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1.
Yunguang Lu 《Proceedings of the American Mathematical Society》2002,130(5):1339-1343
This paper is concerned with the Hölder estimates of weak solutions of the Cauchy problem for the general degenerate parabolic equations
with the initial data , where the diffusion function can be a constant on a nonzero measure set, such as the equations of two-phase Stefan type. Some explicit Hölder exponents of the composition function with respect to the space variables are obtained by using the maximum principle.
with the initial data , where the diffusion function can be a constant on a nonzero measure set, such as the equations of two-phase Stefan type. Some explicit Hölder exponents of the composition function with respect to the space variables are obtained by using the maximum principle.
2.
Nakao Hayashi Elena I. Kaikina Pavel I. Naumkin 《Transactions of the American Mathematical Society》2006,358(3):1165-1185
We study large time asymptotics of small solutions to the Cauchy problem for nonlinear damped wave equations with a critical nonlinearity
where 0,$"> and space dimensions . Assume that the initial data
where \frac{n}{2},$"> weighted Sobolev spaces are
Also we suppose that
where
Then we prove that there exists a positive such that the Cauchy problem above has a unique global solution satisfying the time decay property
for all 0,$"> where
where 0,$"> and space dimensions . Assume that the initial data
where \frac{n}{2},$"> weighted Sobolev spaces are
Also we suppose that
0,\int u_{0}\left( x\right) dx>0, \end{displaymath}">
where
Then we prove that there exists a positive such that the Cauchy problem above has a unique global solution satisfying the time decay property
for all 0,$"> where
3.
Ismail Kombe 《Proceedings of the American Mathematical Society》2004,132(9):2683-2691
In this paper we consider the following initial value problem:
where and . Nonexistence of positive solutions is analyzed.
where and . Nonexistence of positive solutions is analyzed.
4.
Andrea Pascucci 《Transactions of the American Mathematical Society》2003,355(3):901-924
We study the interior regularity properties of the solutions to the degenerate parabolic equation,
which arises in mathematical finance and in the theory of diffusion processes.
which arises in mathematical finance and in the theory of diffusion processes.
5.
A class of functions and the corresponding solutions of
are obtained as a special case of the solutions of
where is defined as .
6.
Takeshi Okano 《Proceedings of the American Mathematical Society》2002,130(6):1603-1605
7.
O. Dosly J. R. Graef J. Jaros 《Proceedings of the American Mathematical Society》2003,131(9):2859-2867
Oscillation properties of solutions of the forced second order linear difference equation
are investigated. The authors show that if the forcing term does not oscillate, in some sense, too rapidly, then the oscillation of the unforced equation implies oscillation of the forced equation. Some results illustrating this statement and extensions to the more general half-linear equation
are also given.
are investigated. The authors show that if the forcing term does not oscillate, in some sense, too rapidly, then the oscillation of the unforced equation implies oscillation of the forced equation. Some results illustrating this statement and extensions to the more general half-linear equation
1, \end{displaymath}">
are also given.
8.
Richard F. Bass Moritz Kassmann 《Transactions of the American Mathematical Society》2005,357(2):837-850
We consider harmonic functions with respect to the operator
Under suitable conditions on we establish a Harnack inequality for functions that are nonnegative and harmonic in a domain. The operator is allowed to be anisotropic and of variable order.
Under suitable conditions on we establish a Harnack inequality for functions that are nonnegative and harmonic in a domain. The operator is allowed to be anisotropic and of variable order.
9.
C. Lederman J. L. Vá zquez N. Wolanski 《Transactions of the American Mathematical Society》2001,353(2):655-692
We investigate the uniqueness and agreement between different kinds of solutions for a free boundary problem in heat propagation that in classical terms is formulated as follows: to find a continuous function defined in a domain and such that
0\}. \end{displaymath}">
We also assume that the interior boundary of the positivity set, \nobreak 0\}$">, so-called free boundary, is a regular hypersurface on which the following conditions are satisfied:
Here denotes outward unit spatial normal to the free boundary. In addition, initial data are specified, as well as either Dirichlet or Neumann data on the parabolic boundary of . This problem arises in combustion theory as a limit situation in the propagation of premixed flames (high activation energy limit).
The problem admits classical solutions only for good data and for small times. Several generalized concepts of solution have been proposed, among them the concepts of limit solution and viscosity solution. We investigate conditions under which the three concepts agree and produce a unique solution.
10.
Nakao Hayashi Pavel I. Naumkin Yasuko Yamazaki 《Proceedings of the American Mathematical Society》2002,130(3):779-789
We consider the derivative nonlinear Schrödinger equations
where the coefficient satisfies the time growth condition
is a sufficiently small constant and the nonlinear interaction term consists of cubic nonlinearities of derivative type
where and . We suppose that the initial data satifsfy the exponential decay conditions. Then we prove the sharp decay estimate , for all , where . Furthermore we show that for there exist the usual scattering states, when and the modified scattering states, when
where the coefficient satisfies the time growth condition
is a sufficiently small constant and the nonlinear interaction term consists of cubic nonlinearities of derivative type
where and . We suppose that the initial data satifsfy the exponential decay conditions. Then we prove the sharp decay estimate , for all , where . Furthermore we show that for there exist the usual scattering states, when and the modified scattering states, when
11.
G. A. Karagulyan 《Proceedings of the American Mathematical Society》2007,135(10):3133-3141
We show that for any infinite set of unit vectors in the maximal operator defined by is not bounded in .
12.
D. G. De Figueiredo Y. H. Ding 《Transactions of the American Mathematical Society》2003,355(7):2973-2989
We study existence and multiplicity of solutions of the elliptic system
where , is a smooth bounded domain and . We assume that the nonlinear term
where , , and . So some supercritical systems are included. Nontrivial solutions are obtained. When is even in , we show that the system possesses a sequence of solutions associated with a sequence of positive energies (resp. negative energies) going toward infinity (resp. zero) if 2$"> (resp. ). All results are proved using variational methods. Some new critical point theorems for strongly indefinite functionals are proved.
where , is a smooth bounded domain and . We assume that the nonlinear term
where , , and . So some supercritical systems are included. Nontrivial solutions are obtained. When is even in , we show that the system possesses a sequence of solutions associated with a sequence of positive energies (resp. negative energies) going toward infinity (resp. zero) if 2$"> (resp. ). All results are proved using variational methods. Some new critical point theorems for strongly indefinite functionals are proved.
13.
On a Liouville-type theorem and the Fujita blow-up phenomenon 总被引：3，自引：0，他引：3
The main purpose of this paper is to obtain the well-known results of H.Fujita and K.Hayakawa on the nonexistence of nontrivial nonnegative global solutions for the Cauchy problem for the equation
with on the half-space as a consequence of a new Liouville theorem of elliptic type for solutions of () on . This new result is in turn a consequence of other new phenomena established for nonlinear evolution problems. In particular, we prove that the inequality
has no nontrivial solutions on when We also show that the inequality
has no nontrivial nonnegative solutions for , and it has no solutions on bounded below by a positive constant for 1.$">
with on the half-space as a consequence of a new Liouville theorem of elliptic type for solutions of () on . This new result is in turn a consequence of other new phenomena established for nonlinear evolution problems. In particular, we prove that the inequality
has no nontrivial solutions on when We also show that the inequality
has no nontrivial nonnegative solutions for , and it has no solutions on bounded below by a positive constant for 1.$">
14.
Louis Jeanjean Kazunaga Tanaka 《Proceedings of the American Mathematical Society》2003,131(8):2399-2408
We study a mountain pass characterization of least energy solutions of the following nonlinear scalar field equation in :
where . Without the assumption of the monotonicity of , we show that the mountain pass value gives the least energy level.
where . Without the assumption of the monotonicity of , we show that the mountain pass value gives the least energy level.
15.
Peter C. Gibson 《Transactions of the American Mathematical Society》2002,354(12):4703-4749
We solve the following physically motivated problem: to determine all finite Jacobi matrices and corresponding indices such that the Green's function
is proportional to an arbitrary prescribed function . Our approach is via probability distributions and orthogonal polynomials.
(where square brackets denote the equivalence class consisting of scalar multiples). This enables us to construct the solution set as a fibration over a connected, semi-algebraic coordinate base. The end result is a wealth of explicit constructions for Jacobi matrices. These reveal precise geometric information about the solution set, and provide the basis for new existence theorems.
is proportional to an arbitrary prescribed function . Our approach is via probability distributions and orthogonal polynomials.
We introduce what we call the auxiliary polynomial of a solution in order to factor the map
(where square brackets denote the equivalence class consisting of scalar multiples). This enables us to construct the solution set as a fibration over a connected, semi-algebraic coordinate base. The end result is a wealth of explicit constructions for Jacobi matrices. These reveal precise geometric information about the solution set, and provide the basis for new existence theorems.
16.
Norimichi Hirano Naoki Shioji 《Proceedings of the American Mathematical Society》2006,134(9):2585-2592
Let , let and let be a bounded domain with a smooth boundary . Our purpose in this paper is to consider the existence of solutions of the problem: where
17.
Norbert Peyerimhoff Evangelia Samiou 《Proceedings of the American Mathematical Society》2004,132(5):1525-1529
We show that the Cheeger isoperimetric constant of a solvable simply connected Lie group with Lie algebra is
18.
Philippe Poulin 《Proceedings of the American Mathematical Society》2007,135(1):77-85
It is well known that the Green function of the standard discrete Laplacian on , exhibits a pathological behavior in dimension . In particular, the estimate fails for . This fact complicates the study of the scattering theory of discrete Schrödinger operators. Molchanov and Vainberg suggested the following alternative to the standard discrete Laplacian, and conjectured that the estimate holds for all . In this paper we prove this conjecture.
19.
In this paper we give asymptotic estimates of the least energy solution of the functional
as goes to infinity. Here is a smooth bounded domain of . Among other results we give a positive answer to a question raised by Chen, Ni, and Zhou (2000) by showing that .
as goes to infinity. Here is a smooth bounded domain of . Among other results we give a positive answer to a question raised by Chen, Ni, and Zhou (2000) by showing that .
20.
We study the Cauchy problem for the nonlinear degenerate parabolic equation of second order
and present regularity results for the viscosity solutions.
and present regularity results for the viscosity solutions.