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1.
This article deals with the global existence and blow-up of positive solution of a nonlinear diffusion equation with nonlocal source and nonlocal nonlinear boundary condition. We investigate the influence of the reaction terms, the weight functions and the nonlinear terms in the boundary conditions on global existence and blow up for this equation. Moreover, we establish blow-up rate estimates under some appropriate hypotheses.  相似文献
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This paper deals with the blow-up properties of the positive solutions to a degenerate parabolic system with localized sources and nonlocal boundary conditions. We investigate the influence of the reaction terms, the weight functions, local terms and localized source on the blow-up properties. We will show that the weight functions play the substantial roles in determining whether the solutions will blow-up or not, and obtain the blow-up conditions and its blow-up rate estimate.  相似文献
4.
We are concerned with the Cauchy problem of the new integrable three-component system with cubic nonlinearity. We establish the local well- posedness in a range of the Besov spaces. Then the precise blow-up scenario for strong solutions to the system is derived.  相似文献
5.
In this paper, we deal with the global existence and nonexistence of solutions to a nonlinear diffusion system coupled via nonlinear boundary flux. By constructing various kinds of sub- and super-solutions and using the basic properties of M-matrix, we give the necessary and sufficient conditions for global existence of nonnegative solutions. The critical curve of Fujita type is conjectured with the aid of some new results, which extend the recent results of Wang et al. [Nonlinear Anal. 71 (2009) 2134-2140] and Li et al. [J. Math. Anal. Appl. 340 (2008) 876-883] to more general equations.  相似文献
6.
In this paper, we consider the positive solution of the Cauchy problem for the following doubly degenerate parabolic equation
ut-div(|?u|p ?um)=uqu_t-{\rm div}(|\nabla u|^{p} \nabla u^m)=u^q  相似文献
7.
This paper is concerned with the Cauchy problem of the modified Hunter‐Saxton equation. The local well‐posedness of the model equation is obtained in Besov spaces (which generalize the Sobolev spaces Hs) by using Littlewood‐Paley decomposition and transport equation theory. Moreover, the local well‐posedness in critical case (with ) is considered.  相似文献
8.
In this paper, we consider the positive solution of the Cauchy problem for the following doubly degenerate parabolic equation
$$u_t-{\rm div}(|\nabla u|^{p} \nabla u^m)=u^q$$
with p > 0, q > 1,m > 1, and initial value decaying at infinity and give a new secondary critical exponent for the existence of global and nonglobal solutions. Furthermore, the large time behavior and the life spans of solutions are also studied.
  相似文献
9.
In this paper, we are concerned with the Cauchy problem of the generalized Camassa–Holm equation. Using a Galerkin-type approximation scheme, it is shown that this equation is well-posed in Sobolev spaces Hs, s>3/2 for both the periodic and the nonperiodic case in the sense of Hadamard. That is, the data-to-solution map is continuous. Furthermore, it is proved that this dependence is sharp by showing that the solution map is not uniformly continuous. The nonuniform dependence is proved using the method of approximate solutions and well-posedness estimates. Moreover, it is shown that the solution map for the generalized Camassa–Holm equation is Hölder continuous in Hr-topology. Finally, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time.  相似文献
10.
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