共查询到20条相似文献,搜索用时 31 毫秒
1.
This paper deals with the Cauchy problem ut − uxx + up = 0; − ∞ < x < + ∞, t>0, u(x, 0) = u0(x); − ∞ < x < + ∞, where 0 < p < 1 and u0(x) is continuous, nonnegative, and bounded. In this case, solutions are known to vanish in a finite time T, and interfaces separating the regions where u(x, t) > 0 and u(x, t) = 0 appear when t is close to T. We describe here all possible asymptotic behaviours of solutions and interfaces near an extinction point as the extinction time is approached. We also give conditions under which some of these behaviours actually occur. 相似文献
2.
Fucai Li 《Applied Mathematics Letters》2004,17(12):686
This paper deals with the higher-order Kirchhoff-type equation with nonlinear dissipationutt+(∫Ω׀Dmu׀2dx)q(−Δ)mu+ut׀ut׀r=׀u׀pu,xΩ,t>0,in a bounded domain, where m < 1 is a positive integer, q, p, r < 0 arepositive constants. We obtain that the solution exists globally if p ≤ r, while ifp > max r, 2q , then for any initial data with negative initial energy, the solution blowsup at finite time in Lp+2 norm. 相似文献
3.
In this paper we study the existence of periodic solutions of the fourth-order equations uiv − pu″ − a(x)u + b(x)u3 = 0 and uiv − pu″ + a(x)u − b(x)u3 = 0, where p is a positive constant, and a(x) and b(x) are continuous positive 2L-periodic functions. The boundary value problems (P1) and (P2) for these equations are considered respectively with the boundary conditions u(0) = u(L) = u″(0) = u″(L) = 0. Existence of nontrivial solutions for (P1) is proved using a minimization theorem and a multiplicity result using Clark's theorem. Existence of nontrivial solutions for (P2) is proved using the symmetric mountain-pass theorem. We study also the homoclinic solutions for the fourth-order equation uiv + pu″ + a(x)u − b(x)u2 − c(x)u3 = 0, where p is a constant, and a(x), b(x), and c(x) are periodic functions. The mountain-pass theorem of Brezis and Nirenberg and concentration-compactness arguments are used. 相似文献
4.
Li Jiayu 《Journal of Functional Analysis》1991,100(2)
We derive the gradient estimates and Harnack inequalities for positive solutions of nonlinear parabolic and nonlinear elliptic equations (Δ − ∂/∂t) u(x, t) + h(x, t)uα(x, t) = 0 and Δu + b · u + huα = 0 on Riemannian manifolds. We also obtain a theorem of Liouville type for positive solutions of the nonlinear elliptic equation. 相似文献
5.
In this piece of work, we introduce a new idea and obtain stability interval for explicit difference schemes of O(k2+h2) for one, two and three space dimensional second-order hyperbolic equations utt=a(x,t)uxx+α(x,t)ux-2η2(x,t)u,utt=a(x,y,t)uxx+b(x,y,t)uyy+α(x,y,t)ux+β(x,y,t)uy-2η2(x,y,t)u, and utt=a(x,y,z,t)uxx+b(x,y,z,t)uyy+c(x,y,z,t)uzz+α(x,y,z,t)ux+β(x,y,z,t)uy+γ(x,y,z,t)uz-2η2(x,y,z,t)u,0<x,y,z<1,t>0 subject to appropriate initial and Dirichlet boundary conditions, where h>0 and k>0 are grid sizes in space and time coordinates, respectively. A new idea is also introduced to obtain explicit difference schemes of O(k2) in order to obtain numerical solution of u at first time step in a different manner. 相似文献
6.
E. Kimchi 《Journal of Approximation Theory》1978,24(4):350-360
Let {u0, u1,… un − 1} and {u0, u1,…, un} be Tchebycheff-systems of continuous functions on [a, b] and let f ε C[a, b] be generalized convex with respect to {u0, u1,…, un − 1}. In a series of papers ([1], [2], [3]) D. Amir and Z. Ziegler discuss some properties of elements of best approximation to f from the linear spans of {u0, u1,…, un − 1} and {u0, u1,…, un} in the Lp-norms, 1 p ∞, and show (under different conditions for different values of p) that these properties, when valid for all subintervals of [a, b], can characterize generalized convex functions. Their methods of proof rely on characterizations of elements of best approximation in the Lp-norms, specific for each value of p. This work extends the above results to approximation in a wider class of norms, called “sign-monotone,” [6], which can be defined by the property: ¦ f(x)¦ ¦ g(x)¦,f(x)g(x) 0, a x b, imply f g . For sign-monotone norms in general, there is neither uniqueness of an element of best approximation, nor theorems characterizing it. Nevertheless, it is possible to derive many common properties of best approximants to generalized convex functions in these norms, by means of the necessary condition proved in [6]. For {u0, u1,…, un} an Extended-Complete Tchebycheff-system and f ε C(n)[a, b] it is shown that the validity of any of these properties on all subintervals of [a, b], implies that f is generalized convex. In the special case of f monotone with respect to a positive function u0(x), a converse theorem is proved under less restrictive assumptions. 相似文献
7.
For the equation K(t)u
xx
+ u
tt
− b
2
K(t)u = 0 in the rectangular domain D = “(x, t)‖ 0 < x < 1, −α < t < β”, where K(t) = (sgnt)|t|
m
, m > 0, and b > 0, α > 0, and β > 0 are given real numbers, we use the spectral method to obtain necessary and sufficient conditions for the unique solvability
of the boundary value problem u(0, t) = u(1, t), u
x
(0, t) = u
x
(1, t), −α ≤ t ≤ β, u(x, β) = φ(x), u(x,−α) = ψ(x), 0 ≤ x ≤ 1. 相似文献
8.
We investigate the large-time behaviour of solutions to the nonlinear heat-conduction equation with absorption ut = Δ(uσ + 1) − uβ in Q = RN × (0, ∞) (E) with N 1, σ > 0 and critical absorption exponent β = σ + 1 + 2/N; the initial function u(x, 0) = 0 is assumed to be integrable, nonnegative and compactly supported. We prove that u converges as t → ∞ to a unique self-similar function which is a contracted version of one of the asymptotic profiles of the nonabsorptive problem ut = Δ(uσ + 1), the same for any initial data. The cornerstone of the proof is a result about ω-limits of (infinite-dimensional) asymptotical dynamical systems. Combining this result with an asymptotic evaluation of the mass function as well as typical PDE estimates gives the behaviour of (E) for large times.Similar unusual asymptotic behaviour is obtained for the equation ut = div(¦Du¦σ Du) − uβ with same conditions on σ and u(x, 0) and critical value for β = σ + 1 + (σ + 2)/N. 相似文献
9.
The main purpose of this article is to establish nearly optimal results concerning the rate of almost everywhere convergence of the Gauss–Weierstrass, Abel–Poisson, and Bochner–Riesz means of the one-dimensional Fourier integral. A typical result for these means is the following: If the function f belongs to the Besov spaceBsp, p, 1<p<∞, 0<s<1, thenTmtf (x)−f(x)=ox(ts) a.e. ast→0+. 相似文献
10.
A. G. Ramm 《Applied Mathematics Letters》1988,1(3)
Let·(σ(x)u)= 0 in D R3, where D is a bounded domain with a smooth boundary. Suppose that σ ≥ m> 0, σ H3(D), where Hℓ is the Sobolev space. Let the set {u, σuN} be given on Γ for all u H3/2(Γ), where uN is the normal derivative of u on Γ. 相似文献
11.
For all integers m3 and all natural numbers a1,a2,…,am−1, let n=R(a1,a2,…,am−1) represent the least integer such that for every 2-coloring of the set {1,2,…,n} there exists a monochromatic solution to
a1x1+a2x2++am−1xm−1=xm.