共查询到20条相似文献,搜索用时 343 毫秒
1.
This paper deals with the solutions defined for all time of the KPP equation ut = uxx + f(u), 0 < u(x,t) < 1, (x,t) ∈ ℝ2, where ƒ is a KPP‐type nonlinearity defined in [0,1]: ƒ(0) = ƒ(1) = 0, ƒ′(0) > 0, ƒ′(1) < 0, ƒ > 0 in (0,1), and ƒ′(s) ≤ ƒ′(0) in [0,1]. This equation admits infinitely many traveling‐wave‐type solutions, increasing or decreasing in x. It also admits solutions that depend only on t. In this paper, we build four other manifolds of solutions: One is 5‐dimensional, one is 4‐dimensional, and two are 3‐dimensional. Some of these new solutions are obtained by considering two traveling waves that come from both sides of the real axis and mix. Furthermore, the traveling‐wave solutions are on the boundary of these four manifolds. © 1999 John Wiley & Sons, Inc. 相似文献
2.
We show that any entropy solution u of a convection diffusion equation ?t u + div F(u)-Df(u) = b{\partial_t u + {\rm div} F(u)-\Delta\phi(u) =b} in Ω × (0, T) belongs to C([0,T),L1loc(W)){C([0,T),L^1_{\rm loc}({\Omega}))} . The proof does not use the uniqueness of the solution. 相似文献
3.
A. V. Kazeykina 《Computational Mathematics and Mathematical Physics》2010,50(4):690-710
The asymptotic behavior of the solution to the Cauchy problem for the Korteweg-de Vries-Burgers equation u
t
+ (f(u))
x
+ au
xxx
− bu
xx
= 0 as t → ∞ is analyzed. Sufficient conditions for the existence and local stability of a traveling-wave solution known in the case
of f(u) = u
2 are extended to the case of an arbitrary sufficiently smooth convex function f(u). 相似文献
4.
Lorenzo Bertini Alberto De Sole Davide Gabrielli Giovanni Jona‐Lasinio Claudio Landim 《纯数学与应用数学通讯》2011,64(5):649-696
Consider the viscous Burgers equation ut + f(u)x = εuxx on the interval [0,1] with the inhomogeneous Dirichlet boundary conditions u(t,0) = ρ0, u(t,1) = ρ1. The flux f is the function f(u) = u(1 − u), ε > 0 is the viscosity, and the boundary data satisfy 0 < ρ0 < ρ1 < 1. We examine the quasi‐potential corresponding to an action functional arising from nonequilibrium statistical mechanical models associated with the above equation. We provide a static variational formula for the quasi‐potential and characterize the optimal paths for the dynamical problem. In contrast with previous cases, for small enough viscosity, the variational problem defining the quasi‐potential admits more than one minimizer. This phenomenon is interpreted as a nonequilibrium phase transition and corresponds to points where the superdifferential of the quasi‐potential is not a singleton. © 2011 Wiley Periodicals, Inc. 相似文献
5.
I. V. Filimonova 《Journal of Mathematical Sciences》2007,143(4):3415-3428
One considers a semilinear parabolic equation u
t
= Lu − a(x)f(u) or an elliptic equation u
tt
+ Lu − a(x)f(u) = 0 in a semi-infinite cylinder Ω × ℝ+ with the nonlinear boundary condition
, where L is a uniformly elliptic divergent operator in a bounded domain Ω ∈ ℝn; a(x) and b(x) are nonnegative measurable functions in Ω. One studies the asymptotic behavior of solutions of such boundary-value problems
for t → ∞.
__________
Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 26, pp. 368–389, 2007. 相似文献
6.
Xiaojing Yang 《Mathematische Nachrichten》2004,268(1):102-113
In this paper, the boundedness of all solutions of the nonlinear differential equation (φp(x′))′ + αφp(x+) – βφp(x–) + f(x) = e(t) is studied, where φp(u) = |u|p–2 u, p ≥ 2, α, β are positive constants such that = 2w–1 with w ∈ ?+\?, f is a bounded C5 function, e(t) ∈ C6 is 2πp‐periodic, x+ = max{x, 0}, x– = max{–x, 0}. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
7.
In this paper, we consider the following nonlinear fractional three-point boundary-value problem:
*20c D0 + a u(t) + f( t,u(t) ) = 0, 0 < t < 1, u(0) = u¢(0) = 0, u¢(1) = ò0h u(s)\textds, \begin{array}{*{20}{c}} {D_{0 + }^\alpha u(t) + f\left( {t,u(t)} \right) = 0,\,\,\,\,0 < t < 1,} \\ {u(0) = u'(0) = 0,\,\,\,\,u'(1) = \int\limits_0^\eta {u(s){\text{d}}s,} } \\ \end{array} 相似文献
8.
We derive an algorithm for solving the initial value problem for ut = ½σ2uxx + f(u)ux. The approach is based on the representation of the solution to the above equation in the form of the functional of Brownian motion. For small σ we get the approximation for ut = f(u)ux. A comparison with the random choice method is illustrated by the numerical example. 相似文献
9.
Rossitza I. Semerdjieva 《Mathematische Nachrichten》2002,237(1):89-104
Let k(y) > 0, 𝓁(y) > 0 for y > 0, k(0) = 𝓁(0) = 0 and limy → 0k(y)/𝓁(y) exists; then the equation L(u) ≔ k(y)uxx – ∂y(𝓁(y)uy) + a(x, y)ux = f(x, y, u) is strictly hyperbolic for y > 0 and its order degenerates on the line y = 0. Consider the boundary value problem Lu = f(x, y, u) in G, u|AC = 0, where G is a simply connected domain in ℝ2 with piecewise smooth boundary ∂G = AB∪AC∪BC; AB = {(x, 0) : 0 ≤ x ≤ 1}, AC : x = F(y) = ∫y0(k(t)/𝓁(t))1/2dt and BC : x = 1 – F(y) are characteristic curves. Existence of generalized solution is obtained by a finite element method, provided f(x, y, u) satisfies Carathéodory condition and |f(x, y, u)| ≤ Q(x, y) + b|u| with Q ∈ L2(G), b = const > 0. It is shown also that each generalized solution is a strong solution, and that fact is used to prove uniqueness under the additional assumption |f(x, y, u1) – f(x, y, u2| ≤ C|u1 – u2|, where C = const > 0. 相似文献
10.
We report a new unconditionally stable implicit alternating direction implicit (ADI) scheme of O(k2 + h2) for the difference solution of linear hyperbolic equation utt + 2αut + β2u = uxx + uyy + f(x, y, t), αβ ≥ 0, 0 < x, y < 1, t > 0 subject to appropriate initial and Dirichlet boundary conditions, where α > 0 and β ≥ 0 are real numbers. The resulting system of algebraic equations is solved by split method. Numerical results are provided to demonstrate the efficiency and accuracy of the method. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 684–688, 2001 相似文献
11.
Evangelos A. Latos Dimitrios E. Tzanetis 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(2):137-151
We investigate the behaviour of solution u = u(x, t; λ) at λ = λ* for the non-local porous medium equation ${u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2}
12.
We consider solutions of the singular diffusion equation t, = (um?1 ux)x, m ≦ 0, associated with the flux boundary condition limx→?∞ (um?1ux)x = λ > 0. The evolutions defined by this problem depend on both m and λ. We prove existence and stability of traveling wave solutions, parameterized by λ. Each traveling wave is stable in its appropriate evolution. These traveling waves are in L1 for ?1 < m ≦ 0, but have non-integrable tails for m ≦ ?1. We also show that these traveling waves are the same as those for the scalar conservation law ut = ?[f(u)]x + uxx for a particular nonlinear convection term f(u) = f(u;m, λ). © 1993 John Wiley & Sons, Inc. 相似文献
13.
Zongming Guo 《Mathematische Nachrichten》2004,267(1):12-36
The structure of nontrivial nonnegative solutions to singularly perturbed quasilinear Dirichlet problems of the form –?Δpu = f(u) in Ω, u = 0 on ?Ω, Ω ? R N a bounded smooth domain, is studied as ? → 0+, for a class of nonlinearities f(u) satisfying f(0) = f(z1) = f(z2) = 0 with 0 < z1 < z2, f < 0 in (0, z1), f > 0 in (z1, z2) and f(u)/up–1 = –∞. It is shown that there are many nontrivial nonnegative solutions with spike‐layers. Moreover, the measure of each spike‐layer is estimated as ? → 0+. These results are applied to the study of the structure of positive solutions of the same problems with f changing sign many times in (0,∞). Uniqueness of a solution with a boundary‐layer and many positive intermediate solutions with spike‐layers are obtained for ? sufficiently small. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
14.
This article deals with a class of nonlocal and degenerate quasilinear parabolic equation u t = f(u)(Δu + a∫Ω u(x, t)dx ? u) with homogeneous Dirichlet boundary conditions. The local existence of positive classical solutions is proved by using the method of regularization. The global existence of positive solutions and blow-up criteria are also obtained. Furthermore, it is shown that, under certain conditions, the solutions have global blow-up property. When f(s) = s p , 0 < p ≤ 1, the blow-up rate estimates are also obtained. 相似文献
15.
Junping Shi 《纯数学与应用数学通讯》2002,55(7):815-830
We prove that Δu + f(u) = 0 has a unique entire solution u(x, y) on ?2 which has the same sign as the function xy, where f is a balanced bistable function like f(u) = u ? u3. But we neither assume f is odd nor assume the monotonicity properties of f(u)/u. Our result generalizes a previous result by Dang, Fife, and Peletier [12]. Our approach combines bifurcation methods and recent results on the qualitative properties for elliptic equations in unbounded domains by Berestycki, Caffarelli and Nirenberg [5, 6]. © 2002 Wiley Periodicals, Inc. 相似文献
16.
Xiaojing Yang 《Mathematische Nachrichten》2004,276(1):89-102
In this paper, we consider the unboundedness of solutions of the following differential equation (φp(x′))′ + (p ? 1)[αφp(x+) ? βφp(x?)] = f(x)x′ + g(x) + h(x) + e(t) where φp(u) = |u|p? 2 u, p > 1, x± = max {±x, 0}, α and β are positive constants satisfying with m, n ∈ N and (m, n) = 1, f and g are continuous and bounded functions such that limx→±∞g(x) ? g(±∞) exists and h has a sublinear primitive, e(t) is 2πp‐periodic and continuous. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
17.
Maria E. Schonbek 《Mathematische Annalen》2006,336(3):505-538
This paper considers the existence and large time behavior of solutions to the convection-diffusion equation u
t
−Δu+b(x)·∇(u|u|
q
−1)=f(x, t) in ℝ
n
×[0,∞), where f(x, t) is slowly decaying and q≥1+1/n (or in some particular cases q≥1). The initial condition u
0 is supposed to be in an appropriate L
p
space. Uniform and nonuniform decay of the solutions will be established depending on the data and the forcing term.This work is partially supported by an AMO Grant 相似文献
18.
A. ada 《Mathematical Methods in the Applied Sciences》1996,19(3):217-233
The mixed-Neumann problem for the non-linear wave equation □u−a(u)(∣∂tu)∣2−∣∇u∣2 = fε(z) is studied. The function fε(z) = ∑k∈Kfk(z,ε−1ϕk(z),ε), ε∈[0,1], K is finite, fk(z,θk,ε) are 2π-periodic with respect to θk. The existence of solution uε on a domain z = (t,x,y)∈[0,T]×ℝ+×ℝd, d = 1 or 2, is proved when ε is sufficiently small; T does not depend on ε. By the non-linear geometric optics method the asymptotic (with respect to ε→0) solution ũ ε is constructed. The estimation for the rest ε2rε = uε−ũε is derived and the limit rε, ε→0, is studied. 相似文献
19.
《数学物理学报(B辑英文版)》1999,19(4):382-390
Using the shooting argument and an approximating method, this paper is concerned with the existence of fast-decay ground state of p-Laplacian equation: Δpu + f(u) = 0, in Rn, where f(u) behaves just like f(u) = uq – us, as s > q >np/(n – p) – 1. 相似文献
20.
H. Stetkær 《Aequationes Mathematicae》1997,54(1-2):144-172
Summary We produce complete solution formulas of selected functional equations of the formf(x +y) ±f(x + σ (ν)) = Σ
I
2
=1
g
l
(x)h
l
(y),x, y∈G, where the functionsf,g
1,h
1 to be determined are complex valued functions on an abelian groupG and where σ:G→G is an involution ofG. The special case of σ=−I encompasses classical functional equations like d’Alembert’s, Wilson’s first generalization of it, Jensen’s equation and
the quadratic equation. We solve these equations, the equation for symmetric second differences in product form and similar
functional equations for a general involution σ. 相似文献
|