共查询到20条相似文献,搜索用时 156 毫秒
1.
Nakao Hayashi Pavel I. Naumkin Jean-Claude Saut 《Communications in Mathematical Physics》1999,201(3):577-590
We study the large time asymptotic behavior of solutions to the generalized Kadomtsev-Petviashvili (KP) equations $ \left\{\alignedat2 &u_t + u_{xxx} + \sigma\partial_x^{-1}u_{yy}= - (u^{\rho})_x, &;&;\qquad (t,x,y) \in {\bold R}\times {\bold R}^2,\\ \vspace{.5\jot} &u(0,x,y) = u_0 (x,y),&;&; \qquad (x,y) \in{\bold R}^2, \endalignedat \right. \TAG KP $ \left\{\alignedat2 &u_t + u_{xxx} + \sigma\partial_x^{-1}u_{yy}= - (u^{\rho})_x, &;&;\qquad (t,x,y) \in {\bold R}\times {\bold R}^2,\\ \vspace{.5\jot} &u(0,x,y) = u_0 (x,y),&;&; \qquad (x,y) \in{\bold R}^2, \endalignedat \right. \TAG KP where = 1 or = m 1. When = 2 and = m 1, (KP) is known as the KPI equation, while = 2, = + 1 corresponds to the KPII equation. The KP equation models the propagation along the x-axis of nonlinear dispersive long waves on the surface of a fluid, when the variation along the y-axis proceeds slowly [10]. The case = 3, = m 1 has been found in the modeling of sound waves in antiferromagnetics [15]. We prove that if S 3 is an integer and the initial data are sufficiently small, then the solution u of (KP) satisfies the following estimates: ||u(t)||¥ £ C (1 + |t|)-1 (log(2+|t|))k, ||ux(t)||¥ £ C (1 + |t|)-1 \|u(t)\|_\infty \le C (1 + |t|)^{-1} (\log (2+|t|))^{\kappa}, \|u_x(t)\|_\infty \le C (1 + |t|)^{-1} for all t ] R, where s = 1 if = 3 and s = 0 if S 4. We also find the large time asymptotics for the solution. 相似文献
2.
Manuel del Pino Patricio L. Felmer Peter Sternberg 《Communications in Mathematical Physics》2000,210(2):413-446
We examine the asymptotic behavior of the eigenvalue w(h) and corresponding eigenfunction associated with the variational problem m(h) o infy ? H1(W;C ) \fracòW \abs(i?+hA)y2 dx dy òW\absy2 dx dy \mu(h)\equiv\inf_{\psi\in H^{1}(\Omega;{\bf C} )} \frac{\int_{\Omega } \abs{(i\nabla+h{\bf A})\psi}^{2}\,dx\,dy} {\int_{\Omega }\abs{\psi}^{2}\,dx\,dy} in the regime h>>1. Here A is any vector field with curl equal to 1. The problem arises within the Ginzburg-Landau model for superconductivity with the function w(h) yielding the relationship between the critical temperature vs. applied magnetic field strength in the transition from normal to superconducting state in a thin mesoscopic sample with cross-section W ì \R2\Omega\subset\R^{2}. We first carry out a rigorous analysis of the associated problem on a half-plane and then rigorously justify some of the formal arguments of [BS], obtaining an expansion for w while also proving that the first eigenfunction decays to zero somewhere along the sample boundary ?W\partial \Omega when z is not a disc. For interior decay, we demonstrate that the rate is exponential. 相似文献
3.
Let S
2 be the 2-dimensional unit sphere and let J
α
denote the nonlinear functional on the Sobolev space H
1(S
2) defined by
$J_\alpha(u) = \frac{\alpha}{16\pi}\int_{S^2}|\nabla u|^2\, d\mu_0 + \frac{1}{4\pi} \int_{S^2} u\, d \mu_0 -{\rm ln} \int_{S^2} e^{u}
\, \frac{d \mu_0}{4\pi},$J_\alpha(u) = \frac{\alpha}{16\pi}\int_{S^2}|\nabla u|^2\, d\mu_0 + \frac{1}{4\pi} \int_{S^2} u\, d \mu_0 -{\rm ln} \int_{S^2} e^{u}
\, \frac{d \mu_0}{4\pi}, 相似文献
4.
This paper considers Hardy–Lieb–Thirring inequalities for higher order differential operators. A result for general fourth-order
operators on the half-line is developed, and the trace inequality
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