Uniqueness of large solutions for non-monotone nonlinearities
The paper is concerned with the autonomous nonlinear second-order differential equation: \[ u''=f(u),\quad 0\leq t<T \] with the condition \(u'(0)=0\). More precisely, the authors investigate the existence and uniqueness of a large solution, i.e., a positive solution that explodes at the time \(T\): \(u(T)=+\infty\). The authors starts by estimating the maximum interval of existence and its bound \(T_{\max}\) by proving important properties of \(T_{\max}\). With further assumptions on \(T_{\max}\) together with \(f(0)=0\), \(f(u)>0\), \(f'(u)\geq 0\) for positive \(u\), existence and uniqueness of a large solution (among others) is established. The originality of the paper resides in that no monotonicity assumption on the nonlinearity \(f\) is required.
scientific article; zbMATH DE number 7030184
scientific article