共查询到20条相似文献,搜索用时 11 毫秒
1.
In this paper, we consider a degenerate reaction-diffusion equation
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Tong Zhang 《Applied mathematics and computation》2010,217(2):801-810
By constructing different auxiliary functions and using Hopf’s maximum principle, the sufficient conditions for the blow-up and global solutions are presented for nonlinear parabolic equation ut = ∇(a(u)b(x)c(t)∇u) + f(x, u, q, t) with different kinds of boundary conditions. The upper bounds of the “blow-up time” and the “upper estimates” of global solutions are provided. Finally, some examples are presented as the application of the obtained results. 相似文献
3.
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
4.
Tomasz Ma?olepszy Wojciech Okrasiñski 《Journal of Mathematical Analysis and Applications》2010,366(1):372-384
The problem of the estimating of a blow-up time for solutions of Volterra nonlinear integral equation with convolution kernel is studied. New estimates, lower and upper, are found and, moreover, the procedure for the improvement of the lower estimate is presented. Main results are illustrated by examples. The new estimates are also compared with some earlier ones related to a shear band model. 相似文献
5.
Alexander Gladkov Kwang Ik Kim 《Journal of Mathematical Analysis and Applications》2008,338(1):264-273
In this paper, we consider a semilinear heat equation ut=Δu+c(x,t)up for (x,t)∈Ω×(0,∞) with nonlinear and nonlocal boundary condition and nonnegative initial data where p>0 and l>0. We prove global existence theorem for max(p,l)?1. Some criteria on this problem which determine whether the solutions blow up in a finite time for sufficiently large or for all nontrivial initial data or the solutions exist for all time with sufficiently small or with any initial data are also given. 相似文献
6.
Steven Dejak Zhou Gang Israel Michael Sigal Shuangcai Wang 《Advances in Applied Mathematics》2008,40(4):433-481
We study the blow-up of solutions of nonlinear heat equations in dimension 1. We show that for an open set of even initial data which are characterized roughly by having maxima at the origin, the solutions blow up in finite time and at a single point. We find the universal blow-up profile and remainder estimates. Our results extend previous results in several directions and our techniques differ from the techniques previously used for this problem. In particular, they do not rely on maximum principle. 相似文献
7.
We prove the global existence of analytic solutions to the Cauchy problem for the cubic Schrödinger equation in space dimension n?3 for sufficiently small data with exponential decay at infinity. Minimal regularity assumption regarding scaling invariance is imposed on the Cauchy data. 相似文献
8.
Blow-up analysis for a system of heat equations with nonlinear flux which obey different laws 总被引:1,自引:0,他引:1
We consider a system of heat equations ut=Δu and vt=Δv in Ω×(0,T) completely coupled by nonlinear boundary conditions We prove that the solutions always blow up in finite time for non-zero and non-negative initial values. Also, the blow-up only occurs on ∂Ω with for p,q>0, 0≤α<1 and 0≤β<p. 相似文献
9.
We consider the Cauchy–Goursat initial characteristic problem for nonlinear wave equations with power nonlinearity. Depending on the power of nonlinearity and the parameter in an equation we investigate the problem on existence and nonexistence of global solutions of the Cauchy–Goursat problem. The question on local solvability of the problem is also considered. 相似文献
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This paper discusses a class of second-order derivative nonlinear Schrödinger equations which are used to describe the upper-hybrid oscillation propagation. By establishing a variational problem, applying the potential well argument and the concavity method, we prove that there exists a sharp condition for global existence and blow-up of the solutions to the nonlinear Schrödinger equation. In addition, we also answer the question: how small are the initial data, the global solutions exist? 相似文献
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Sining Zheng Xianfa Song Zhaoxin Jiang 《Journal of Mathematical Analysis and Applications》2004,298(1):308-324
We establish the critical Fujita exponents for degenerate parabolic equations coupled via nonlinear boundary flux and then determine the blow-up rates and the blow-up sets for the nonglobal solutions. 相似文献
15.
W. Mydlarczyk W. Okrasiński C.A. Roberts 《Journal of Mathematical Analysis and Applications》2005,301(1):208-218
A system of nonlinear Volterra integral equations with convolution kernels is considered. Estimates are given for the blow-up time when conditions are such that the solution is known to become unbounded in finite time. For two examples that arise in combustion problems, numerical estimates of blow-up time are presented. Additionally, the asymptotic behavior of the blow-up solution in the key limit is established for the power-law and exponential nonlinearity cases. 相似文献
16.
Hailiang Zhang 《Journal of Applied Mathematics and Computing》2010,32(2):535-545
This paper deals with the blow-up of positive solutions for a nonlinear parabolic equation subject to mixed boundary condition. We obtain the conditions under which the solutions may exist globally or blow up in a finite time by a new approach. Moreover, upper estimates of “blow-up time”, blow-up rate and global solutions are obtained also. The results improve and extend importantly the findings obtained by A. Friedman and R. Sperb. 相似文献
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In this paper, we study the 3D compressible magnetohydrodynamic equations. We obtain a blow up criterion for the local strong solutions just in terms of the gradient of the velocity, similar to the Beal-Kato-Majda criterion (see J.T. Beal, T. Kato and A. Majda (1984) [1]) for the ideal incompressible flow. In addition, initial vacuum is allowed in our case. 相似文献
19.
Hamid Ghidouche Philippe Souplet Domingo Tarzia 《Proceedings of the American Mathematical Society》2001,129(3):781-792
We consider a one-phase Stefan problem for the heat equation with a nonlinear reaction term. We first exhibit an energy condition, involving the initial data, under which the solution blows up in finite time in norm. We next prove that all global solutions are bounded and decay uniformly to 0, and that either: (i) the free boundary converges to a finite limit and the solution decays at an exponential rate, or (ii) the free boundary grows up to infinity and the decay rate is at most polynomial. Finally, we show that small data solutions behave like (i).
20.
Mami Suzuki 《Annali di Matematica Pura ed Applicata》2008,187(2):353-368
There is no general existence theorem for solutions for nonlinear difference equations, so we must prove the existence of
solutions in accordance with models one by one. In our work, we found theorems for the existence of analytic solutions of
the following nonlinear second order difference equation,
where f(x,y) is an entire function of x, y. The main work of the present paper is obtaining representations of analytic general solutions of the difference equation
with new methods of complex analysis.
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