The tangential oblique derivative problem for second order quasilinear parabolic operators

Abstract

We present the precise blow-up scenario for the generalized Camassa–Holm equation. We also prove a blow-up result showing that the equation has smooth solutions in which singularities develop in finite time.     

Geometry of unbounded domains,poincaré inequalities and stability in semilinear parabolic equations

We investigate thc close relations existing between certain geometric properties of domains Ω of R^N, the validity of Poincark inequalities in Ω, and the behavior of solutions of semilinear parabolic equations. For the equation u_t-△u=|u|^p-1 we obtain a purely geometric, necessary and sufficient condition on Ω, for the 0 solution to be asymptotically (and exponentially) stable in L^r(ω)1<r<∞ when r is supercritical(r>N(p-1)/2 . The condition is that the inradius of Ω be finite. The result is different for r critical. For the equation u_t-△u=u^p-μ|u|^q,q≥p>1,μ>0 we prove that the finiteness of the inradius is a necessary and sufficient condition for global existence and boundedness of all nonnegative solutions.     

L p c -concentration of blow-up solutions for the biharmonic nonlinear Schrödinger equation

This article is concerned with the blow-up solutions of the biharmonic Schrödinger equation with L ²-super critical nonlinearity. We obtain the nonexistence of strong limit of L ^{p _c} -norm and L ^{p _c} -concentration properties of the radially symmetric blow-up solutions, where L ^{p _c} is the invariant Lebesgue space.     

Existence and blow-up of solutions for a porous medium equation with nonlocal boundary condition

In this article, a porous medium equation with nonlocal boundary condition and a localized source is studied. The results of the existence of global solutions or blow-up of solutions are given. The blow-up rate estimates are also obtained under some conditions.     

A MATHEMATICAL MODEL OF ENTERPRISE COMPETITIVE ABILITY AND PERFORMANCE THROUGH EMDEN-FOWLER EQUATION （Ⅱ）

In this paper we work with the ordinary equation u＇＇ - u2 （u ＋ ） = 0 and ob- tain some interesting phenomena concerning, blow-up, blow-up rate, life-span of solutions to those equations.     

Blow-up of the Solution for a Class of Porous Medium Equation with Positive Initial Energy

This paper deals with a class of porous medium equation

ut = Δum + f(u)$ut\hspace{0.17em}=\hspace{0.17em}\text{Δ}{u}^m\hspace{0.17em}+\hspace{0.17em}f\left(u\right)$

with homogeneous Dirichlet boundary conditions. The blow-up criteria is established by using the method of energy under the suitable condition on the function f(u).