where and . Nonexistence of positive solutions is analyzed.
$ \left\{ {l} u_{tt}+2\alpha u_{t}-\beta u_{xx}=-\lambda \left| u\right| ^{\sigma}u,\text{ }x\in \Omega ,t >0 , \\ u(0,x)=\phi \left( x\right) ,\text{}u_{t}(0,x)=\psi \left( x\right) ,\text{ }x\in \Omega , \right. $ \left\{ \begin{array}{l} u_{tt}+2\alpha u_{t}-\beta u_{xx}=-\lambda \left| u\right| ^{\sigma}u,\text{ }x\in \Omega ,t >0 , \\ u(0,x)=\phi \left( x\right) ,\text{}u_{t}(0,x)=\psi \left( x\right) ,\text{ }x\in \Omega , \end{array} \right. 相似文献
6.
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\}$">, so-called free boundary, is a regular hypersurface on which the following conditions are satisfied: Here denotes outward unit spatial normal to the free boundary. In addition, initial data are specified, as well as either Dirichlet or Neumann data on the parabolic boundary of . This problem arises in combustion theory as a limit situation in the propagation of premixed flames (high activation energy limit). The problem admits classical solutions only for good data and for small times. Several generalized concepts of solution have been proposed, among them the concepts of limit solution and viscosity solution. We investigate conditions under which the three concepts agree and produce a unique solution. 7.
In this paper, we first construct ``viscosity' solutions (in the Crandall-Lions sense) of fully nonlinear elliptic equations of the form
In fact, viscosity solutions are surprisingly weak. Since candidates for solutions are just continuous, we only require that the ``test' polynomials (those tangent from above or below to the graph of at a point ) satisfy the correct inequality only if . That is, we simply disregard those test polynomials for which . Nevertheless, this is enough, by an appropriate use of the Alexandroff-Bakelman technique, to prove existence, regularity and, in two dimensions, for , (0$">) and constant boundary conditions on a convex domain, to prove that there is only one convex patch. 8.
Pé ter Komjá th Mikló s Laczkovich 《Proceedings of the American Mathematical Society》2002,130(5):1487-1491
(GCH) For every cardinal there exists such that for every , there are such that .
9.
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 10.
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. 11.
A class of functions and the corresponding solutions of are obtained as a special case of the solutions of where is defined as . 12.
Jerome A. Goldstein Gaston M. N'Gué ré kata 《Proceedings of the American Mathematical Society》2005,133(8):2401-2408
We are concerned with the semilinear differential equation in a Banach space ,
where generates an exponentially stable -semigroup and is a function of the form . Under appropriate conditions on and , and using the Schauder fixed point theorem, we prove the existence of an almost automorphic mild solution to the above equation. 13.
Consider the Kirchhoff type equation \begin{equation}\label{eq0.1}-\left(a+b\int_{\mathbb{R}^{N}}|\nabla u|^{2}\,dx\right) \Delta u=\left(\frac{1}{|x|^\mu}*F(u)\right)f(u)\ \ \mbox{in}\ \mathbb{R}^N, \ \ u\in D^{1,2}(\mathbb{R}^N), ~~~~~~(0.1)\end{equation}where $a>0$, $b\geq0$, $0<\mu<\min\{N, 4\}$ with $N\geq 3$, $f: \mathbb{R}\to\mathbb{R}$ is a continuous function and $F(u)=\int_0^u f(t)\,dt$. Under some general assumptions on $f$, we establish the existence of a nontrivial spherically symmetric solution for problem (0.1). The proof is mainly based on mountain pass approach and a scaling technique introduced by Jeanjean. 相似文献
14.
Athanassios G. Kartsatos Igor V. Skrypnik 《Transactions of the American Mathematical Society》2000,352(10):4603-4640
We consider the general initial-boundary value problem
(1) (2) (3) where is a bounded open set in with sufficiently smooth boundary. The problem (1)-(3) is first reduced to the analogous problem in the space with zero initial condition and The resulting problem is then reduced to the problem where the operator satisfies Condition This reduction is based on a priori estimates which are developed herein for linear parabolic operators with coefficients in Sobolev spaces. The local and global solvability of the operator equation are achieved via topological methods developed by I. V. Skrypnik. Further applications are also given involving relevant coercive problems, as well as Galerkin approximations. 15.
S. Prashanth 《Proceedings of the American Mathematical Society》2007,135(1):201-209
Let denote the closure of in the norm Let and define the constants and Let We consider the following problem for
16.
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. 17.
In this paper, we study the well-posedness of an initial-boundary-value problem (IBVP) for the Boussinesq equation on a bounded domain,\begin{cases} &u_{tt}-u_{xx}+(u^2)_{xx}+u_{xxxx}=0,\quad x\in (0,1), \;\;t>0,\\ &u(x,0)=\varphi(x),\;\;\; u_t(x,0)=ψ(x),\\ &u(0,t)=h_1(t),\;\;\;u(1,t)=h_2(t),\;\;\;u_{xx}(0,t)=h_3(t),\;\;\;u_{xx}(1,t)=h_4(t).\\ \end{cases} It is shown that the IBVP is locally well-posed in the space $H^s (0,1)$ for any $s\geq 0$ with the initial data $\varphi,$ $\psi$ lie in $H^s(0,1)$ and $ H^{s-2}(0,1)$, respectively, and the naturally compatible boundary data $h_1,$ $h_2$ in the space $H_{loc}^{(s+1)/2}(\mathbb{R}^+)$, and $h_3 $, $h_4$ in the the space of $H_{loc}^{(s-1)/2}(\mathbb{R}^+)$ with optimal regularity. 相似文献
18.
Cristian E. Gutié rrez Federico Tournier 《Transactions of the American Mathematical Society》2006,358(11):4843-4872
Let be a strictly convex domain and let be a convex function such that det in . The linearized Monge-Ampère equation is
19.
Stefan Friedl 《Proceedings of the American Mathematical Society》2005,133(3):647-653
Let be a number field closed under complex conjugation. Denote by the Witt group of hermitian forms over . We find full invariants for detecting non-zero elements in . This group plays an important role in topology in the work done by Casson and Gordon.
20.
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:
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. |