Global existence for one-dimensional hyperbolic equation with power type nonlinearity. (Q2059003)

The author considers the equation \(\partial_t^2u(t,x)+\partial_tu(t,x)-\partial_x(a(x)\partial_x u(t,x))=|u(t,x)|^p\), \(t>0\), \(x\in\mathbb{R}\), where \(3<p<5\), \(0<m<a(x)<M<+\infty\) and \(\frac{M-m}{m}<\frac{2(p-3)}{5-p}\). For sufficiently small initial conditions satisfying the special condition in the infinity, the Cauchy problem to the equation admits the unique global weak solution \(u(t,x)\). Furthemore, there are proved the estimates of \(u(t,x)\), which give its behavior for \(t\) tending to infinity.