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1.
Shu-Yu Hsu 《Mathematische Annalen》2003,325(4):665-693
We prove that the solution u of the equation u
t
=Δlog u, u>0, in (Ω\{x
0})×(0,T), Ω⊂ℝ2, has removable singularities at {x
0}×(0,T) if and only if for any 0<α<1, 0<a<b<T, there exist constants ρ0, C
1, C
2>0, such that C
1
|x−x
0|α≤u(x,t)≤C
2|x−x
0|−α holds for all 0<|x−x
0|≤ρ0 and a≤t≤b. As a consequence we obtain a sufficient condition for removable singularities at {∞}×(0,T) for solutions of the above equation in ℝ2×(0,T) and we prove the existence of infinitely many finite mass solutions for the equation in ℝ2×(0,T) when 0≤u
0∉L
1
(ℝ2) is radially symmetric and u
0L
loc
1(ℝ2).
Received: 16 December 2001 / Revised version: 20 May 2002 / Published online: 10 February 2003
Mathematics Subject Classification (1991): 35B40, 35B25, 35K55, 35K65 相似文献
2.
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
3.
Huashui Zhan 《Applications of Mathematics》2008,53(6):521-533
We study the large time asymptotic behavior of solutions of the doubly degenerate parabolic equation u
t
= div(u
m−1|Du|
p−2
Du) − u
q
with an initial condition u(x, 0) = u
0(x). Here the exponents m, p and q satisfy m + p ⩾ 3, p > 1 and q > m + p − 2.
The paper was supported by NSF of China (10571144), NSF for youth of Fujian province in China (2005J037) and NSF of Jimei
University in China. 相似文献
4.
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
5.
An approach to inverse problems based on the boundary control theory is developed. The dynamic problem to recover a density
of an inhomogeneous string via its free endopoint oscillations generated by an instantaneous force source is proposed. The
problem is to determine the coefficient ρ(x)>0 in the equation ρ(x)utt(x, t)−uxx(x, t)=0(x, t>0) with the conditions u|<0=0, ux(0, t)=δ(t) by using a known function (response) u(0, t)=r(t) (t>0). The authors propose an algorithm based upon the approach
and demonstrate its numerical efficiency in the test problems including those for nonmonotone ρ(x)'s. Bibliography: 12 titles.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 186, pp. 37–49, 1990.
Translated by T. N. Surkova. 相似文献
6.
Consider the Cauchy problem ∂u(x, t)/∂t = ℋu(x, t) (x∈ℤd, t≥ 0) with initial condition u(x, 0) ≡ 1 and with ℋ the Anderson Hamiltonian ℋ = κΔ + ξ. Here Δ is the discrete Laplacian, κ∈ (0, ∞) is a diffusion constant,
and ξ = {ξ(x): x∈ℤ
d
} is an i.i.d.random field taking values in ℝ. G?rtner and Molchanov (1990) have shown that if the law of ξ(0) is nondegenerate,
then the solution u is asymptotically intermittent.
In the present paper we study the structure of the intermittent peaks for the special case where the law of ξ(0) is (in the
vicinity of) the double exponential Prob(ξ(0) > s) = exp[−e
s
/θ] (s∈ℝ). Here θ∈ (0, ∞) is a parameter that can be thought of as measuring the degree of disorder in the ξ-field. Our main result
is that, for fixed x, y∈ℤ
d
and t→∈, the correlation coefficient of u(x, t) and u(y, t) converges to ∥w
ρ∥−2
ℓ2Σz ∈ℤd
w
ρ(x+z)w
ρ(y+z). In this expression, ρ = θ/κ while w
ρ:ℤd→ℝ+ is given by w
ρ = (v
ρ)⊗
d
with v
ρ: ℤ→ℝ+ the unique centered ground state (i.e., the solution in ℓ2(ℤ) with minimal l
2-norm) of the 1-dimensional nonlinear equation Δv + 2ρv log v = 0. The uniqueness of the ground state is actually proved only for large ρ, but is conjectured to hold for any ρ∈ (0, ∞).
empty
It turns out that if the right tail of the law of ξ(0) is thicker (or thinner) than the double exponential, then the correlation
coefficient of u(x, t) and u(y, t) converges to δ
x, y
(resp.the constant function 1). Thus, the double exponential family is the critical class exhibiting a nondegenerate correlation
structure.
Received: 5 March 1997 / Revised version: 21 September 1998 相似文献
7.
In this paper the Cauchy problem for the following nonhomogeneous Burgers’ equation is considered : (1)u
t
+uu
x
=μu
xx
−kx,x ∈R,t > 0, where μ and k are positive constants. Since the nonhomogeneous term kx does not belong to any Lp(R) space, this type of equation is beyond usual Sobolev framework in some sense. By Hopf-Cole transformation, (1) takes the
form (2)ϕ
t
−ϕ
xx
= −x
2
ϕ. With the help of the Hermite polynomials and their properties, (1) and (2) are solved exactly. Moreover, the large time
behavior of the solutions is also considered, similar to the discussion in Hopf’s paper. Especially, we observe that the nonhomogeneous
Burgers’ equation (1) is nonlinearly unstable. 相似文献
8.
Yves Belaud 《Journal of Mathematical Sciences》2010,171(1):1-8
We are dealing with the first vanishing time for solutions of the Cauchy–Neumann problem for the semilinear parabolic equation
∂
t
u − Δu + a(x)u
q
= 0, where
a(x) \geqslant d0exp( - \fracw( | x | )| x |2 ) a(x) \geqslant {d_0}\exp \left( { - \frac{{\omega \left( {\left| x \right|} \right)}}{{{{\left| x \right|}^2}}}} \right) , d
0 > 0, 1 > q > 0, and ω is a positive continuous radial function. We give a Dini-like condition on the function ω which implies that any solution of the above equation vanishes in finite time. The proof is derived from semi-classical limits
of some Schr¨odinger operators. 相似文献
9.
Shu-Yu Hsu 《Mathematische Annalen》2006,334(1):153-197
Let a1,a2, . . . ,am ∈ ℝ2, 2≤f ∈ C([0,∞)), gi ∈ C([0,∞)) be such that 0≤gi(t)≤2 on [0,∞) ∀i=1, . . . ,m. For any p>1, we prove the existence and uniqueness of solutions of the equation ut=Δ(logu), u>0, in satisfying and logu(x,t)/log|x|→−f(t) as |x|→∞, logu(x,t)/log|x−ai|→−gi(t) as |x−ai|→0, uniformly on every compact subset of (0,T) for any i=1, . . . ,m under a mild assumption on u0 where We also obtain similar existence and uniqueness of solutions of the above equation in bounded smooth convex domains of ℝ2 with prescribed singularities at a finite number of points in the domain. 相似文献
10.
Emilien Tarquini 《Monatshefte für Mathematik》2007,243(1):333-339
In this paper we consider the Gross-Pitaevskii equation iu
t
= Δu + u(1 − |u|2), where u is a complex-valued function defined on
\Bbb RN×\Bbb R{\Bbb R}^N\times{\Bbb R}
, N ≥ 2, and in particular the travelling waves, i.e., the solutions of the form u(x, t) = ν(x
1 − ct, x
2, …, x
N
), where
c ? \Bbb Rc\in{\Bbb R}
is the speed. We prove for c fixed the existence of a lower bound on the energy of any non-constant travelling wave. This bound provides a non-existence
result for non-constant travelling waves of fixed speed having small energy. 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}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
12.
Emilien Tarquini 《Monatshefte für Mathematik》2007,151(4):333-339
In this paper we consider the Gross-Pitaevskii equation iu
t
= Δu + u(1 − |u|2), where u is a complex-valued function defined on
, N ≥ 2, and in particular the travelling waves, i.e., the solutions of the form u(x, t) = ν(x
1 − ct, x
2, …, x
N
), where
is the speed. We prove for c fixed the existence of a lower bound on the energy of any non-constant travelling wave. This bound provides a non-existence
result for non-constant travelling waves of fixed speed having small energy. 相似文献
13.
Explicit exact solutions for a new generalized Hamiltonian amplitude equation with nonlinear terms of any order 总被引:1,自引:0,他引:1
Making use of a proper transformation and a generalized ansatz, we consider a new generalized Hamiltonian amplitude equation with nonlinear terms of any order, iux + utt + (|u|p + |u|2p)u + uxt = 0. As a result, many explicit exact solutions, which include kink-shaped soliton solutions, bell-shaped soliton solutions, periodic wave solutions, the combined formal solitary wave solutions and rational solutions, are obtained.Received: April 4, 2002 相似文献
14.
Jorge García-Melián 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2009,31(2):594-607
In this paper we consider the boundary blow-up problem Δpu = a(x)uq in a smooth bounded domain Ω of
\mathbbRN{\mathbb{R}}^N, with u = +∞ on ∂Ω. Here Dpu = div(|?u|p-2?u)\Delta_{p}u = {\rm div}(|\nabla u|^{p-2}\nabla u) is the well-known p-Laplacian operator with p > 1, q > p − 1, and a(x) is a nonnegative weight function which can be singular on ∂Ω. Our results include existence, uniqueness and exact boundary
behavior of positive solutions. 相似文献
15.
Mohamed Ben Ayed Khalil El Mehdi 《NoDEA : Nonlinear Differential Equations and Applications》2006,13(4):485-509
This paper is concerned with a biharmonic equation under the Navier boundary condition
, u > 0 in Ω and u = Δu = 0 on ∂Ω, where Ω is a smooth bounded domain in
, n ≥ 5, and ε > 0. We study the asymptotic behavior of solutions of (P
−ε) which are minimizing for the Sobolev quotient as ε goes to zero. We show that such solutions concentrate around a point
x
0 ∈Ω as ε → 0, moreover x
0 is a critical point of the Robin’s function. Conversely, we show that for any nondegenerate critical point x
0 of the Robin’s function, there exist solutions of (P
−ε) concentrating around x
0 as ε → 0. Finally we prove that, in contrast with what happened in the subcritical equation (P
−ε), the supercritical problem (P
+ε) has no solutions which concentrate around a point of Ω as ε → 0.
Work finished when the authors were visiting Mathematics Department of the University of Roma “La Sapienza”. They would like
to thank the Mathematics Department for its warm hospitality. The authors also thank Professors Massimo Grossi and Filomena
Pacella for their constant support. 相似文献
16.
S. Staněk 《Ukrainian Mathematical Journal》2008,60(2):277-298
We present existence principles for the nonlocal boundary-value problem (φ(u(p−1)))′=g(t,u,...,u(p−1), αk(u)=0, 1≤k≤p−1, where p ≥ 2, π: ℝ → ℝ is an increasing and odd homeomorphism, g is a Carathéodory function that is either regular or has singularities in its space variables, and α
k: C
p−1[0, T] → ℝ is a continuous functional. An application of the existence principles to singular Sturm-Liouville problems (−1)n(φ(u(2n−)))′=f(t,u,...,u(2n−1)), u(2k)(0)=0, αku(2k)(T)+bku(2k=1)(T)=0, 0≤k≤n−1, is given.
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 2, pp. 240–259, February, 2008. 相似文献
17.
N. A. Chalkina 《Moscow University Mathematics Bulletin》2011,66(6):231-234
Sufficient conditions for the existence of an inertial manifold are found for the equation u
tt
+ 2γu
t
− Δu = f(u, u
t
), u = u(x, t), x ∈ Ω ⋐ ℝ
N
, u|
∂Ω = 0, t > 0 under the assumption that the function f satisfies the Lipschitz condition. 相似文献
18.
We discuss and outline proofs of some recent results on application of singular perturbation techniques for solutions in entire
space of the Allen-Cahn equation Δu + u − u
3 = 0. In particular, we consider a minimal surface Γ in
\mathbb R9{\mathbb {R}^9} which is the graph of a nonlinear entire function x
9 = F(x
1, . . . , x
8), found by Bombieri, De Giorgi and Giusti, the BDG surface. We sketch a construction of a solution to the Allen Cahn equation
in
\mathbb R9{\mathbb {R}^9} which is monotone in the x9 direction whose zero level set lies close to a large dilation of Γ, recently obtained by M. Kowalczyk and the authors. This
answers a long standing question by De Giorgi in large dimensions (1978), whether a bounded solution should have planar level
sets. We sketch two more applications of the BDG surface to related questions, respectively in overdetermined problems and
in eternal solutions to the flow by mean curvature for graphs. 相似文献
19.
We study a periodic problem for the equation u
tt−uxx=g(x, t), u(x, t+T)=u(x, t), u(x+ω, t)= =u(x, t), ℝ2 and establish conditions of the existence and uniqueness of the classical solution.
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 4, pp. 558–565, April, 1997. 相似文献
20.
The main purpose of this paper is to analyze the asymptotic behavior of the radial solution of Hénon equation −Δu = |x|
α
u
p−1, u > 0, x ∈ B
R
(0) ⊂ ℝ
n
(n ⩾ 3), u = 0, x ∈ ∂B
R
(0), where $
p \to p(\alpha ) = \frac{{2(n + \alpha )}}
{{n - 2}}
$
p \to p(\alpha ) = \frac{{2(n + \alpha )}}
{{n - 2}}
from left side, α > 0. 相似文献