共查询到20条相似文献,搜索用时 15 毫秒
1.
Xinglong Wu 《Journal of Functional Analysis》2011,260(5):1428-1445
We mainly study the Cauchy problem of the periodic generalized Degasperis-Procesi equation. First, we establish the local well-posedness for the equation. Second, we give the precise blow-up scenario, a conservation law and prove that the equation has smooth solutions which blow up in finite time. Finally, we investigate the blow-up rate for the blow-up solutions. 相似文献
2.
Weisheng Niu Xiaotong Sun Xiaojuan Chai 《Journal of Mathematical Analysis and Applications》2010,364(2):508-521
By introducing a stress multiplier we derive a family of Burgers-like equations. We investigate the blow-up phenomena of the equations both on the real line R and on the circle S to get a comparison with the Degasperis-Procesi equation. On the line R, we first establish the local well-posedness and the blow-up scenario. Then we use conservation laws of the equations to get the estimate for the L∞-norm of the strong solutions, by which we prove that the solutions to the equations may blow up in the form of wave breaking for certain initial profiles. Analogous results are provided in the periodic case. Especially, we find differences between the Burgers-like equations and the Degasperis-Procesi equation, see Remark 4.1. 相似文献
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Joachim Escher 《Journal of Functional Analysis》2006,241(2):457-485
In this paper we study several qualitative properties of the Degasperis-Procesi equation. We first established the precise blow-up rate and then determine the blow-up set of blow-up strong solutions to this equation for a large class of initial data. We finally prove the existence and uniqueness of global weak solutions to the equation provided the initial data satisfies appropriate conditions. 相似文献
5.
Lili Du 《Journal of Mathematical Analysis and Applications》2006,324(1):304-320
This paper deals with the global existence and blow-up of nonnegative solution of the degenerate reaction-diffusion system with nonlinear localized sources involved a product with local terms. We investigate the influence of localized sources and local terms on global existence and blow up for this system. Moreover, we establish the precise blow-up estimates. Finally, for the special case p1=p2=0, we show the blow-up set is whole region and the uniform blow-up profiles are obtained. These extend a resent work of Chen and Xie in [Y. Chen, C. Xie, Blow-up for a porous medium equation with a localized source, Appl. Math. Comput. 159 (2004) 79-93], which considered the single equation with localized sources. 相似文献
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In this article, we study the following initial value problem for the nonlinear equation 相似文献
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In this paper, we study the blow-up profiles of the nonlocal dispersal equation. More precisely, we prove that the positive solution of nonlocal dispersal equation has different blow-up profiles, depending on the refuge domain. 相似文献
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This paper is devoted to studying the modified b-family of equations with cubic nonlinearity, called the b-family of FORQ/MCH equations, which includes the cubic Camassa–Holm equation (also called Fokas–Olver–Rosenau–Qiao equation) as a special case. We first study the local well-posedness for the Cauchy problem of the equation, and then make good use of fine structure of the equation, we derive the precise blow-up scenario and a new blow-up result with respect to initial data. Finally, peakon solutions are derived. 相似文献
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In this paper we investigates the blow-up properties of the positive solutions to a porous medium equation with nonlocal reaction source and with nonlocal boundary condition, we obtain the blow-up condition and its blow-up rate estimate. 相似文献
10.
Juliá n Ferná ndez Bonder Julio D. Rossi 《Proceedings of the American Mathematical Society》2001,129(1):139-144
In this paper, we study the blow-up problem for positive solutions of a semidiscretization in space of the heat equation in one space dimension with a nonlinear flux boundary condition and a nonlinear absorption term in the equation. We obtain that, for a certain range of parameters, the continuous problem has blow-up solutions but the semidiscretization does not and the reason for this is that a spurious attractive steady solution appears.
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Qilin Liu Youpeng Chen Chunhong Xie 《Journal of Mathematical Analysis and Applications》2003,285(2):487-505
In this paper, we investigate the blowup properties of the positive solutions to the following nonlocal degenerate parabolic equation
12.
L.E. Payne 《Journal of Mathematical Analysis and Applications》2007,335(1):371-376
In this paper we consider two different initial-boundary value problems in temperature dependent viscous flow when the temperature equation has a nonlinear heat source term. When blow-up occurs we derive lower bounds for the blow-up time in each case. 相似文献
13.
José A. Carrillo María d. M. González Maria P. Gualdani Maria E. Schonbek 《偏微分方程通讯》2013,38(3):385-409
In this paper we analyze the global existence of classical solutions to the initial boundary-value problem for a nonlinear parabolic equation describing the collective behavior of an ensemble of neurons. These equations were obtained as a diffusive approximation of the mean-field limit of a stochastic differential equation system. The resulting nonlocal Fokker-Planck equation presents a nonlinearity in the coefficients depending on the probability flux through the boundary. We show by an appropriate change of variables that this parabolic equation with nonlinear boundary conditions can be transformed into a non standard Stefan-like free boundary problem with a Dirac-delta source term. We prove that there are global classical solutions for inhibitory neural networks, while for excitatory networks we give local well-posedness of classical solutions together with a blow up criterium. Surprisingly, we will show that the spectrum for the operator in the linear case, that corresponding to a system of uncoupled networks, does not give any information about the large time asymptotic behavior. 相似文献
14.
The finite time blow-up of solutions to a nonlinear Timoshenko-type equation with variable exponents is studied. More concretely, we prove that the solutions blow up in finite time with positive initial energy. Also, the existence of finite time blow-up solutions with arbitrarily high initial energy is established. Meanwhile, the upper and lower bounds of the blow-up time are derived. These results deepen and generalize the ones obtained in [Nonlinear Anal. Real World Appl., 61: Paper No. 103341, 2021]. 相似文献
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Kazuhiro Ishige 《Journal of Differential Equations》2005,212(1):114-128
We consider the blow-up problem of a semilinear heat equation,
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We study blow-up of radially symmetric solutions of the nonlinear heat equation ut=Δu+|u|p−1u either on RN or on a finite ball under the Dirichlet boundary conditions. We assume that N?3 and . Our first goal is to analyze a threshold behavior for solutions with initial data u0=λv, where v∈C∩H1 and v?0, v?0. It is known that there exists λ?>0 such that the solution converges to 0 as t→∞ if 0<λ<λ?, while it blows up in finite time if λ?λ?. We show that there exist at most finitely many exceptional values λ1=λ?<λ2<λk such that, for all λ>λ? with λ≠λj (j=1,2,…,k), the blow-up is complete and of type I with a flat local profile. Our method is based on a combination of the zero-number principle and energy estimates. In the second part of the paper, we employ the very same idea to show that the constant solution κ attains the smallest rescaled energy among all non-zero stationary solutions of the rescaled equation. Using this result, we derive a sharp criterion for no blow-up. 相似文献
17.
Ting Cheng 《Journal of Differential Equations》2008,244(4):766-802
The blow-up rate estimate for the solution to a semilinear parabolic equation ut=Δu+V(x)|u|p−1u in Ω×(0,T) with 0-Dirichlet boundary condition is obtained. As an application, it is shown that the asymptotic behavior of blow-up time and blow-up set of the problem with nonnegative initial data u(x,0)=Mφ(x) as M goes to infinity, which have been found in [C. Cortazar, M. Elgueta, J.D. Rossi, The blow-up problem for a semilinear parabolic equation with a potential, preprint, arXiv: math.AP/0607055, July 2006], is improved under some reasonable and weaker conditions compared with [C. Cortazar, M. Elgueta, J.D. Rossi, The blow-up problem for a semilinear parabolic equation with a potential, preprint, arXiv: math.AP/0607055, July 2006]. 相似文献
18.
Wenying Chen 《Journal of Mathematical Analysis and Applications》2011,379(1):351-359
In this paper we consider a new integrable equation (the Degasperis-Procesi equation) derived recently by Degasperis and Procesi (1999) [3]. Analogous to the Camassa-Holm equation, this new equation admits blow-up phenomenon and infinite propagation speed. First, we give a proof for the blow-up criterion established by Zhou (2004) in [12]. Then, infinite propagation speed for the Degasperis-Procesi equation is proved in the following sense: the corresponding solution u(x,t) with compactly supported initial datum u0(x) does not have compact x-support any longer in its lifespan. Moreover, we show that for any fixed time t>0 in its lifespan, the corresponding solution u(x,t) behaves as: u(x,t)=L(t)e−x for x?1, and u(x,t)=l(t)ex for x?−1, with a strictly increasing function L(t)>0 and a strictly decreasing function l(t)<0 respectively. 相似文献
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The generalized Jang equation was introduced in an attempt to prove the Penrose inequality in the setting of general initial data for the Einstein equations. In this paper we give an extensive study of this equation, proving existence, regularity, and blow-up results. In particular, precise asymptotics for the blow-up behavior are given, and it is shown that blow-up solutions are not unique. 相似文献