共查询到20条相似文献,搜索用时 15 毫秒
1.
Huabing Jia Wei Xu Xiaoshan Zhao Zhanguo Li 《Journal of Mathematical Analysis and Applications》2008,339(2):982-995
This paper considers nonlinear diffusion equations with x-dependent convection and source terms: ut=(Dx(u)ux)+Q(x,u)ux+P(x,u). The functional separation of variables of the equations is studied by using the generalized conditional symmetry approach. We formulate conditions for such equations which admit the functionally separable solutions. As a consequence, some exact solutions to the resulting equations are constructed. Finally, we consider a special case for the equations which admit the functionally separable solutions when the convection and source terms are independent of x. 相似文献
2.
Separation of variables of a generalized porous medium equation with nonlinear source 总被引:2,自引:0,他引:2
P.G. Estévez Changzheng QuShunli Zhang 《Journal of Mathematical Analysis and Applications》2002,275(1):44-59
This paper considers a general form of the porous medium equation with nonlinear source term: ut=(D(u)uxn)x+F(u), n≠1. The functional separation of variables of this equation is studied by using the generalized conditional symmetry approach. We obtain a complete list of canonical forms for such equations which admit the functional separable solutions. As a consequence, some exact solutions to the resulting equations are constructed, and their behavior are also investigated. 相似文献
3.
Joo-Paulo Dias Mrio Figueira Luis Sanchez 《Mathematical Methods in the Applied Sciences》1998,21(12):1107-1113
In this paper we consider the Cauchy problem for the equation ∂u/∂t + u ∂u/∂x + u/x = 0 for x > 0, t ⩾ 0, with u(x, 0) = u0−(x) for x < x0, u(x, 0) = u0+(x) for x > x0, u0−(x0) > u0+(x0). Following the ideas of Majda, 1984 and Lax, 1973, we construct, for smooth u0− and u0+, a global shock front weak solution u(x, t) = u−(x, t) for x < ϕ(t), u(x, t) = u+(x, t) for x > ϕ(t), where u− and u+ are the strong solutions corresponding (respectively) to u0− and u0+ and the curve t → ϕ(t) is defined by dϕ/dt (t) = 1/2[u−(ϕ(t), t) + u+(ϕ(t), t)], t ⩾ 0 and ϕ(0) = x0. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd. 相似文献
4.
G. Gripenberg S.-O. Londen J. Prüss 《Mathematical Methods in the Applied Sciences》1997,20(16):1427-1448
It is proved that there is a (weak) solution of the equation ut=a*uxx+b*g(ux)x+f, on ℝ+ (where * denotes convolution over (−∞, t)) such that ux is locally bounded. Emphasis is put on having the assumptions on the initial conditions as weak as possible. The kernels a and b are completely monotone and if a(t)=t−α, b(t)=t−β, and g(ξ)∼sign(ξ)∣ξ∣γ for large ξ, then the main assumption is that α>(2γ+2)/(3γ+1)β+(2γ−2)/(3γ+1). © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd. 相似文献
5.
Maisa Khader 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(12):3945-3963
We study the long time behavior of solutions for damped wave equations with absorption. These equations are generally accepted as models of wave propagation in heterogeneous media with space-time dependent friction a(t,x)ut and nonlinear absorption |u|p−1u (Ikawa (2000) [17]). We consider 1<p<(n+2)/(n−2) and separable a(t,x)=λ(x)η(t) with λ(x)∼(1+|x|)−α and η(t)∼(1+t)−β satisfying conditions (A1) or (A2) which are given. The main results are precise decay estimates for the energy, L2 and Lp+1 norms of solutions. We also observe the following behavior: if α∈[0,1), β∈(−1,1) and 0<α+β<1, there are three different regions for the decay of solutions depending on p; if α∈(−∞,0) and β∈(−1,1), there are only two different regions for the decay of the solutions depending on p. 相似文献
6.
《Applied Mathematics Letters》2002,15(6):727-734
This paper is concerned with the asymptotic behaviors of the solutions to the initial-boundary value problem for scalar viscous conservations laws ut + f(u)x = uxx on [0, 1], with the boundary condition u(0, t) = u−(t) → u−, u(1, t) = u+(t) → u+, as t → +∞ and the initial data u(x,0) = u0(x) satisfying u0(0) = u−(0), u0(1) = u+(1), where u± are given constants, u− ≠ u+ and f is a given function satisfying f″(u) > 0 for u under consideration. By means of an elementary energy estimates method, both the global existence and the asymptotic behavior are obtained. When u− 〈 u+, which corresponds to rarefaction waves in inviscid conservation laws, no smallness conditions are needed. While for u− > u+, which corresponds to shock waves in inviscid conservation laws, it is established for weak shock waves, that is, |u−−u+| is small. Moreover, when u±(t) ≡ u±, t ≥ 0, exponential decay rates are both obtained. 相似文献
7.
N. G. Khoma 《Ukrainian Mathematical Journal》1998,50(11):1755-1764
In three spaces, we obtain exact classical solutions of the boundary-value periodic problem u
tt−a
2
u
xx=g(x,t), u(0,t)=u(π,t)=0, u(x,t+T)=u(x,t)=0, x,t∈ĝ
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 11, pp. 1537–1544, November, 1998. 相似文献
8.
Yacheng Liu 《Journal of Mathematical Analysis and Applications》2008,338(2):1169-1187
In this paper we study Cauchy problem of generalized double dispersion equations utt−uxx−uxxtt+uxxxx=f(u)xx, where f(u)=p|u|, p>1 or u2k, . By introducing a family of potential wells we not only get a threshold result of global existence and nonexistence of solutions, but also obtain the invariance of some sets and vacuum isolating of solutions. In addition, the global existence and finite time blow up of solutions for problem with critical initial conditions E(0)=d, I(u0)?0 or I(u0)<0 are proved. 相似文献
9.
Patrick S. Hagan 《Studies in Applied Mathematics》1981,64(1):57-88
We consider the class of equations ut=f(uxx, ux, u) under the restriction that for all a,b,c. We first consider this equation over the unbounded domain ? ∞ < x < + ∞, and we show that very nearly every bounded nonmonotonic solution of the form u(t, x)=?(x?ct) is unstable to all nonnegative and all nonpositive perturbations. We then extend these results to nonmonotonic plane wave solutions u(t, x, y)=?(x?ct) of ut = F(uxx, uxy, ux, uy, u). Finally, we consider the class of equations ut=f(uxx, ux, u) over the bounded domain 0 < x < 1 with the boundary conditions u(t, x)=A at x=0 and u(t, x)=B at x=1, and we find the stability of all steady solutions u(t, x)=?(x). 相似文献
10.
11.
Paschalis Karageorgis P.J. McKenna 《Nonlinear Analysis: Theory, Methods & Applications》2010,73(2):367-2727
Focusing on the fourth-order wave equation utt+Δ2u+f(u)=0, we prove the existence of ground state solutions u=u(x+ct) for an optimal range of speeds c∈Rn and a variety of nonlinearities f. 相似文献
12.
Zhifei Zhang 《Journal of Mathematical Analysis and Applications》2010,363(2):549-558
We discuss the existence of periodic solutions to the wave equation with variable coefficients utt−div(A(x)∇u)+ρ(x,ut)=f(x,t) with Dirichlet boundary condition. Here ρ(x,v) is a function like ρ(x,v)=a(x)g(v) with g′(v)?0 where a(x) is nonnegative, being positive only in a neighborhood of a part of the domain. 相似文献
13.
In this paper, we consider one-dimensional nonlinear Schrödinger equation iut−uxx+V(x)u+f(2|u|)u=0 on [0,π]×R under the boundary conditions a1u(t,0)−b1ux(t,0)=0, a2u(t,π)+b2ux(t,π)=0, , for i=1,2. It is proved that for a prescribed and analytic positive potential V(x), the above equation admits small-amplitude quasi-periodic solutions corresponding to d-dimensional invariant tori of the associated infinite-dimensional dynamical system. 相似文献
14.
In this paper, we study the existence, uniqueness and asymptotic stability of travelling wavefronts of the following equation:
ut(x,t)=D[u(x+1,t)+u(x-1,t)-2u(x,t)]-du(x,t)+b(u(x,t-r)), 相似文献
15.
Luiz Gustavo Farah 《Journal of Differential Equations》2010,249(8):1968-1985
We prove that the initial value problem (IVP) for the critical generalized KdV equation ut+uxxx+x(u5)=0 on the real line is globally well-posed in Hs(R) if s>3/5 with the appropriate smallness assumption on the initial data. 相似文献
16.
Xiaojing Yang 《Journal of Differential Equations》2002,183(1):108-131
In this paper, the boundedness of all solutions of the nonlinear equation (?p(x′))′+(p-1)[α?p(x+)−β?p(x−)]+f(x)+g(x)=e(t) is discussed, where e(t)∈C7 is 2πp-periodic, f,g are bounded C6 functions, ?p(u)=∣u∣p−2u, p?2,α,β are positive constants, x+=max{x,0},x−=max{−x,0}. 相似文献
17.
In this paper, we study the nonlinear one-dimensional periodic wave equation with x-dependent coefficients u(x)ytt−(ux(x)yx)+g(x,t,y)=f(x,t) on (0,π)×R under the boundary conditions a1y(0,t)+b1yx(0,t)=0, a2y(π,t)+b2yx(π,t)=0 ( for i=1,2) and the periodic conditions y(x,t+T)=y(x,t), yt(x,t+T)=yt(x,t). Such a model arises from the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. A main concept is the notion “weak solution” to be given in Section 2. For T is the rational multiple of π, we prove some important properties of the weak solution operator. Based on these properties, the existence and regularity of weak solutions are obtained. 相似文献
18.
A new approach to group classification problems and more general investigations on transformational properties of classes
of differential equations is proposed. It is based on mappings between classes of differential equations, generated by families
of point transformations. A class of variable coefficient (1+1)-dimensional semilinear reaction–diffusion equations of the
general form f(x)u
t
=(g(x)u
x
)
x
+h(x)u
m
(m≠0,1) is studied from the symmetry point of view in the framework of the approach proposed. The singular subclass of the equations
with m=2 is singled out. The group classifications of the entire class, the singular subclass and their images are performed with
respect to both the corresponding (generalized extended) equivalence groups and all point transformations. The set of admissible
transformations of the imaged class is exhaustively described in the general case m≠2. The procedure of classification of nonclassical symmetries, which involves mappings between classes of differential equations,
is discussed. Wide families of new exact solutions are also constructed for equations from the classes under consideration
by the classical method of Lie reductions and by generation of new solutions from known ones for other equations with point
transformations of different kinds (such as additional equivalence transformations and mappings between classes of equations). 相似文献
19.
Bin Guo 《Journal of Mathematical Analysis and Applications》2011,374(2):374-384
The authors of this paper study the existence and uniqueness of weak solutions of the initial and boundary value problem for ut=div((uσ+d0)|∇u|p(x,t)−2∇u)+f(x,t). Localization property of weak solutions is also discussed. 相似文献
20.
We establish a maximum principle for viscosity subsolutions and supersolutions of equations of the form ut+F(t,dxu)=0, u(0,x)=u0(x), where is a bounded uniformly continuous function, M is a Riemannian manifold, and . This yields uniqueness of the viscosity solutions of such Hamilton-Jacobi equations. 相似文献