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1.
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.$">
with on the half-space as a consequence of a new Liouville theorem of elliptic type for solutions of () on . This new result is in turn a consequence of other new phenomena established for nonlinear evolution problems. In particular, we prove that the inequality
has no nontrivial solutions on when We also show that the inequality
has no nontrivial nonnegative solutions for , and it has no solutions on bounded below by a positive constant for 1.$">
2.
Joe Kamimoto Haseo Ki Young-One Kim 《Proceedings of the American Mathematical Society》2000,128(1):189-194
We show that all the zeros of the Fourier transforms of the functions , , are real and simple. Then, using this result, we show that there are infinitely many polynomials such that for each the translates of the function
generate . Finally, we discuss the problem of finding the minimum number of monomials , , which have the property that the translates of the functions , , generate , for a given .
3.
A class of functions and the corresponding solutions of
are obtained as a special case of the solutions of
where is defined as .
4.
Daisuke Hirata 《Proceedings of the American Mathematical Society》2005,133(6):1823-1827
In this note we consider the global regularity of smooth solutions to the vector-valued Cauchy problem
We show that if , the gradient-blowup phenomenon occurs in finite time for suitably chosen vanishing at infinity. We also present a simple example of the -blowup solutions for for any 0$">, if .
We show that if , the gradient-blowup phenomenon occurs in finite time for suitably chosen vanishing at infinity. We also present a simple example of the -blowup solutions for for any 0$">, if .
5.
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 .
6.
Yun-guang Lu Christian Klingenberg 《Proceedings of the American Mathematical Society》2003,131(11):3511-3516
In this paper we contrast two approaches for proving the validity of relaxation limits of systems of balance laws
In one approach this is proven under some suitable stability condition; in the other approach, one adds artificial viscosity to the system
and lets and together with for a suitable large constant . We illustrate the usefulness of this latter approach by proving the convergence of a relaxation limit for a system of mixed type, where a subcharacteristic condition is not available.
In one approach this is proven under some suitable stability condition; in the other approach, one adds artificial viscosity to the system
and lets and together with for a suitable large constant . We illustrate the usefulness of this latter approach by proving the convergence of a relaxation limit for a system of mixed type, where a subcharacteristic condition is not available.
7.
Stefano Bianchini Rinaldo M. Colombo 《Proceedings of the American Mathematical Society》2002,130(7):1961-1973
We consider the dependence of the entropic solution of a hyperbolic system of conservation laws
on the flux function . We prove that the solution is Lipschitz continuous w.r.t. the norm of the derivative of the perturbation of . We apply this result to prove the convergence of the solution of the relativistic Euler equation to the classical limit.
on the flux function . We prove that the solution is Lipschitz continuous w.r.t. the norm of the derivative of the perturbation of . We apply this result to prove the convergence of the solution of the relativistic Euler equation to the classical limit.
8.
G. Fonseca F. Linares G. Ponce 《Proceedings of the American Mathematical Society》2003,131(6):1847-1855
We discuss results regarding global existence of solutions for the critical generalized Korteweg-de Vries equation,
The theory established shows the existence of global solutions in Sobolev spaces with order below the one given by the energy space , i.e. solutions corresponding to data , 3/4$">, with , where is the solitary wave solution of the equation.
The theory established shows the existence of global solutions in Sobolev spaces with order below the one given by the energy space , i.e. solutions corresponding to data , 3/4$">, with , where is the solitary wave solution of the equation.
9.
Sergei Yu. Vasilovsky 《Proceedings of the American Mathematical Society》1999,127(12):3517-3524
The algebra of all matrices over a field has a natural -grading . In this paper graded identities of the -graded algebra over a field of characteristic zero are studied. It is shown that all the -graded polynomial identities of follow from the following:
10.
Let , , be a bounded smooth connected open set and be a map satisfying the hypotheses (H1)-(H4) below. Let with , in and with be two weak solutions of
Suppose that in . Then we show that u_1$"> in under the following assumptions: either u_1$"> on , or on and in . We also show a measure-theoretic version of the Strong Comparison Principle.
Suppose that in . Then we show that u_1$"> in under the following assumptions: either u_1$"> on , or on and in . We also show a measure-theoretic version of the Strong Comparison Principle.
11.
By using Krasnoselskii's fixed point theorem, we prove that the following periodic species Lotka-Volterra competition system with multiple deviating arguments has at least one positive periodic solution provided that the corresponding system of linear equations has a positive solution, where and are periodic functions with Furthermore, when and , , are constants but , remain -periodic, we show that the condition on is also necessary for to have at least one positive periodic solution.
12.
norm in some a priori estimates for non-divergence form equations in Carnot groups Donatella Danielli Nicola Garofalo Duy-Minh Nhieu 《Proceedings of the American Mathematical Society》2003,131(11):3487-3498
Let be a group of Heisenberg type with homogeneous dimension . For every we construct a non-divergence form operator and a non-trivial solution to the Dirichlet problem: in , on . This non-uniqueness result shows the impossibility of controlling the maximum of with an norm of when . Another consequence is the impossiblity of an Alexandrov-Bakelman type estimate such as
where is the dimension of the horizontal layer of the Lie algebra and is the symmetrized horizontal Hessian of .
where is the dimension of the horizontal layer of the Lie algebra and is the symmetrized horizontal Hessian of .
13.
Jay Kovats 《Proceedings of the American Mathematical Society》2002,130(4):1055-1064
We use Bernstein's technique to show that for any fixed , strong solutions of the uniformly parabolic equation in are real analytic in . Here, is a bounded domain and the coefficients are measurable. We also use Bernstein's technique to obtain interior estimates for pure second derivatives of solutions of the fully nonlinear, uniformly parabolic, concave equation in , where is measurable in .
14.
Ross G. Pinsky 《Proceedings of the American Mathematical Society》2002,130(6):1673-1679
Let and let be a continuous, nonincreasing function on satisfying . Consider the heat equation in the exterior of a time-dependent shrinking disk in the plane:
If there exist constants and a constant 0$"> such that , for sufficiently large , then . The same result is also shown to hold when is replaced by , where . Also, a discrepancy is noted between the asymptotics for the above forward heat equation and the corresponding backward one. The method used is probabilistic.
0.\end{split}\end{displaymath}">
If there exist constants and a constant 0$"> such that , for sufficiently large , then . The same result is also shown to hold when is replaced by , where . Also, a discrepancy is noted between the asymptotics for the above forward heat equation and the corresponding backward one. The method used is probabilistic.
15.
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
16.
Let be a uniformly smooth real Banach space and let be a mapping with . Suppose is a generalized Lipschitz generalized -quasi-accretive mapping. Let and be real sequences in [0,1] satisfying the following conditions: (i) ; (ii) ; (iii) ; (iv) Let be generated iteratively from arbitrary by
where is defined by and is an arbitrary bounded sequence in . Then, there exists such that if the sequence converges strongly to the unique solution of the equation . A related result deals with approximation of the unique fixed point of a generalized Lipschitz and generalized -hemi-contractive mapping.
where is defined by and is an arbitrary bounded sequence in . Then, there exists such that if the sequence converges strongly to the unique solution of the equation . A related result deals with approximation of the unique fixed point of a generalized Lipschitz and generalized -hemi-contractive mapping.
17.
Y. Gordon A. E. Litvak C. Schü tt E. Werner 《Proceedings of the American Mathematical Society》2006,134(12):3665-3675
For a given sequence of real numbers , we denote the th smallest one by . Let be a class of random variables satisfying certain distribution conditions (the class contains Gaussian random variables). We show that there exist two absolute positive constants and such that for every sequence of real numbers and every , one has
-
where are independent random variables from the class . Moreover, if , then the left-hand side estimate does not require independence of the 's. We provide similar estimates for the moments of as well. 18.
Ali Taheri 《Proceedings of the American Mathematical Society》2003,131(10):3101-3107
Let be a bounded starshaped domain. In this note we consider critical points of the functional
where of class satisfies the natural growth
for some and 0$">, is suitably rank-one convex and in addition is strictly quasiconvex at . We establish uniqueness results under the extra assumption that is stationary at with respect to variations of the domain. These statements should be compared to the uniqueness result of Knops & Stuart (1984) in the smooth case and recent counterexamples to regularity produced by Müller & Sverák (2003).
where of class satisfies the natural growth
for some and 0$">, is suitably rank-one convex and in addition is strictly quasiconvex at . We establish uniqueness results under the extra assumption that is stationary at with respect to variations of the domain. These statements should be compared to the uniqueness result of Knops & Stuart (1984) in the smooth case and recent counterexamples to regularity produced by Müller & Sverák (2003).
19.
J. M. A. M. van Neerven 《Proceedings of the American Mathematical Society》2002,130(8):2325-2333
Let be a -semigroup with generator on a Banach space . Let be a fixed element. We prove the following individual stability results.
(i) Suppose is an ordered Banach space with weakly normal closed cone and assume there exists such that for all . If the local resolvent admits a bounded analytic extension to the right half-plane 0\}$">, then for all and we have
(ii) Suppose is a rearrangement invariant Banach function space over with order continuous norm. If is an element such that defines an element of , then for all and we have
20.
H. D. Voulov 《Proceedings of the American Mathematical Society》2003,131(7):2155-2160
An open problem posed by G. Ladas is to investigate the difference equation
where are any nonnegative real numbers with 0$">. We prove that there exists a positive integer such that every positive solution of this equation is eventually periodic of period .
where are any nonnegative real numbers with 0$">. We prove that there exists a positive integer such that every positive solution of this equation is eventually periodic of period .