共查询到20条相似文献,搜索用时 109 毫秒
1.
Weikui Ye 《Applicable analysis》2020,99(8):1300-1315
ABSTRACTWe first establish the local well-posedness for a generalized Degasperis-Procesi equation in nonhomogeneous Besov spaces. Then we present a global existence result for the equation. Moreover, we obtain a blow-up criteria and provide a sufficient condition for strong solutions to blow up in finite time. 相似文献
2.
Weisheng Niu 《Journal of Mathematical Analysis and Applications》2011,374(1):166-177
We first establish the local well-posedness for the nonuniform weakly dissipative b-equation which includes both the weakly dissipative Camassa-Holm equation and the weakly dissipative Degasperis-Procesi equation as its special cases. We then study the blow-up phenomena and the long time behavior of the solutions. Two blow-up results are established for certain initial profiles. Moreover, two sufficient conditions for the decay of the solutions are presented. 相似文献
3.
Alessia Ascanelli 《Annali dell'Universita di Ferrara》2006,52(2):199-209
Abstract We consider the Cauchy problem for a second order equation of hyperbolic type. This equation degenerates in two different
ways. On one hand, the coefficients have a bad behavior with respect to time: there is a blow-up phenomenon in the first time
derivative of the principal part’s coefficients, that is the derivative vanishes at the time t=0. On the other hand, the equation is weakly hyperbolic and the multiplicity of the roots is not constant, but zeroes are
of finite order.
Here we overcome the blow-up problem and, moreover, the finitely degeneration of the Cauchy problem allows us to give an appropriate
Levi condition on the lower order terms in order to get C∞ well posedness of the Cauchy problem.
Keywords: Cauchy problem, Hyperbolic equations, Levi conditions 相似文献
4.
In this paper, we study the Cauchy problem of a weakly dissipative μ-Hunter–Saxton equation. We first establish the local well-posedness for the weakly dissipative μ-Hunter–Saxton equation by Kato's semigroup theory. Then, we derive the precise blow-up scenario for strong solutions to the equation. Moreover, we present some blow-up results for strong solutions to the equation. Finally, we give two global existence results to the equation. 相似文献
5.
L. E. Payne G. A. Philippin S. Vernier Piro 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2010,50(4):999-1007
We consider the blow-up of the solution to a semilinear heat equation with nonlinear boundary condition. We establish conditions
on nonlinearities sufficient to guarantee that u(x, t) exists for all time t > 0 as well as conditions on data forcing the solution u(x, t) to blow up at some finite time t*. Moreover, an upper bound for t* is derived. Under somewhat more restrictive conditions, lower bounds for t* are also derived. 相似文献
6.
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
7.
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
8.
In this paper, we study blow-up solutions of the Cauchy problem to the L2 critical nonlinear Schrdinger equation with a Stark potential. Using the variational characterization of the ground state for nonlinear Schrdinger equation without any potential, we obtain some concentration properties of blow-up solutions, including that the origin is the blow-up point of the radial blow-up solutions, the phenomenon of L2-concentration and rate of L2-concentration of blow-up solutions. 相似文献
9.
We obtain a blow-up result for solutions to a semi-linear wave equation with scale-invariant dissipation and mass and power non-linearity, in the case in which the model has a “wave like” behavior. We perform a change of variables that transforms our starting equation in a strictly hyperbolic semi-linear wave equation with time-dependent speed of propagation. Applying Kato's lemma we prove a blow-up result for solutions to the transformed equation under some assumptions on the initial data. The limit case, that is, when the exponent p is exactly equal to the upper bound of the range of admissible values of p yielding blow-up needs special considerations. In this critical case an explicit integral representation formula for solutions of the corresponding linear Cauchy problem in 1d is derived. Finally, carrying out the inverse change of variables we get a non-existence result for global (in time) solutions to the original model. 相似文献
10.
S. I. Pokhozhaev 《Differential Equations》2011,47(4):488-493
We consider the problem of finite-time blow-up of solutions of a class of initial-boundary value problems for the Korteweg-de
Vries equation. By using the method of optimal test functions corresponding to the boundary conditions, we obtain blow-up
conditions for local (with respect to t > 0) solutions and estimate the blow-up time. 相似文献
11.
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
12.
13.
Ni Xiang 《Nonlinear Analysis: Theory, Methods & Applications》2010,73(12):3940-3946
We consider the exact asymptotic behavior of Γ-subharmonic solutions to boundary blow-up problems for the complex Hessian equation in bounded domains Ω. 相似文献
14.
We propose a method for regularizing the Korteweg–de Vries equation near, rather than on, a blow-up surface. This allows showing that for sufficiently small initial data at x = 0, a blow-up surface exists nearby and is an analytic manifold. 相似文献
15.
Fernando Quirós Julio D. Rossi 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2004,50(4):357-362
We consider the heat equation in the half-line with
Dirichlet boundary data which blow up in finite time. Though the
blow-up set may be any interval [0,a],
a ? [0,¥]a\in[0,\infty]
depending on the Dirichlet data, we prove that the
effective
blow-up set, that is, the set of points
x 3 0x\ge0
where the solution behaves like u(0,t), consists always only of the
origin.
As an application of our results we consider a system of two heat
equations with a nontrivial nonlinear flux coupling at the
boundary. We show that by prescribing the non-linearities the two
components may have different blow-up sets. However, the effective
blow-up sets do not depend on the coupling and coincide with the
origin for both components. 相似文献
16.
Jakow Baris 《Applicable analysis》2013,92(11):1339-1345
This article deals with blow-up solutions of the Cauchy–Dirichlet problem for system of semilinear heat equations with quadratic non-linearities. Sufficient conditions for the existence of blow-up solutions are established. Sets of initial values for these solutions as well as upper bounds for corresponding blow-up time are determined. Furthermore, an application to the Lotka-Volterra system with diffusion is also discussed. The result of this article may be considered as a continuation and a generalization of the results obtained in (Baris, J., Baris, P. and Ruchlewicz, B., 2006, On blow-up solutions of nonautonomous quadratic differential systems. Differential Equations, 42, 320–326; Baris, J., Baris, P. and Wawiórko, E., 2006, Asymptotic behaviour of solutions of Lotka-Volterra systems. Nonlinear Analysis: Real World Applications, 7, 610–618; Baris, J., Baris, P. and Ruchlewicz, B., 2006, On blow-up solutions of quadratic systems of differential equations. Sovremennaya Matematika. Fundamentalnye Napravleniya, 15, 29–35 (in Russian); Baris, J. and Wawiórko, E., On blow-up solutions of polynomial Kolmogorov systems. Nonlinear Analysis: Real World Applications, to appear). 相似文献
17.
In this paper we work with the ordinary diffential equation u′′ u3 = 0 and obtain some interesting phenomena concerning blow-up, blow-up rate, life-spann, zeros and critical points of solutions to this equation. 相似文献
18.
Huan Zhang 《Applicable analysis》2020,99(6):976-999
ABSTRACTA blow-up analysis for a nonlocal reaction-diffusion system with time-dependent coefficients is investigated under null Dirichlet boundary conditions. Based on the Kaplan's method, comparison principle and modified differential inequality technique, we establish a blow-up criteria and derive the bounds for the blow-up time under the appropriate measures in whole-dimensional space. 相似文献
19.
This paper is concerned with the blow-up solutions of the Cauchy problem for Gross-Pitaevskii equation.In terms of Merle and Raphёel's arguments as well as Carles' transformation,the limiting profiles of blow-up solutions are obtained.In addition,the nonexistence of a strong limit at the blow-up time and the existence of L2 profile outside the blow-up point for the blow-up solutions are obtained. 相似文献
20.
Paul Godin 《Journal of Differential Equations》2002,183(1):224-238
We give a complete discussion of the C∞ or analytic regularity of blow-up curves for Cauchy problems or some mixed problems for the Liouville equation in one space dimension. In the case of mixed problems, the regularity results depend on the boundary condition: actually, we show the existence of a sequence of boundary conditions for which the regularity of the blow-up curve is better than in the general case. 相似文献