In the present work, we establish an optimal estimate for the electric potential difference between closely adjacent spherical perfect conductors in n dimensional space (n ≥ 2). This result indicates that electric fields blow up as a pair of spherical perfect conductors approach each other, and provides the lower bound with the optimal blowup rate which was recently established by Bao, Li and Y...

We consider the 1D nonlinear Schrödinger equation (NLS) with focusing point nonlinearity, (0.1) i∂tψ + ∂ 2 xψ + δ|ψ|p−1ψ = 0, where δ = δ(x) is the delta function supported at the origin. In the L supercritical setting p > 3, we construct self-similar blow-up solutions belonging to the energy space Lx ∩Ḣ x. This is reduced to finding outgoing solutions of a certain stationary profile equation. ...

A Kirchhoff equation type with memory term competing a logarithmic source is considered. By using potential well theory, we obtained the global existence of solution for initial data in stability set created from Nehari Manifold and prove blow up results instability set.

For some deterministic nonlinear PDEs on the torus whose solutions may blow up in finite time, we show that, under suitable conditions term, blow-up is delayed by multiplicative noise of transport type a certain scaling limit. The main result applied to 3D Keller–Segel, Fisher–KPP, and 2D Kuramoto–Sivashinsky equations, yielding long-time existence for large initial data with high probability.

We consider complex-valued solutions of the three-dimensional NavierStokes system without external forcing on R. We show that there exists an open set in the space of 10-parameter families of initial conditions such that for each family from this set there are values of parameters for which the solution develops blow up in finite time.

The nonemptiness problem for nondeterministic automata on infinite words can be reduced to a sequence of reachability queries. The length of a shortest witness to the nonemptiness is then polynomial in the automaton. Nonemptiness algorithms for alternating automata translate them to nondeterministic automata. The exponential blow-up that the translation involves is justified by lower bounds for...