We consider the 1D nonlinear Schrodinger equation (NLS) with focusing point nonlinearity, \begin{document}$ \begin{equation} i\partial_t\psi + \partial_x^2\psi \delta|\psi|^{p-1}\psi = 0, \;\;\;\;\;\;(0.1)\end{equation} $\end{document} where {\delta} {\delta}(x) is delta function supported at origin. In L^2 supercritical setting p>3 , we construct self-similar blow-up solutions belonging to ene...

The present paper is concerned with the Cauchy problem { ∂tu = ∆u + u in R × (0,∞), u(x, 0) = u0(x) ≥ 0 in R , with p,m > 1. A solution u with bounded initial data is said to blow up at a finite time T if lim supt↗T ‖u(t)‖L∞(RN ) = ∞. For N ≥ 3 we obtain, in a certain range of values of p, weak solutions which blow up at several times and become bounded in intervals between these blow-up times....

In this paper, I consider nonlinear parabolic problems under nonlinear boundary conditions. I establish respectively the conditions on nonlinearities to guarantee that ( , ) u x t exists globally or blows up at some finite time. If blow-up occurs, an upper bound for the blow-up time is derived, under somewhat more restrictive conditions, lower bounds for the blow-up time are also derived.