We obtain some sufficient conditions under which solutions to a nonlinear parabolic equation of second order with nonlinear boundary conditions tend to zero or blow up in a finite time. We also give the asymptotic behavior of solutions which tend to zero as t→ ∞. Finally, we obtain the asymptotic behavior near the blow-up time of certain blow-up solutions and describe their blow-up set.

We consider the critical nonlinear Schrödinger equation iut = −∆u − |u| 4 N u with initial condition u(0, x) = u0 in dimension N . For u0 ∈ H1, local existence in time of solutions on an interval [0, T ) is known, and there exists finite time blow up solutions, that is u0 such that limt→T<+∞ |ux(t)|L2 = +∞. This is the smallest power in the nonlinearity for which blow up occurs, and is critical...

A class of stretched solutions of the equations for three-dimensional, incompressible, ideal magneto-hydrodynamics (MHD) is studied. In Elsasser variables, V ± = U ± B, these solutions have the form V ± = (v±, v± 3 ) where v± = v±(x, y, t) and v± 3 (x, y, z, t) = zγ±(x, y, t) + β±(x, y, t). Twodimensional partial differential equations for γ±, v± and β± are obtained that are valid in a tubular ...

we discuss the asymptotic behaviour of solutions for the nonlocal quasilinear hyperbolic problem of kirchhoff type [ u_{tt}-phi (x)||nabla u(t)||^{2}delta u+delta u_{t}=|u|^{a}u,, x in mathbb{r}^{n} ,,tgeq 0;,]with initial conditions $u(x,0) = u_0 (x)$ and $u_t(x,0) = u_1 (x)$, in the case where $n geq 3, ; delta geq 0$ and $(phi (x))^{-1} =g (x)$  is a positive function lying in $l^{n/2}(mathb...

In this paper we consider a system of equations that describes a class of mass-conserving aggregation phenomena, including gravitational collapse and bacterial chemotaxis. In spatial dimensions strictly larger than two, and under the assumptions of radial symmetry, it is known that this system has at least two stable mechanisms of singularity formation (see, e.g., Brenner M P et al 1999 Nonline...

Many audio and multimedia applications would benefit from the ability to classify and search for audio based on its characteristics. The audio analysis, search, and classification engine described here reduces sounds to perceptual and acoustical features. This lets users search or retrieve sounds by any one feature or a combination of them, by specifying previously learned classes based on thes...