共查询到20条相似文献,搜索用时 31 毫秒
1.
In this paper, we investigate the blow-up rate of solutions of diffusion equations with nonlocal nonlinear reaction terms. For large classes of equations, we prove that the solutions have global blow-up and that 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
2.
We consider the blow-up of solutions of equations of the form ut=div(ρ(|∇u|2) grad u)+f(u) by means of a differential inequality technique. A lower bound for blow-up time is determined if blow-up does occur as well as a criterion for blow-up and conditions which ensure that blow-up cannot occur. 相似文献
3.
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
5.
In this paper, we study the regularity of generalized solutions u(x,t) for the n -dimensional quasi-linear parabolic diffraction problem. By using various estimates and Steklov average methods, we prove that (1): for almost all t the first derivatives ux(x,t) are Hölder continuous with respect to x up to the inner boundary, on which the coefficients of the equation are allowed to be discontinuous; and (2): the first derivative ut(x,t) is Hölder continuous with respect to (x,t) across the inner boundary. 相似文献
6.
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
7.
In this paper a localized porous medium equation ut=u r(Δu+ af(u( x0,t))) is considered. It is shown that under certain conditions solutions of the above equation blow up in finite time for large a or large initial data while there exist global positive solutions to the above equation for small a or small initial data. Moreover, it is also shown that all global positive solutions of the above equation are uniformly bounded, and this differs from that of a porous medium equation with a local source. 相似文献
9.
The existence of solutions of degenerate quasilinear pseudoparabolic equations, where the term ∂tu is replace by ∂tb(u) , with memory terms and quasilinear variational inequalities is shown. The existence of solutions of equations is proved under the assumption that the nonlinear function b is monotone and a gradient of a convex, continuously differentiable function. The uniqueness is proved for Lipschitz-continuous elliptic parts. The existence of solutions of quasilinear variational inequalities is proved under stronger assumptions, namely, the nonlinear function defining the elliptic part is assumed to be a gradient and the function b to be Lipschitz continuous. 相似文献
10.
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
13.
By means of Mawhin’s continuation theorem, a class of p-Laplacian type differential equation with a deviating argument of the form (φp(x′(t)))′+f(x(t))x′(t)+β(t)g(t,x(t−τ(t,|x|∞)))=e(t) is studied. A new result, related to β(t) and the deviating argument τ(t, |x|∞) , is obtained. It is significant that the growth degree with respect to the variable x in g(t,x) is allowed to be greater than p−1 , which could be achieved infrequently in previous papers. 相似文献
14.
This paper is concerned with special regularity properties of the solutions to the Maxwell–Landau–Lifshitz (MLL) system describing ferromagnetic medium. Besides the classical results on the boundedness of ∂tm, ∂tE and ∂tH in the spaces L ∞(I,L 2(Ω)) and L 2(I,W 1,2(Ω)) we derive also estimates in weighted Sobolev spaces. This kind of estimates can be used to control the Taylor remainder when estimating the error of a numerical scheme. 相似文献
19.
In the well-known work of P.-L. Lions [The concentration–compactness principle in the calculus of variations, The locally compact case, part 1. Ann. Inst. H. Poincaré, Analyse Non Linéaire 1 (1984) 109–1453] existence of positive solutions to the equation -Δu+u=b(x)u p-1, u>0 , u∈H 1(R N) , p∈(2,2N/(N-2)) was proved under assumption b(x)? b∞? lim|x|→∞b(x) . In this paper we prove the existence for certain functions b satisfying the reverse inequality b(x)< b∞. For any periodic lattice L in R N and for any b∈C(R N) satisfying b(x)< b∞, b∞>0 , there is a finite set Y⊂L and a convex combination b Y of b(·-y) , y∈Y , such that the problem -Δu+u=b Y(x)u p-1 has a positive solution u∈H 1(R N) . 相似文献
|