Wide oscillation finite time blow up for solutions to nonlinear fourth-order differential equations (Q1944700)

The paper studies in finite time oscillating blow up solutions to the fourth-order nonlinear ordinary differential equation \[ w''''(s)+k w''(s)+f(w(s))=0, \tag{1} \] where \(k\in\mathbb R\) and \(f\) is a locally Lipschitz function. ``The primary purpose of the paper is to contribute to a better understanding of the qualitative properties of solutions to (1) when the nonlinearity \(f\) satisfies \[ f\in\mathrm{Lip}_{\mathrm{loc}}(\mathbb R), \quad f(t)t>0\text{ for }t\in\mathbb R\setminus \{0\}. \] Further assumptions on \(f\) are needed in the sequel, although the prototype nonlinearity investigated here is \[ f(t)=\alpha|t|^{q-1}t+|t|^{p-1}t,\quad p>q\geq 1,~ \alpha\geq 0. \] The second, and probably most ambitious, purpose of the paper is to connect the phenomena which hold for (1) with several classes of fourth-order partial differential equations.'' The authors also provide several open problems and numerical results which describe this oscillating blow up.