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1.
On existence, uniform decay rates and blow up for solutions of the 2-D wave equation with exponential source 总被引:1,自引:0,他引:1
Claudianor O. Alves Marcelo M. Cavalcanti 《Calculus of Variations and Partial Differential Equations》2009,34(3):377-411
This paper is concerned with the study of the nonlinear damped wave equation
where Ω is a bounded domain of having a smooth boundary ∂Ω = Γ. Assuming that g is a function which admits an exponential growth at the infinity and, in addition, that h is a monotonic continuous increasing function with polynomial growth at the infinity, we prove both: global existence as
well as blow up of solutions in finite time, by taking the initial data inside the potential well. Moreover, optimal and uniform
decay rates of the energy are proved for global solutions.
The author is Supported by CNPq 300959/2005-2, CNPq/Universal 472281/2006-2 and CNPq/Casadinho 620025/2006-9.
Research of Marcelo M. Cavalcanti partially supported by the CNPq Grant 300631/2003-0. 相似文献
2.
We study the semiflow defined by a semilinear parabolic equation with a singular square potential . It is known that the Hardy-Poincaré inequality and its improved versions, have a prominent role on the definition of the
natural phase space. Our study concerns the case 0 < μ ≤ μ*, where μ* is the optimal constant for the Hardy-Poincaré inequality. On a bounded domain of , we justify the global bifurcation of nontrivial equilibrium solutions for a reaction term f(s) = λs − |s|2γ
s, with λ as a bifurcation parameter. We remark some qualitative differences of the branches in the subcritical case μ < μ* and the critical case μ = μ*. The global bifurcation result is used to show that any solution , initiating form initial data tends to the unique nonnegative equilibrium. 相似文献
3.
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
4.
In this paper, we prove that if is a radially symmetric, sign-changing stationary solution of the nonlinear heat equation
in the unit ball of , N ≥ 3, with Dirichlet boundary conditions, then the solution of (NLH) with initial value blows up in finite time if |λ − 1| > 0 is sufficiently small and if α is subcritical and sufficiently close to 4/(N − 2).
F. Dickstein was partially supported by CNPq (Brazil). 相似文献
5.
The asymptotic behavior of viscosity solutions to the Cauchy–Dirichlet problem for the degenerate parabolic equation u
t
= Δ∞
u in Ω × (0,∞), where Δ∞ stands for the so-called infinity-Laplacian, is studied in three cases: (i) and the initial data has a compact support; (ii) Ω is bounded and the boundary condition is zero; (iii) Ω is bounded and the boundary condition is non-zero. Our method of proof is based on the comparison principle and barrier function
arguments. Explicit representations of separable type and self-similar type of solutions are also established. Moreover, in
case (iii), we propose another type of barrier function deeply related to a solution of .
Goro Akagi was supported by the Shibaura Institute of Technology grant for Project Research (no. 2006-211459, 2007-211455),
and the grant-in-aid for young scientists (B) (no. 19740073), Ministry of Education, Culture, Sports, Science and Technology.
Petri Juutinen was supported by the Academy of Finland project 108374. Ryuji Kajikiya was supported by the grant-in-aid for
scientific research (C) (no. 16540179), Ministry of Education, Culture, Sports, Science and Technology. 相似文献
6.
We consider the problem
where Ω is a bounded smooth domain in , 1 < p< + ∞ if N = 2, if N ≥ 3 and ε is a parameter. We show that if the mean curvature of ∂Ω is not constant then, for ε small enough, such a problem
has always a nodal solution u
ε with one positive peak and one negative peak on the boundary. Moreover, and converge to and , respectively, as ε goes to zero. Here, H denotes the mean curvature of ∂Ω.
Moreover, if Ω is a ball and , we prove that for ε small enough the problem has nodal solutions with two positive peaks on the boundary and arbitrarily
many negative peaks on the boundary.
The authors are supported by the M.I.U.R. National Project “Metodi variazionali e topologici nello studio di fenomeni non
lineari”. 相似文献
7.
Futoshi Takahashi 《Calculus of Variations and Partial Differential Equations》2007,29(4):509-520
We continue to study the asymptotic behavior of least energy solutions to the following fourth order elliptic problem (E
p
): as p gets large, where Ω is a smooth bounded domain in R
4
. In our earlier paper (Takahashi in Osaka J. Math., 2006), we have shown that the least energy solutions remain bounded uniformly
in p and they have one or two “peaks” away form the boundary. In this note, following the arguments in Adimurthi and Grossi (Proc.
AMS 132(4):1013–1019, 2003) and Lin and Wei (Comm. Pure Appl. Math. 56:784–809, 2003), we will obtain more sharper estimates
of the upper bound of the least energy solutions and prove that the least energy solutions must develop single-point spiky
pattern, under the assumption that the domain is convex. 相似文献
8.
Daniele Castorina Pierpaolo Esposito Berardino Sciunzi 《Calculus of Variations and Partial Differential Equations》2009,34(3):279-306
The behavior of the “minimal branch” is investigated for quasilinear eigenvalue problems involving the p-Laplace operator, considered in a smooth bounded domain of , and compactness holds below a critical dimension N
#. The nonlinearity f(u) lies in a very general class and the results we present are new even for p = 2. Due to the degeneracy of p-Laplace operator, for p ≠ 2 it is crucial to define a suitable notion of semi-stability: the functional space we introduce in the paper seems to
be the natural one and yields to a spectral theory for the linearized operator. For the case p = 2, compactness is also established along unstable branches satisfying suitable spectral information. The analysis is based
on a blow-up argument and stronger assumptions on the nonlinearity f(u) are required.
Authors are partially supported by MIUR, project “Variational methods and nonlinear differential equations”. 相似文献
9.
Norimichi Hirano 《NoDEA : Nonlinear Differential Equations and Applications》2009,16(2):159-188
In this paper, we consider the multiple existence of nonradial positive solutions of coupled nonlinear Schr?dinger system
where μ1, μ2 > 0 with and β < 0.
It is known that the solutions of (P) is not necessarily radial [12]. We show that problem (P) has multiple nonradial solutions
in case that |β| is sufficiently small.
相似文献
10.
Pierpaolo Esposito Juncheng Wei 《Calculus of Variations and Partial Differential Equations》2009,34(3):341-375
For the Neumann sinh-Gordon equation on the unit ball
we construct sequence of solutions which exhibit a multiple blow up at the origin, where λ ± are positive parameters. It answers partially an open problem formulated in Jost et al. [Calc Var Partial Diff Equ 31(2):263–276].
The research of the first named author is supported by M. U. R. S. T., project “Variational methods and nonlinear differential
equations”. The research of the second named author is supported by an Earmarked grant from RGC of Hong Kong. 相似文献
11.
We consider a two-dimensional convection model augmented with the rotational Coriolis forcing, centrifugal forcing as well
as the quadratic potential
with a fixed Ω > 0 being the rotational frequency. This model arises in the semiclassical limit of the Gross–Pitaevskii equation for Bose–Einstein condensates in a rotational frame. We investigate whether the action of dispersive rotational
forcing complemented with the underlying potential prevents the generic finite time breakdown of the free nonlinear convection.
We show that the rotating equations admit global smooth solutions for and only for a subset of generic initial configurations.
Thus, the global regularity depends on whether the initial configuration crosses an intrinsic critical threshold, which is
quantified in terms of the initial spectral gap associated with the 2 × 2 initial velocity gradient, λ 2 (0) − λ 1 (0), λ j (0)=λ j (∇ x U0) as well as the initial divergence, divx (U0).
We also prove that for the case of isotropic trapping potential the smooth velocity field is periodic if and only if the ratio
of the rotational frequency and the potential frequency is a rational number. The critical thresholds are also established
for the case of repulsive potential. Finally the position density and the velocity field are explicitly recorded along the
deformed flow map.
Received: November 12, 2003; revised: May 4, 2004 相似文献
12.
Marc Briane Juan Casado–Díaz 《Calculus of Variations and Partial Differential Equations》2008,33(4):463-492
In this paper we study the limit, in the sense of the Γ-convergence, of sequences of two-dimensional energies of the type
, where A
n
is a symmetric positive definite matrix-valued function and μ
n
is a nonnegative Borel measure (which can take infinite values on compact sets). Under the sole equicoerciveness of A
n
we prove that the limit energy belongs to the same class, i.e. its reads as , where is a diffusion independent of μ
n
and μ is a nonnegative Borel measure which does depend on . This compactness result extends in dimension two the ones of [11,23] in which A
n
is assumed to be uniformly bounded. It is also based on the compactness result of [7] obtained for sequences of two-dimensional
diffusions (without zero-order term). Our result does not hold in dimension three or greater, since nonlocal effects may appear.
However, restricting ourselves to three-dimensional diffusions with matrix-valued functions only depending on two coordinates,
the previous two-dimensional result provides a new approach of the nonlocal effects. So, in the periodic case we obtain an
explicit formula for the limit energy specifying the kernel of the nonlocal term. 相似文献
13.
Bernhard Hein 《Calculus of Variations and Partial Differential Equations》2007,28(2):249-273
In
, n < 7, we treat the quasilinear, degenerate parabolic initial and boundary value problem which is the natural parabolic extension of Huisken and Ilmanen’s weak inverse mean curvature flow (IMCF). We prove long time existence and partial uniqueness of Lipschitz continuous weak solutions u(x,t) and show C
1,α-regularity for the sets ∂{x| u(x,t) < z }. Our approach offers a new approximation for weak solutions of the IMCF starting from a class of interesting and easily obtainable initial values; for these, the above sets are shown to converge against corresponding surfaces of the IMCF as t → ∞ globally in Hausdorff distance and locally uniformly with respect to the C
1,α-norm.Research partially supported by the DFG, SFB 382 at Tübingen University 相似文献
14.
Maurizio Grasselli Giulio Schimperna Antonio Segatti Sergey Zelik 《Journal of Evolution Equations》2009,9(2):371-404
We study the modified Cahn–Hilliard equation proposed by Galenko et al. in order to account for rapid spinodal decomposition
in certain glasses. This equation contains, as additional term, the second-order time derivative of the (relative) concentration
multiplied by a (small) positive coefficient . Thus, in absence of viscosity effects, we are in presence of a Petrovsky type equation and the solutions do not regularize
in finite time. Many results are known in one spatial dimension. However, even in two spatial dimensions, the problem of finding
a unique solution satisfying given initial and boundary conditions is far from being trivial. A fairly complete analysis of
the 2D case has been recently carried out by Grasselli, Schimperna and Zelik. The 3D case is still rather poorly understood
but for the existence of energy bounded solutions. Taking advantage of this fact, Segatti has investigated the asymptotic
behavior of a generalized dynamical system which can be associated with the equation. Here we take a step further by establishing
the existence and uniqueness of a global weak solution, provided that is small enough. More precisely, we show that there exists such that well-posedness holds if (suitable) norms of the initial data are bounded by a positive function of which goes to + ∞ as tends to 0. This result allows us to construct a semigroup on an appropriate (bounded) phase space and, besides, to prove the existence of a global attractor. Finally, we show a regularity
result for the attractor by using a decomposition method and we discuss the existence of an exponential attractor.
相似文献
15.
Reika Fukuizumi Tohru Ozawa 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2005,56(6):1000-1011
Exponential decay estimates are obtained for complex-valued solutions to nonlinear elliptic equations in
where the linear term is given by Schr?dinger operators H = − Δ + V with nonnegative potentials V and the nonlinear term is given by a single power with subcritical Sobolev exponent in the attractive case. We describe specific
rates of decay in terms of V, some of which are shown to be optimal. Moreover, our estimates provide a unified understanding of two distinct cases in
the available literature, namely, the vanishing potential case V = 0 and the harmonic potential case V(x) = |x|2.
Dedicated to Professor Jun Uchiyama on the occasion of his sixtieth birthday
Received: May 4, 2004 相似文献
16.
Zhian Wang 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2006,57(3):399-418
We derive the optimal decay rates of solution to the Cauchy problem for a set of nonlinear evolution equations with ellipticity
and dissipative effects
with initial data
where α and ν are positive constants such that α < 1, ν < α(1 − α), which is a special case of (1.1). We show that the solution
to the system decays with the same rate to that of its associated homogenous linearized system. The main results are obtained
by the use of Fourier analysis and interpolation inequality under some suitable restrictions on coefficients α and ν. Moreover,
we discuss the asymptotic behavior of the solution to general system (1.1) at the end.
The research was supported by the F. S. Chia Scholarship of the University of Alberta.
Received: January 27, 2005; revised: April 27, 2005 相似文献
17.
Kin Ming Hui 《Mathematische Annalen》2007,339(2):395-443
We prove the existence of a unique solution of the following Neumann problem , u > 0, in (a, b) × (0, T), u(x, 0) = u
0(x) ≥ 0 in (a, b), and , where if m < 0, if m = 0, and
m≤ 0, , and the case −1 < m ≤ 0, , for some constant p > 1 − m. We also obtain a similar result in higher dimensions. As a corollary we will give a new proof of a result of A. Rodriguez
and J.L. Vazquez on the existence of infinitely many finite mass solutions of the above equation in for any −1 < m ≤ 0. We also obtain the exact decay rate of the solution at infinity. 相似文献
18.
THEBLOW┐UPPROPERTYFORASYSTEMOFHEATEQUATIONSWITHNONLINEARBOUNDARYCONDITIONSLINZHIGUI,XIECHUNHONGANDWANGMINGXINAbstract.Thispap... 相似文献
19.
This paper is concerned with the existence of a global attractor for a semiflow governed by the weak solutions to a nonlinear one-dimensional thermoviscoelasticity with a non-convex free energy density. The constitutive assumptions for the Helmholtz free energy include the model for the study of martensitic phase transitions in shape memory alloys. To describe physically phase transitions between different configurations of crystal lattices, we work in a framework in which the strain u belongs to L∞. New approaches are introduced and more delicate estimates are derived to establish the crucial L∞-estimate of strain u in deriving the compactness of the orbit of the semiflow and the existence of an absorbing set. 相似文献
20.
Jérôme Droniou Juan-Luis Vázquez 《Calculus of Variations and Partial Differential Equations》2009,34(4):413-434
We study the existence and uniqueness of solutions of the convective–diffusive elliptic equation
posed in a bounded domain , with pure Neumann boundary conditions
Under the assumption that with p = N if N ≥ 3 (resp. p > 2 if N = 2), we prove that the problem has a solution if ∫Ω
f
dx = 0, and also that the kernel is generated by a function , unique up to a multiplicative constant, which satisfies a.e. on Ω. We also prove that the equation
has a unique solution for all ν > 0 and the map is an isomorphism of the respective spaces. The study is made in parallel with the dual problem, with equation
The dependence on the data is also examined, and we give applications to solutions of nonlinear elliptic PDE with measure
data and to parabolic problems. 相似文献