1_ 0(a(x)g(u_ x))_ xu_ t dx=- \int^ 1_ 0(a(x)g(u_ x))u_{tx} dx. \]
sufficient conditions for asymptotic stability
scientific article; zbMATH DE number 62827
scientific article