Strong regularizing effect of a gradient term in the heat equation with the Hardy potential

We deal with the following parabolic problem,

     

Regional, Single Point, and Global Blow-Up for the Fourth-Order Porous Medium Type Equation with Source

Blow-up behaviour for the fourth-order quasilinear porous medium equation with source,ut=-（｜u｜^nu）xxxx＋｜u｜^p-1u in R×R＋,where n 〉 0, p 〉 1, is studied. Countable and finite families of similarity blow-up patterns of the form us（x,t）=（T-t）^-1/p-1f（y）,where y=x/（T-t）^β＇ β=p-（n＋1）/4（p-1）,which blow-up as t → T^- 〈∞ are described. These solutions explain key features of regional （for p = n＋1）, single point （for p 〉 n＋1）, and global （for p ∈ （1,n＋1））blowup. The concepts and various variational, bifurcation, and numerical approaches for revealing the structure and multiplicities of such blow-up patterns are presented.     

Dispersive Blow-Up Ⅱ.Schr(o)dinger-Type Equations,Optical and Oceanic Rogue Waves

Addressed here is the occurrence of point singularities which owe to the fo-cusing of short or long waves,a phenomenon labeled dispersive blow-up.The context of this investigation is linear and nonlinear,strongly dispersive equations or systems of equa-tions.The present essay deals with linear and nonlinear Schr(o)dinger equations,a class of fractional order SchrSdinger equations and the linearized water wave equations,with and without surface tension. Commentary about how the results may bear upon the formation of rogue waves in fluid and optical environments is also included.     

Blow-up rate for critical nonlinear Schrödinger equation with Stark potential

We consider the blow-up solutions of the Cauchy problem for the critical nonlinear Schrödinger equation with Stark potential and the sharp lower and upper bounds of blow-up rate are established.     

Critical curves for fast diffusive polytropic filtration equations coupled through boundary

This article deals with the critical curve of a fast diffusive polytropic filtration system coupled at the boundary condition. By constructing the self-similar supersolution and subsolution, we obtain the critical global existence curve. The critical curve of Fujita type is conjectured with the aid of some new results.