共查询到20条相似文献,搜索用时 15 毫秒
1.
We study an energy conserving distributed parameter system described by a nonlinear string equation with the input and output at the boundary. We prove the existence of global smooth solutions to this quasilinear hyperbolic system if the initial data and the boundary input are small. If, moreover, the input function becomes zero after some finite time, then the state trajectories decay exponentially. 相似文献
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This paper deals with the blow-up properties of positive solutions to a nonlinear parabolic equation with a localized reaction source and a nonlocal boundary condition. Under certain conditions, the blowup criteria is established. Furthermore, when f(u)=up, 0<p?1, the global blowup behavior is shown, and the blowup rate estimates are also obtained. 相似文献
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Yue Liu 《Mathematische Annalen》2006,335(3):717-735
Considered herein are the problems of the existence of global solutions and the formation of singularities for a new nonlinear
shallow water wave equation derived by Dullin, Gottward and Holm. Blow-up can occur only in the form of wave-breaking. A wave-breaking
mechanism for solutions with certain initial profiles is described in detail and the exact blow-up rate is established. The
blow-up set for a class of initial profiles and lower bounds of the existence time of the solution are also determined. 相似文献
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In this paper, the authors discuss the global existence and blow-up of the solution to an evolution p-Laplace system with nonlinear sources and nonlinear boundary condition. The authors first establish the local existence of solutions, then give a necessary and sufficient condition on the global existence of the positive solution. 相似文献
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In this paper, we consider initial boundary value problem of the generalized Boussinesq equation with nonlinear interior source and boundary absorptive terms. We establish firstly the local existence of solutions by standard Galerkin method. Then we prove both the global existence of the solution and a general decay of the energy functions under some restrictions on the initial data. We also prove a blow-up result for solutions with positive and negative initial energy respectively. 相似文献
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Shun-Tang Wu 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2012,63(1):65-106
In this paper, a viscoelastic equation with nonlinear boundary damping and source terms of the form $$\begin{array}{llll}u_{tt}(t)-\Delta u(t)+\displaystyle\int\limits_{0}^{t}g(t-s)\Delta u(s){\rm d}s=a\left\vert u\right\vert^{p-1}u,\quad{\rm in}\,\Omega\times(0,\infty), \\ \qquad\qquad\qquad\qquad\qquad u=0,\,{\rm on}\,\Gamma_{0} \times(0,\infty),\\ \dfrac{\partial u}{\partial\nu}-\displaystyle\int\limits_{0}^{t}g(t-s)\frac{\partial}{\partial\nu}u(s){\rm d}s+h(u_{t})=b\left\vert u\right\vert ^{k-1}u,\quad{\rm on} \ \Gamma_{1} \times(0,\infty) \\ \qquad\qquad\qquad\qquad u(0)=u^{0},u_{t}(0)=u^{1},\quad x\in\Omega, \end{array}$$ is considered in a bounded domain ??. Under appropriate assumptions imposed on the source and the damping, we establish both existence of solutions and uniform decay rate of the solution energy in terms of the behavior of the nonlinear feedback and the relaxation function g, without setting any restrictive growth assumptions on the damping at the origin and weakening the usual assumptions on the relaxation function g. Moreover, for certain initial data in the unstable set, the finite time blow-up phenomenon is exhibited. 相似文献
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We study the global and blow-up solutions for a strong degenerate reaction–diffusion system modeling the interactions of two biological species. The local existence and uniqueness of a classical solution are established. We further give the critical exponent for reaction and absorption terms for the existence of global and blow-up solutions. We show that the solution may blow up if the intraspecific competition is weak. This supports ecologist A.J. Nicholson’s conclusion that intraspecific competition is the main factor regulating population size. 相似文献
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主要研究在Dirichlet边界条件或Neumann边界条件下的一类非局部非线性的扩散方程问题.在适当的假设下,证明解的存在性、唯一性、比较原则、以及解对初边值条件的连续依赖性,并就给定的初边值条件,证明解在有限时刻全局爆破. 相似文献
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This paper concerns with a nonlinear degenerate parabolic system coupled via nonlocal sources, subjecting to homogeneous Dirichlet boundary condition. The main aim of this paper is to study conditions on the global existence and/or blow-up in finite time of solutions, and give the estimates of blow-up rates of blow-up solutions. 相似文献
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Zongming Guo 《Journal of Differential Equations》2006,228(2):486-506
The singularly perturbed boundary blow-up problem
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Global existence and blow-up for the solutions to nonlinear parabolic-elliptic system modelling chemotaxis 总被引:1,自引:0,他引:1
In this paper we study the global in-time and blow-up solutionsfor the simplified KellerSegel system modelling chemotaxis.We prove that there is a critical number which determines theoccurrence of blowup in the two-dimensional case for 1 <p < 2. In three- or higher-dimensional cases, we show thatthe radial symmetrical solution will blow up if 1 < p <N/N2 (N 3) for non-negative initial value. 相似文献
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Global existence and blow-up of solutions for a system of nonlinear viscoelastic wave equations with damping and source 总被引:1,自引:0,他引:1
In this paper we investigate the global existence and finite time blow-up of solutions to the system of nonlinear viscoelastic wave equations in Ω×(0,T) with initial and Dirichlet boundary conditions, where Ω is a bounded domain in . Under suitable assumptions on the functions gi(), , the initial data and the parameters in the equations, we establish several results concerning local existence, global existence, uniqueness and finite time blow-up property. 相似文献
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Kazuhiro Ishige Tatsuki Kawakami 《Calculus of Variations and Partial Differential Equations》2010,39(3-4):429-457
We consider the heat equation with a nonlinear boundary condition, $$(P) \left\{\begin{array}{ll} \partial_t u = \Delta u, & x \in \Omega, \quad t > 0, \\ \partial_\nu u=u^p, & x \in \partial \Omega,\quad t > 0,\\ u (x,0) = \phi (x),& x\in\Omega, \end{array}\right.$$ where ${\Omega = \{x = (x^{\prime},x_N) \in {\bf R}^{N} : x_N > 0\}, N \ge 2, \partial_t = \partial{/}\partial t , \partial_\nu = -\partial{/}\partial x_{N}}$ , p > 1 + 1/N, and (N ? 2)p < N. In this paper we give a complete classification of the large time behaviors of the nonnegative global solutions of (P). 相似文献
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Tatsuki Kawakami 《Journal of Mathematical Analysis and Applications》2010,368(1):320-329
We consider the initial-boundary value problem for the heat equation with a nonlinear boundary condition:
20.
Jorge García-Melin 《Journal of Mathematical Analysis and Applications》2009,360(2):530-536
In this paper we prove the uniqueness of the positive solution for the boundary blow-up problem where Ω is a C2 bounded domain in , under the hypotheses that f(t) is nondecreasing in t>0 and f(t)/tp is increasing for large t and some p>1. We also consider the uniqueness of a related problem when the equation includes a nonnegative weight a(x). 相似文献