On the Generalized Porous Medium Equation in Fourier-Besov Spaces

We study a kind of generalized porous medium equation with fractional Laplacian and abstract pressure term. For a large class of equations corresponding to the form: $u_t+\nu \Lambda^{\beta}u=\nabla\cdot(u\nabla Pu)$, we get their local well-posedness in Fourier-Besov spaces for large initial data. If the initial data is small, then the solution becomes global. Furthermore, we prove a blowup criterion for the solutions.     

三维空间中带非线性阻尼项的等熵欧拉方程组轴对称解的爆破

研究了三维空间中带非线性阻尼项的可压缩等熵欧拉方程组Dirichlet初边值问题.采用泛函方法,定义几种不同的泛函,当初始速度足够大时分别得到了经典解在某一时间内必定爆破的结论.由于出现了非线性阻尼项,较之线性阻尼的情形,经典解爆破的难度随之增加.     

The Nonlinear Schr¨odinger Equations with Combined Nonlinearities of Power-Type and Hartree-Type

The primary goal of this paper is to present a comprehensive study of the nonlinear Schr?dinger equations with combined nonlinearities of the power-type and Hartree-type. Under certain structural conditions, the authors are able to provide a complete picture of how the nonlinear Schr?dinger equations with combined nonlinearities interact in the given energy space. The method used in the paper is based upon the Morawetz estimates and perturbation principles.     

Blow‐up behavior of the Solution for the problem in a subdiffusive mediums

The diffusion problem in a subdiffusive medium is formulated by using the fractional differential operator. In this paper, we consider a fractional differential equation with concentrated source. The existence of the solution in a finite time is given. The finite time blow‐up criteria for the solution of the problem is established, and the location of the blow‐up point is investigated.     

On dynamics of the system of two coupled nonlinear Schrödinger in

In this paper, we give a new proof of the scattering and blow‐up theory of the two coupled nonlinear Schrödinger system via establishing the corresponding interaction Morawetz estimate and scattering criterion. The method of this paper simplifies the proof in Xu, and the result of the paper improves the result in Xu.     

Nonexistence of Type Ⅱ Blowup for Heat Equation with Exponential Nonlinearity

This paper deals with the blowup behavior of the radially symmetric solution of the nonlinear heat equation ut = ?u + e~u in R~N. The authors show the nonexistence of type II blowup under radial symmetric case in the lower supercritical range 3 ≤ N ≤ 9,and give a sufficient condition for the occurrence of type I blowup. The result extends that of Fila and Pulkkinen(2008) in a finite ball to the whole space.     

Asymptotic stability and blowup of solutions for a class of viscoelastic inverse problem with boundary feedback

In this paper, we consider a nonlinear viscoelastic inverse problem with memory in the boundary. Under some suitable conditions on the coefficients, relaxation function, and initial data, we proved stability of solutions when the integral overdetermination tends to zero as time goes to infinity. Furthermore, we show that there are solutions under some conditions on initial data that blow up in finite time. Copyright © 2015 John Wiley & Sons, Ltd.     

Gromov–Witten Invariants of Blow-ups Along Surfaces

In this paper, using the gluing formula of Gromov–Witten invariants for symplectic cutting developed by Li and Ruan, we established some relations between Gromov–Witten invariants of a semipositive symplectic manifold M and its blow-ups along a smooth surface.     

The nonsymmetric case of the Keller-Segel model in chemotaxis: some recent results

Looking at the nonsymmetric case of a reaction-diffusion model known as the Keller-Segel model, we summarize known facts concerning (global in time) existence and prove new blowup results for solutions of this system of two strongly coupled parabolic partial differential equations. We show in Section 4, Theorem 4, that if the solution blows up under a condition on the initial data, blowup takes place at the boundary of a smooth domain . Using variational techniques we prove in Section 5 the existence of nontrivial stationary solutions in a special case of the system. Received April 2000