Blow-up Kneser solutions to nonlinear higher-order differential equations
A solution \( u : (0,\infty) \to \mathbb{R}\) to the differential equation \[ u^{(n)}= f(t,u,\dots, u^{(n-1)}),\qquad n \geq 2, \tag{*} \] where \( f : (0, \infty)\times \mathbb{R}^ n \to \mathbb{R}\) is continuous in \(x_1,\dots,x_n\) for almost all \(t \in (0,\infty)\) and measurable in \(t\) for all \((x_1,\dots, x_n) \in \mathbb{R}^n,\) \(f(t, 0,\dots, 0) = 0, (-1)^n f (t, x_1,\ldots, x_n) x_1 \geq 0\) whenever \( t > 0\) and \((-1)^{ i-1}x_i x_1 \geq 0\), \(i = 1,\dots,n\), is said to be Kneser if \((-1)^i u^{(i)}(t) u (t)\geq 0\) for \(t >0\), \(i = 0, \ldots, n-1\). If \(\lim_{t \to 0} u (t) = \infty\), then \(u\) is called a blow-up solution. The author obtains sufficient conditions for the existence of blow-up Kneser solutions to (*) and global two-sided estimates on such solutions.
scientific article; zbMATH DE number 1758749