共查询到20条相似文献,搜索用时 30 毫秒
1.
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
2.
Alemdar Hasanov 《Applied mathematics and computation》2003,140(2-3):501-515
An inverse polynomial method of determining the unknown leading coefficient k=k(x) of the linear Sturm–Liouville operator Au=−(k(x)u′(x))′+q(x)u(x), x(0,1), is presented. As an additional condition only two measured data at the boundary (x=0,x=1) are used. In absence of a singular point (u′(x)≠0,u″(x)≠0,x[0,1]) the inverse problem is classified as a well-conditioned . If there exists at least one singular point, then the inverse problem is classified as moderately ill-conditioned (u′(x0)=0,x0(0,1);u′(x)≠0,x≠x0;u″(x)≠0,x[0,1]) and severely ill-conditioned (u′(x0)=u″(x0)=0,x0(0,1);u′(x)≠0,u″(x)≠0,x≠x0). For each of the cases direct problem solution is approximated by corresponding polynomials and the inverse problem is reformulated as a Cauchy problem for to the first order differential equation with respect the unknown function k=k(x). An approximate analytical solution of the each Cauchy problems are derived in explicit form. Numerical simulations all the above cases are given for noise free and noisy data. An accuracy of the presented approach is demonstrated on numerical test solutions. 相似文献
3.
Wei Dong 《Journal of Mathematical Analysis and Applications》2004,290(2):469-480
In this paper, we consider positive solutions of the logistic type p-Laplacian equation −Δpu=a(x)|u|p−2u−b(x)|u|q−1u, xRN (N2). We show that under rather general conditions on a(x) and b(x) for large |x|, the behavior of the positive solutions for large |x| can be determined. This is then used to show that there is a unique positive solution. Our results improve the corresponding ones in J. London Math. Soc. (2) 64 (2001) 107–124 and J. Anal. Math., in press. 相似文献
4.
The n × n generalized Pascal matrix P(t) whose elements are related to the hypergeometric function 2F1(a, b; c; x) is presented and the Cholesky decomposition of P(t) is obtained. As a result, it is shown that
is the solution of the Gauss's hypergeometric differential equation, . where a and b are any nonnegative integers. Moreover, a recurrence relation for generating the elements of P(t) is given. 相似文献
Full-size image
x(1 − x)y″ + [1 + (a + b − 1)x]y′ − ABY = 0
5.
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
6.
We consider the nonnegative solutions to the nonlinear degenerate parabolic equation ut = (D(x, t)um − 1ux)x − b(x, t)up with m > 1, 0 < p < 1, and positive D(x, t), b(x, t). After obtaining the uniqueness and Hölder regularity results, we investigate the dependence of such phenomena as extinction in finite time and instantaneous shrinking of the support on the behaviour of D(x, t) and b(x, t). 相似文献
7.
Anthony Leung 《Journal of Mathematical Analysis and Applications》1975,50(3):560-578
Asymptotic expansions as ε → 0+ or x → ∞ for fundamental systems of solutions for ε2u″(x) − p(x) u(x) = 0 were obtained by Evgrafov and Fedoryuk on unbounded canonical domains with neighborhoods deleted around turning points. When p(x) is a polynomial, lateral connection formulas were found by Evgrafov and Fedoryuk, and Leung for two fundamental systems of solutions with known expansions in the interior of two different unbounded overlapping canonical regions with common first or second order turning point at their boundaries. In this paper, lateral connection formulas are found when the turning points are of any higher order. Recent result of uniform simplification by Sibuya around higher order turning point is used. 相似文献
8.
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
9.
Let Ω be a plane bounded region. Let U = {Uμ(P):μ(P)εL∞(Ω), uμ ε H22, 0(Ω) and a(P, μ(P))uμ,xx + 2b(P, μ(P))uμ,xy + c(P, μ(P))uμ,vv = ƒ(P) for P ε Ω; here we are given a(P, X), b(P, X), c(P, X) ε L∞(Ω × E1), ƒ(P) ε Lp(Ω) with p > 2, and our partial differential equation is uniformly elliptic. The functions μ(P) are called profiles. We establish sufficient conditions—which when they apply are constructive—that there exist a μ0 ε L∞(Ω) such that uμ0 (P) uμ(P) for all P ε Ω and for each μ ε L∞(Ω). Similar results are obtained for a difference equation and convergence is proved. 相似文献
10.
Let ga(t) and gb(t) be two positive, strictly convex and continuously differentiable functions on an interval (a, b) (−∞ a < b ∞), and let {Ln} be a sequence of linear positive operators, each with domain containing 1, t, ga(t), and gb(t). If Ln(ƒ; x) converges to ƒ(x) uniformly on a compact subset of (a, b) for the test functions ƒ(t) = 1, t, ga(t), gb(t), then so does every ƒ ε C(a, b) satisfying ƒ(t) = O(ga(t)) (t → a+) and ƒ(t) = O(gb(t)) (t → b−). We estimate the convergence rate of Lnƒ in terms of the rates for the test functions and the moduli of continuity of ƒ and ƒ′. 相似文献
11.
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
12.
For a bounded linear injectionCon a Banach spaceXand a closed linear operatorA : D(A) X → Xwhich commutes withCwe prove that (1) the abstract Cauchy problem,u″(t) = Au(t),t R,u(0) = Cx,u′(0) = Cy, has a unique strong solution for everyx,y D(A) if and only if (2)A1 = AD(A2) generates aC1-cosine function onX1(D(A) with the graph norm), if (and only if, in caseAhas nonempty resolvent set) (3)Agenerates aC-cosine function onX. HereC1 = CX1. Under the assumption thatAis densely defined andC−1AC = A, statement (3) is also equivalent to each of the following statements: (4) the problemv″(t) = Av(t) + C(x + ty) + ∫t0 Cg(r) dr,t R,v(0) = v′(0) = 0, has a unique strong solution for everyg L1locandx, y X; (5) the problemw″(t) = Aw(t) + Cg(t),t R,w(0) = Cx,w′(0) = Cy, has a unique weak solution for everyg L1locandx, y X. Finally, as an application, it is shown that for any bounded operatorBwhich commutes withCand has range contained in the range ofC,A + Bis also a generator. 相似文献
13.
In this paper, we study the existence of periodic solutions for a fourth-order p-Laplacian differential equation with a deviating argument as follows:
[φp(u″(t))]″+f(u″(t))+g(u(t−τ(t)))=e(t).