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2.
Using a bifurcation result on noncompact branches of solutions in an abstract setting, we establish the existence of global bifurcation for the following nonlinear equation −div(a|∇u|p−2∇u)−μ0b|u|p−2u=q(λ,x,u,∇u) subject to Dirichlet boundary conditions under certain assumptions on a,b and q when μ0 is not an eigenvalue of the above divergence form. 相似文献
3.
In this paper, we introduce a new method for investigating the rate of blow-up of solutions of diffusion equations with nonlocal nonlinear reaction terms. In some cases, we prove that the solutions have global blow-up and the rate of blow-up is uniform in all compact subsets of the domain. In each case, the blow-up rate of |u(t)|∞ is precisely determined. 相似文献
4.
We consider the simplest possible heat equation for director fields, ut=Δu+|∇u| 2u (|u|=1 ), and construct axially symmetric traveling wave solutions defined in an infinitely long cylinder. The traveling waves have a point singularity of topological degree 0 or 1. 相似文献
5.
Following Coclite, Holden and Karlsen [G.M. Coclite, H. Holden and K.H. Karlsen, Well-posedness for a parabolic-elliptic system, Discrete Contin. Dyn. Syst. 13 (3) (2005) 659–682] and Tian and Fan [Lixin Tian, Jinling Fan, The attractor on viscosity Degasperis-Procesi equation, Nonlinear Analysis: Real World Applications, 2007], we study the dynamical behaviors of the parabolic–elliptic system ut+(f(t,x,u))x+g(t,x,u)+Px−εuxx=0 and −Pxx+P=h(t,x,u,ux)+k(t,x,u) with initial data u|t=0=u0. The existence of global solution to the parabolic–elliptic system in L 2 under the periodic boundary condition is discussed. We also establish the existence of the global attractor of semi-group to solutions on the parabolic–elliptic system in H 2. 相似文献
10.
We consider the Cauchy problem for the generalized Ostrovsky equation where f(u)=|u| ρ−1u if ρ is not an integer and f(u)=u ρ if ρ is an integer. We obtain the L∞ time decay estimates and the large time asymptotics of small solutions under suitable conditions on the initial data and the order of the nonlinearity. 相似文献
16.
In this paper we prove local well-posedness in L 2(R) and H 1(R) for the generalized sixth-order Boussinesq equation utt= uxx+β uxxxx+ uxxxxxx+ (|u|αu)xx. Our proof relies in the oscillatory integrals estimates introduced by Kenig et al. (1991) [14]. We also show that, under suitable conditions, a global solution for the initial value problem exists. In addition, we derive the sufficient conditions for the blow-up of the solution to the problem. 相似文献
20.
The paper deals with the radially symmetric solutions of ut=Δu+u m(x,t)v n(0,t) , vt=Δv+u p(0,t)v q(x,t) , subject to null Dirichlet boundary conditions. For the blow-up classical solutions, we propose the critical exponents for non-simultaneous blow-up by determining the complete and optimal classification for all the non-negative exponents: (i) There exist initial data such that u (v ) blows up alone if and only if m>p+1 (q>n+1 ), which means that any blow-up is simultaneous if and only if m≤p+1 , q≤n+1 . (ii) Any blow-up is u (v ) blowing up with v (u ) remaining bounded if and only if m>p+1 , q≤n+1 (m≤p+1 , q>n+1 ). (iii) Both non-simultaneous and simultaneous blow-up may occur if and only if m>p+1 , q>n+1 . Moreover, we consider the blow-up rate and set estimates which were not obtained in the previously known work for the same model. 相似文献
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