共查询到20条相似文献,搜索用时 28 毫秒
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Jie Xiao 《Journal of Differential Equations》2006,224(2):277-295
Let u(t,x) be the solution of the heat equation (∂t-Δx)u(t,x)=0 on subject to u(0,x)=f(x) on Rn. The main goal of this paper is to characterize such a nonnegative measure μ on that f(x)?u(t2,x) induces a bounded embedding from the Sobolev space , p∈[1,n) into the Lebesgue space , q∈(0,∞). 相似文献
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In this paper, we consider one-dimensional nonlinear Schrödinger equation iut−uxx+V(x)u+f(2|u|)u=0 on [0,π]×R under the boundary conditions a1u(t,0)−b1ux(t,0)=0, a2u(t,π)+b2ux(t,π)=0, , for i=1,2. It is proved that for a prescribed and analytic positive potential V(x), the above equation admits small-amplitude quasi-periodic solutions corresponding to d-dimensional invariant tori of the associated infinite-dimensional dynamical system. 相似文献
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Let V(x) be a non-negative, bounded potential in RN, N?3 and p supercritical, . We look for positive solutions of the standing-wave nonlinear Schrödinger equation Δu−V(x)u+up=0 in RN, with u(x)→0 as |x|→+∞. We prove that if V(x)=o(−2|x|) as |x|→+∞, then for N?4 and this problem admits a continuum of solutions. If in addition we have, for instance, V(x)=O(|x|−μ) with μ>N, then this result still holds provided that N?3 and . Other conditions for solvability, involving behavior of V at ∞, are also provided. 相似文献
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Mina Jiang 《Journal of Differential Equations》2009,246(1):50-2513
In this paper, we consider the so-called p-system with linear damping on quadrant. We show that for a certain class of given large initial data (v0(x),u0(x)), the corresponding initial-boundary value problem admits a unique global smooth solution (v(x,t),u(x,t)) and such a solution tends time-asymptotically, at the Lp (2?p?∞) optimal decay rates, to the corresponding nonlinear diffusion wave which satisfies (1.9) provided the corresponding prescribed initial error function (V0(x),U0(x)) lies in (H3(R+)∩L1(R+))×(H2(R+)∩L1(R+)). 相似文献
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Giovanna Cerami 《Journal of Mathematical Analysis and Applications》2009,359(1):15-27
This paper is concerned with the problem of finding positive solutions of the equation −Δu+(a∞+a(x))u=|u|q−2u, where q is subcritical, Ω is either RN or an unbounded domain which is periodic in the first p coordinates and whose complement is contained in a cylinder , a∞>0, a∈C(RN,R) is periodic in the first p coordinates, infx∈RN(a∞+a(x))>0 and a(x′,x″)→0 as |x″|→∞ uniformly in x′. The cases a?0 and a?0 are considered and it is shown that, under appropriate assumptions on a, the problem has one solution in the first case and p+1 solutions in the second case when p?N−2. 相似文献
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A maximum principle is proved for the weak solutions of the telegraph equation in space dimension three utt−Δxu+cut+λu=f(t,x), when c>0, λ∈(0,c2/4] and (Theorem 1). The result is extended to a solution and a forcing belonging to a suitable space of bounded measures (Theorem 2). Those results provide a method of upper and lower solutions for the semilinear equation utt−Δxu+cut=F(t,x,u). Also, they can be employed in the study of almost periodic solutions of the forced sine-Gordon equation. A counterexample for the maximum principle in dimension four is given. 相似文献
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Maitere Aguerrea 《Nonlinear Analysis: Theory, Methods & Applications》2010,72(6):2753-1889
We establish the existence of a continuous family of fast positive wavefronts u(t,x)=?(x+ct), ?(−∞)=0, ?(+∞)=κ, for the non-local delayed reaction-diffusion equation . Here 0 and κ>0 are fixed points of g∈C2(R+,R+) and the non-negative K is such that is finite for every real λ. We also prove that the fast wavefronts are non-monotone if . 相似文献
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A nonlinear shallow water equation, which includes the famous Camassa-Holm (CH) and Degasperis-Procesi (DP) equations as special cases, is investigated. The local well-posedness of solutions for the nonlinear equation in the Sobolev space Hs(R) with is developed. Provided that does not change sign, u0∈Hs () and u0∈L1(R), the existence and uniqueness of the global solutions to the equation are shown to be true in u(t,x)∈C([0,∞);Hs(R))∩C1([0,∞);Hs−1(R)). Conditions that lead to the development of singularities in finite time for the solutions are also acquired. 相似文献
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Hongtao Xue 《Journal of Mathematical Analysis and Applications》2011,384(2):439-443
By a sub-supersolution method and a perturbed argument, we improve the earlier results concerning the existence of ground state solutions to a semilinear elliptic problem −Δu+p(x)q|∇u|=f(x,u), u>0, x∈RN, , where q∈(1,2], for some α∈(0,1), p(x)?0, ∀x∈RN, and f:RN×(0,∞)→[0,∞) is a locally Hölder continuous function which may be singular at zero. 相似文献
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Pavol Quittner Frédérique Simondon 《Journal of Mathematical Analysis and Applications》2005,304(2):614-631
We study a priori estimates of positive solutions of the equation t∂u−Δu=λu+a(x)up, x∈Ω, t>0, satisfying the homogeneous Dirichlet boundary conditions. Here Ω is a bounded domain in Rn, λ∈R, p>1 is subcritical, changes sign and a,p satisfy some additional technical hypotheses. Assume that the solution u blows up in a finite time T and the set is connected. Using our a priori bounds, we show that u blows up completely in Ω+ at t=T and the blow-up time T depends continuously on the initial data. 相似文献
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In this paper we prove that the initial value problem of the OST equation ut+uxxx+η(Hux+Huxxx)+uux=0 (x∈R, t?0), where η>0 and H denotes the usual Hilbert transformation, is locally well-posed in the Sobolev space Hs(R) when , and globally well-posed in Hs(R) when s?0. 相似文献
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Vladimir Umanskiy 《Advances in Mathematics》2003,180(1):176-186
Given p≠0 and a positive continuous function g, with g(x+T)=g(x), for some 0<T<1 and all real x, it is shown that for suitable choice of a constant C>0 the functional has a minimizer in the class of positive functions u∈C1(R) for which u(x+T)=u(x) for all x∈R. This minimizer is used to prove the existence of a positive periodic solution y∈C2(R) of two-dimensional Lp-Minkowski problem y1−p(x)(y″(x)+y(x))=g(x), where p∉{0,2}. 相似文献
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Georg Menz 《Journal of Differential Equations》2007,242(1):171-191
First, we consider the linear wave equation utt−uxx+a(x)ut+b(x)u=0 on a bounded interval (0,L)⊂R. The damping function a is allowed to change its sign. If is positive and the spectrum of the operator (∂xx−b) is negative, exponential stability is proved for small . Explicit estimates of the decay rate ω are given in terms of and the largest eigenvalue of (∂xx−b). Second, we show the existence of a global, small, smooth solution of the corresponding nonlinear wave equation utt−σx(ux)+a(x)ut+b(x)u=0, if, additionally, the negative part of a is small enough compared with ω. 相似文献
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We study the following complex Ginzburg-Landau equation with cubic nonlinearity on for under initial and Dirichlet boundary conditions u(x,0)=h(x) for x∈Ω, u(x,t)=Q(x,t) on ∂Ω where h,Q are given smooth functions. Under suitable conditions, we prove the existence of a global solution in H1. Further, this solution approaches to the solution of the NLS limit under identical initial and boundary data as a,b→0+. 相似文献