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1.
Monica Marras Stella Vernier Piro 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2008,59(5):766-779
We investigate the behavior of the solution of a nonlinear heat problem, when Robin conditions are prescribed on the boundary
∂Ω × (t > 0), Ω a bounded R
2 domain. We determine conditions on the geometry and data sufficient to preclude the blow up of the solution and to obtain
an exponential decay bound for the solution and its gradient.
Supported by the University of Cagliari. 相似文献
2.
Lech Zarȩba 《NoDEA : Nonlinear Differential Equations and Applications》2009,16(4):445-467
In this paper we consider the mixed problem for the equation u
tt
+ A
1
u + A
2(u
t
) + g(u
t
) = f(x, t) in unbounded domain, where A
1 is a linear elliptic operator of the fourth order and A
2 is a nonlinear elliptic operator of the second order. Under natural assumptions on the equation coefficients and f we proof existence of a solution. This result contains, as a special case, some of known before theorems of existence. Essentially,
in difference up to previous results we prove theorems of existence without the additional assumption on behavior of solution
at infinity.
相似文献
3.
Our first basic model is the fully nonlinear dual porous medium equation with source
for which we consider the Cauchy problem with given nonnegative bounded initial data u0. For the semilinear case m=1, the critical exponent
was obtained by H. Fujita in 1966. For p ∈(1, p0] any nontrivial solution blows up in finite time, while for p > p0 there exist sufficiently small global solutions. During last thirty years such critical exponents were detected for many
semilinear and quasilinear parabolic, hyperbolic and elliptic PDEs and inequalities. Most of efforts were devoted to equations
with differential operators in divergent form, where classical techniques associated with weak solutions and integration by
parts with a variety of test functions can be applied. Using this fully nonlinear equation, we propose and develop new approaches
to calculating critical Fujita exponents in different functional settings.
The second models with a “semi-divergent” diffusion operator is the thin film equation with source
for which the critical exponent is shown to be
相似文献
4.
We obtain existence results for some strongly nonlinear Cauchy problems posed in
and having merely locally integrable data. The equations we deal with have as principal part a bounded, coercive and pseudomonotone
operator of Leray-Lions type acting on
, they contain absorbing zero order terms and possibly include first order terms with natural growth. For any p > 1 and under
optimal growth conditions on the zero order terms, we derive suitable local a-priori estimates and consequent global existence
results. 相似文献
5.
Fengping Yao 《Archiv der Mathematik》2008,90(5):429-439
In this paper we generalize classical L
p
estimates to Orlicz spaces for the parabolic polyharmonic equations. Our argument is based on the iteration-covering procedure.
Received: 10 September 2007 相似文献
6.
Andrey Shishkov Laurent Véron 《Calculus of Variations and Partial Differential Equations》2008,33(3):343-375
We study the limit behaviour of solutions of with initial data k
δ
0 when k → ∞, where h is a positive nondecreasing function and p > 1. If h(r) = r
β
, β > N(p − 1) − 2, we prove that the limit function u
∞ is an explicit very singular solution, while such a solution does not exist if β ≤ N(p − 1) − 2. If lim
inf
r→ 0
r
2 ln (1/h(r)) > 0, u
∞ has a persistent singularity at (0, t) (t ≥ 0). If , u
∞ has a pointwise singularity localized at (0, 0). 相似文献
7.
The authors localize the blow-up points of positive solutions of the systemu
t
=Δu,v
t
=Δv with conditions
at the boundary of a bounded smooth domain Θ under some restrictions off andg and the initial data (Δu
0, Δν0>c>0).
If Θ is a ball, the hypothesis on the initial data can be removed.
Supported by Universidad de Buenos Aires under grant EX071 and CONICET. 相似文献
8.
Adimurthi Jacques Giacomoni 《NoDEA : Nonlinear Differential Equations and Applications》2005,12(1):1-20
This paper deals with the existence and the behaviour of global connected branches of positive solutions of the problem
We consider a function h which is smooth and changes sign. 相似文献
9.
Nassif Ghoussoub Abbas Moameni 《Calculus of Variations and Partial Differential Equations》2009,36(1):85-118
Selfdual variational calculus is developed further and used to address questions of existence of local and global solutions
for various parabolic semi-linear equations, and Hamiltonian systems of PDEs. This allows for the resolution of such equations
under general time boundary conditions which include the more traditional ones such as initial value problems, periodic and
anti-periodic orbits, but also yield new ones such as “periodic orbits up to an isometry” for evolution equations that may
not have periodic solutions. In the process, we introduce a method for perturbing selfdual functionals in order to induce
coercivity and compactness, without destroying the selfdual character of the system.
N. Ghoussoub was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada. A. Moameni’s
research was supported by a postdoctoral fellowship at the University of British Columbia. 相似文献
10.
Fengjie Li Bingchen Liu Sining Zheng 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2007,58(5):717-735
This paper deals with simultaneous and non-simultaneous blow-up for heat equations coupled via nonlinear boundary fluxes
. It is proved that, if m < q + 1 and n < p + 1, then blow-up must be simultaneous, and that, for radially symmetric and nondecreasing in time solutions, non-simultaneous
blow-up occurs for some initial data if and only if m > q + 1 or n > p + 1. We find three regions: (i) q + 1 < m < p/(p + 1 − n) and n < p+1, (ii) p + 1 < n < q/(q + 1 − m) and m < q+1, (iii) m > q+1 and n > p+1, where both simultaneous and non-simultaneous blow-up are possible. Four different simultaneous blow-up rates are obtained
under different conditions. It is interesting that different initial data may lead to different simultaneous blow-up rates
even for the same values of the exponent parameters.
Supported by the National Natural Science Foundation of China. 相似文献
11.
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
12.
N. M. Ivochkina 《Journal of Fixed Point Theory and Applications》2008,4(1):47-56
We adapt to degenerate m-Hessian evolution equations the notion of m-approximate solutions introduced by N. Trudinger for m-Hessian elliptic equations, and we present close to necessary and sufficient conditions guaranteeing the existence and uniqueness
of such solutions for the first initial boundary value problem.
Dedicated to Professor Felix Browder 相似文献
13.
We consider parabolic variational inequalities having the strong formulation
where
for some admissible initial datum, V is a separable Banach space with separable dual
is an appropriate monotone operator, and
is a convex,
lower semicontinuous functional. Well-posedness of (1) follows from an explicit construction of the related semigroup
Illustrative applications to free boundary problems and to parabolic problems in Orlicz-Sobolev spaces are given. 相似文献
((1)) |
14.
José Maria Gomes 《Archiv der Mathematik》2007,88(3):269-278
Let Ω be a bounded convex domain in
. We consider constrained minimization problems related to the Euler-Lagrange equation
over classes of functions
(Ω) with convex super level sets. We then search for sufficient conditions ensuring that the minimizer obtained is a classical
solution to the above equation.
Supported by ESF activity “Global and geometrical aspects of nonlinear P.D.E.’s.”
Received: 4 April 2006 相似文献
15.
On the existence and stability of periodic orbits in non ideal problems: General results 总被引:1,自引:0,他引:1
Márcio José Horta Dantas José Manoel Balthazar 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2007,58(6):940-958
In this work, motivated by non-ideal mechanical systems, we investigate the following O.D.E.
, where
,
are T periodic functions of t and there is a
0 ∈ Ω such that f ( a
0) = 0 and f ′( a
0) is a nilpotent matrix. When n = 3 and f (x) = (0, q (x
3) , 0) we get results on existence and stability of periodic orbits. We apply these results in a non ideal mechanical system:
the Centrifugal Vibrator. We make a stability analysis of this dynamical system and get a characterization of the Sommerfeld
Effect as a bifurcation of periodic orbits. 相似文献
16.
Caisheng Chen 《Journal of Evolution Equations》2006,6(1):29-43
In this paper, we study the global existence, L∞ estimates and decay estimates of solutions for the quasilinear parabolic system ut = div (|∇ u|m ∇ u) + f(u, v), vt = div (|∇ v|m ∇ v) + g(u,v) with zero Dirichlet boundary condition in a bounded domain Ω ⊂ RN. In particular, we find a critical value for the existence and nonexistence of global solutions to the equation ut = div (|∇ u|m ∇ u) + λ |u|α - 1 u. 相似文献
17.
In this paper we describe and analyze some modified boundary element methods to solve the exterior Dirichlet boundary value
problem for the Helmholtz equation. As in classical combined field integral equations also the proposed approach avoids spurious
modes. Moreover, the stability of related modified boundary element methods can be shown even in the case of Lipschitz boundaries.
The proposed regularization is done based on boundary integral operators which are already included in standard boundary element
formulations. Numerical examples are given to compare the proposed approach with other already existing regularized formulations. 相似文献
18.
We show that, under so called controllable growth conditions, any weak solution in the energy class of the semilinear parabolic
system
is locally regular. Here, A is an elliptic matrix differential operator of order 2m. The result is proved by writing the system as a system with linear growth in u,... , ∇
m
u but with “bad” coefficients and by means of a continuity method, where the time serves as parameter of continuity.
We also give a partial generalization of previous work of the second author and von Wahl to Navier boundary conditions.
Financial support by the Vigoni programme of CRUI (Rome) and DAAD (Bonn) is gratefully acknowledged.
This is the corrected version of the above mentioned article that was published Online First on October 24, 2006; DOI: 10.1007/s00028-006-0265-8.
The footnotes indicate the corrections done.
The online version of the original article can be found at 相似文献
19.
We consider here a class of nonlinear Dirichlet problems, in a bounded domain , of the form
investigating the problem of uniqueness of solutions. The functions (s) and
satisfy rather general assumptions of locally Lipschitz continuity (with possibly exponential growth) and the datum f is in L1(). Uniqueness of solutions is proved both for coercive a(x, s) and for the case of a(x, s) degenerating for s large. 相似文献
20.
Zhongping Li Chunlai Mu Zejian Cui 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2009,60(2):284-298
This paper deals with the critical curves of the fast diffusive polytropic filtration equations coupled via nonlinear boundary
flux. By constructing self-similar super and sub solutions we obtain the critical global existence curve. The critical curve
of Fujita type is conjectured with the aid of some new results.
相似文献