共查询到20条相似文献,搜索用时 998 毫秒
1.
P. Quittner W. Reichel 《Calculus of Variations and Partial Differential Equations》2008,32(4):429-452
Consider the equation −Δu = 0 in a bounded smooth domain , complemented by the nonlinear Neumann boundary condition ∂ν
u = f(x, u) − u on ∂Ω. We show that any very weak solution of this problem belongs to L
∞(Ω) provided f satisfies the growth condition |f(x, s)| ≤ C(1 + |s|
p
) for some p ∈ (1, p*), where . If, in addition, f(x, s) ≥ −C + λs for some λ > 1, then all positive very weak solutions are uniformly a priori bounded. We also show by means of examples that
p* is a sharp critical exponent. In particular, using variational methods we prove the following multiplicity result: if N ∈ {3, 4} and f(x, s) = s
p
then there exists a domain Ω and such that our problem possesses at least two positive, unbounded, very weak solutions blowing up at a prescribed point of
∂Ω provided . Our regularity results and a priori bounds for positive very weak solutions remain true if the right-hand side in the differential
equation is of the form h(x, u) with h satisfying suitable growth conditions. 相似文献
2.
Yu. K. Sabitova 《Russian Mathematics (Iz VUZ)》2009,53(12):41-49
We consider the equation y
m
u
xx
− u
yy
− b
2
y
m
u = 0 in the rectangular area {(x, y) | 0 < x < 1, 0 < y < T}, where m < 0, b ≥ 0, T > 0 are given real numbers. For this equation we study problems with initial conditions u(x, 0) = τ(x), u
y
(x, 0) = ν(x), 0 ≤ x ≤ 1, and nonlocal boundary conditions u(0, y) = u(1, y), u
x
(0, y) = 0 or u
x
(0, y) = u
x
(1, y), u(1, y) = 0 with 0≤y≤T. Using the method of spectral analysis, we prove the uniqueness and existence theorems for solutions to these problems 相似文献
3.
On critical Fujita exponents for heat equations with nonlinear flux conditions on the boundary 总被引:1,自引:0,他引:1
We consider nonnegative solutions of initial-boundary value problems for parabolic equationsu
t=uxx, ut=(um)xxand
(m>1) forx>0,t>0 with nonlinear boundary conditions−u
x=up,−(u
m)x=upand
forx=0,t>0, wherep>0. The initial function is assumed to be bounded, smooth and to have, in the latter two cases, compact support. We prove
that for each problem there exist positive critical valuesp
0,pc(withp
0<pc)such that forp∃(0,p
0],all solutions are global while forp∃(p0,pc] any solutionu≢0 blows up in a finite time and forp>p
csmall data solutions exist globally in time while large data solutions are nonglobal. We havep
c=2,p
c=m+1 andp
c=2m for each problem, whilep
0=1,p
0=1/2(m+1) andp
0=2m/(m+1) respectively.
This work was done during visits of the first author to Iowa State University and the Institute for Mathematics and its Applications
at the University of Minnesota. The second author was supported in part by NSF Grant DMS-9102210. 相似文献
4.
We consider the following singularly perturbed boundary-value problem:
on the interval 0 ≤x ≤ 1. We study the existence and uniqueness of its solutionu(x, ε) having the following properties:u(x, ε) →u
0(x) asε → 0 uniformly inx ε [0, 1], whereu
0(x) εC
∞ [0, 1] is a solution of the degenerate equationf(x, u, u′)=0; there exists a pointx
0 ε (0, 1) such thata(x
0)=0,a′(x
0) > 0,a(x) < 0 for 0 ≤x <x
0, anda(x) > 0 forx
0 <x ≤ 1, wherea(x)=f′
v(x,u
0(x),u′
0(x)).
Translated fromMatematicheskie Zametki, Vol. 67, No. 4, pp. 520–524, April, 2000. 相似文献
5.
An Application of a Mountain Pass Theorem 总被引:3,自引:0,他引:3
We are concerned with the following Dirichlet problem:
−Δu(x) = f(x, u), x∈Ω, u∈H
1
0(Ω), (P)
where f(x, t) ∈C (×ℝ), f(x, t)/t is nondecreasing in t∈ℝ and tends to an L
∞-function q(x) uniformly in x∈Ω as t→ + ∞ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case, an Ambrosetti-Rabinowitz-type condition, that is, for some θ > 2, M > 0,
0 > θF(x, s) ≤f(x, s)s, for all |s|≥M and x∈Ω, (AR)
is no longer true, where F(x, s) = ∫
s
0
f(x, t)dt. As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming
(AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable
conditions on f(x, t) and q(x). Our methods also work for the case where f(x, t) is superlinear in t at infinity, i.e., q(x) ≡ +∞.
Received June 24, 1998, Accepted January 14, 2000. 相似文献
6.
We study the large time behaviour of nonnegative solutions of the Cauchy problemu
t=Δu
m −u
p,u(x, 0)=φ(x). Specifically we study the influence of the rate of decay ofφ(x) for large |x|, and the competition between the diffusion and the absorption term. 相似文献
7.
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}We investigate the behaviour of solution u = u(x, t; λ) at λ = λ* for the non-local porous medium equation ut = (un)xx + lf(u)/(ò-11 f(u)dx)2{u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2} with Dirichlet boundary conditions and positive initial data. The function f satisfies: f(s),−f ′ (s) > 0 for s ≥ 0 and s
n-1
f(s) is integrable at infinity. Due to the conditions on f, there exists a critical value of parameter λ, say λ*, such that for λ > λ* the solution u = u(x, t; λ) blows up globally in finite time, while for λ ≥ λ* the corresponding steady-state problem does not have any solution.
For 0 < λ < λ* there exists a unique steady-state solution w = w(x; λ) while u = u(x, t; λ) is global in time and converges to w as t → ∞. Here we show the global grow-up of critical solution u* = u(x, t; λ*) (u* (x, t) → ∞, as t → ∞ for all x ? (-1,1){x\in(-1,1)}. 相似文献
8.
We study the Cauchy problem for the nonlinear dissipative equations (0.1) uo∂u-αδu + Β|u|2/n
u = 0,x ∃ Rn,t } 0,u(0,x) = u0(x),x ∃ Rn, where α,Β ∃ C, ℜα 0. We are interested in the dissipative case ℜα 0, and ℜδ(α,Β)≥ 0, θ = |∫ u0(x)dx| ⊋ 0, where δ(α, Β) = ##|α|n-1nn/2 / ((n + 1)|α|2 + α2
n/2. Furthermore, we assume that the initial data u0 ∃ Lp are such that (1 + |x|)αu0 ∃ L1, with sufficiently small norm ∃ = (1 + |x|)α u0 1 + u0 p, wherep 1, α ∃ (0,1). Then there exists a unique solution of the Cauchy problem (0.1)u(t, x) ∃ C ((0, ∞); L∞) ∩ C ([0, ∞); L1 ∩ Lp) satisfying the time decay estimates for allt0 u(t)||∞ Cɛt-n/2(1 + η log 〈t〉)-n/2, if hg = θ2/n 2π ℜδ(α, Β) 0; u(t)||∞ Cɛt-n/2(1 + Μ log 〈t〉)-n/4, if η = 0 and Μ = θ4/n 4π)2 (ℑδ(α, Β))2 ℜ((1 + 1/n) υ1-1 υ2) 0; and u(t)||∞ Cɛt-n/2(1 + κ log 〈t〉)-n/6, if η = 0, Μ = 0, κ 0, where υl,l = 1,2 are defined in (1.2), κ is a positive constant defined in (2.31). 相似文献
9.
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
10.
We study the bifurcation diagrams of (classical) positive solutions u with |u |∞ ∈ (0, ∞) of the p -Laplacian Dirichlet problem (φp (u ′(x)))′ + λfq (u (x))) = 0, –1 ≤ x ≤ 1, u (–1) = 0 = u (1), where p > 1, φp (y) = |y |p –2 y, (φp (u ′))′ is the one-dimensional p -Laplacian, λ > 0 is a bifurcation parameter, and the nonlinearity fq (u) = |1 – u |q is defined on [0, ∞) with constant q > 0. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
11.
Alan V. Lair 《Applicable analysis》2013,92(5):431-437
We show that the equation Δu = p(x)f(u) has a positive solution on R N , N ≥ 3, satisfying <artwork name="GAPA31011ei1"> <artwork name="GAPA31011ei2"> if and only if <artwork name="GAPA31011ei3"> when ψ(r) = min{p(x): |x| = r}. The nondecreasing continuous function f satisfies f(0) = 0, f (s) > 0 for s > 0, and sup s ≥ 1 f(s)/s<∞, and the nonnegative continuous function p is required to be asymptotically radial. This extends previous results which required the function p to be constant or radial. 相似文献
12.
Explosive solutions of elliptic equations with absorption and nonlinear gradient term 总被引:2,自引:0,他引:2
Marius Ghergu Constantin Niculescu Vicenţiu Rădulescu 《Proceedings Mathematical Sciences》2002,112(3):441-451
Letf be a non-decreasing C1-function such that
andF(t)/f
2
a(t)→ 0 ast → ∞, whereF(t)=∫
0
t
f(s) ds anda ∈ (0, 2]. We prove the existence of positive large solutions to the equationΔu +q(x)|Δu|
a
=p(x)f(u) in a smooth bounded domain Ω ⊂RN, provided thatp, q are non-negative continuous functions so that any zero ofp is surrounded by a surface strictly included in Ω on whichp is positive. Under additional hypotheses onp we deduce the existence of solutions if Ω is unbounded. 相似文献
13.
This paper studies the asymptotic behavior near the boundary for large solutions of the semilinear equation Δu + au = b(x)f(u) in a smooth bounded domain Ω of ℝN with N ≥ 2, where a is a real parameter and b is a nonnegative smooth function on
. We assume that f(u) behaves like u(ln u)α as u → ∞, for some α > 2. It turns out that this case is more difficult to handle than those where f(u) grows like u
p (p > 1) or faster at infinity. Under suitable conditions on the weight function b(x), which may vanish on ∂Ω, we obtain the first order expansion of the large solutions near the boundary. We also obtain some
uniqueness results.
Research of both authors supported by the Australian Research Council. 相似文献
14.
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
15.
Huiling Li Peter Y. H. Pang Mingxin Wang 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2012,89(1):295-311
We study the existence, uniqueness, and asymptotic behavior of blow-up solutions for a general quasilinear elliptic equation
of the type −Δ
p
u = a(x)u
m
−b(x)f(u) with p > 1 and 0 < m < p−1. The main technical tool is a new comparison principle that enables us to extend arguments for semilinear equations to
quasilinear ones. Indeed, this paper is an attempt to generalize all available results for the semilinear case with p = 2 to the quasilinear case with p > 1. 相似文献
16.
In the paper, we obtain the existence of symmetric or monotone positive solutions and establish a corresponding iterative
scheme for the equation (ϕ
p
(u′))′+q(t)f(u) = 0, 0 < t < 1, where ϕ
p
(s):= |s|
p−2
s, p > 1, subject to nonlinear boundary condition. The main tool is the monotone iterative technique. Here, the coefficient q(t) may be singular at t = 0; 1. 相似文献
17.
We consider the existence and uniqueness of singular solutions for equations of the formu
1=div(|Du|p−2
Du)-φu), with initial datau(x, 0)=0 forx⇑0. The function ϕ is a nondecreasing real function such that ϕ(0)=0 andp>2.
Under a growth condition on ϕ(u) asu→∞, (H1), we prove that for everyc>0 there exists a singular solution such thatu(x, t)→cδ(x) ast→0. This solution is unique and is called a fundamental solution. Under additional conditions, (H2) and (H3), we show the
existence of very singular solutions, i.e. singular solutions such that ∫|x|≤r
u(x,t)dx→∞ ast→0. Finally, for functions ϕ which behave like a power for largeu we prove that the very singular solution is unique. This is our main result.
In the case ϕ(u)=u
q, 1≤q, there are fundamental solutions forq<p*=p-1+(p/N) and very singular solutions forp-1<q<p*. These ranges are optimal.
Dedicated to Professor Shmuel Agmon 相似文献
18.
The existence of positive radial solutions of the equation -din( |Du|p-2Du)=f(u) is studied in annular domains in Rn,n≥2. It is proved that if f(0)≥0, f is somewherenegative in (0,∞), limu→0^ f‘ (u)=0 and limu→∞ (f(u)/u^p-1)=∞, then there is alarge positive radial solution on all annuli. If f(0)≤0 and satisfies certain conditions, then the equation has no radial solution if the annuli are too wide. 相似文献
19.
B. A. Khudaigulyev 《Ukrainian Mathematical Journal》2011,62(12):1989-1999
We consider the following problem of finding a nonnegative function u(x) in a ball B = B(O, R) ⊂ R
n
, n ≥ 3:
- Du = V(x)u, u| ?B = f(x), - \Delta u = V(x)u,\,\,\,\,\,u\left| {_{\partial B} = \phi (x),} \right. 相似文献
20.
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
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