共查询到20条相似文献,搜索用时 15 毫秒
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F. Linares A. Pastor J. Drumond Silva 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2021,38(2):281-300
The main purpose here is the study of dispersive blow-up for solutions of the Zakharov-Kuznetsov equation. Dispersive blow-up refers to point singularities due to the focusing of short or long waves. We will construct initial data such that solutions of the linear problem present this kind of singularities. Then we show that the corresponding solutions of the nonlinear problem present dispersive blow-up inherited from the linear component part of the equation. Similar results are obtained for the generalized Zakharov-Kuznetsov equation. 相似文献
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Luz de Teresa 《偏微分方程通讯》2013,38(1-2):39-72
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In this article, local unique continuation on a line for solutions of the Helmholtz equation is discussed. The fundamental solution of the exterior problem for the Helmholtz equation have a logarithmic singularity which behaves similar to those of the interior problem for the Laplace equation in two dimension. A Hölder-type conditional stability estimate of the proposed exterior problem for the Helmholtz equation is obtained by adopting the complex extension method in Cheng and Yamamoto [J. Cheng and M. Yamamoto, Unique continuation on a line for harmonic functions, Inverse Probl. 14 (1998), pp. 869–882]. Finally, a regularization scheme based on the collocation method is compatible with the Hölder-type stability estimate provided that the line does not intersect the boundary of the domain for both the Laplace and the Helmholtz equations. 相似文献
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Ching-Lung Lin 《偏微分方程通讯》2016,41(5):749-758
A Carleman estimate and the unique continuation of solutions for an anomalous diffusion equation with fractional time derivative of order 0 < α <1 are given. The estimate is derived through some subelliptic estimates for an operator associated to the anomalous diffusion equation using calculus of pseudo-differential operators. 相似文献
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Zehra Arat Azer Khanmamedov Sema Simsek 《Mathematical Methods in the Applied Sciences》2016,39(4):744-761
In this paper, we prove the unique continuation property for the weak solution of the plate equation with the low regular coefficient. Then, we apply this result to study the global attractor for the semilinear plate equation with a localized damping. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
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Giovanni Alessandrini Sergio Vessella 《Proceedings of the American Mathematical Society》2004,132(2):499-501
We consider solutions , in a neighbourhood of , to a parabolic differential equation with variable coefficients depending on space and time variables. We assume that the coefficients in the principal part are Lipschitz continuous and that those in the lower order terms are bounded. We prove that, if vanishes of infinite order at , then .
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Rakesh 《Mathematical Methods in the Applied Sciences》2009,32(2):246-252
We prove unique continuation of solutions of the wave equation along and across lower‐dimensional planes containing the t‐axis. This is a sharpening and a generalization of a result of Cheng, Ding and Yamamoto as well as a simplification of the proof. Copyright © 2008 John Wiley & Sons, Ltd. 相似文献
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In this paper, we study certain unique continuation properties for solutions of the semilinear heat equation t∂u−△u=g(u), with the homogeneous Dirichlet boundary condition, over Ω×(0,T∗). Ω is a bounded, convex open subset of Rd, with a smooth boundary for the subset. The function g:R→R satisfies certain conditions. We establish some observation estimates for (u−v), where u and v are two solutions to the above-mentioned equation. The observation is made over ω×{T}, where ω is any non-empty open subset of Ω, and T is a positive number such that both u and v exist on the interval [0,T]. At least two results can be derived from these estimates: (i) if ‖(u−v)(⋅,T)‖L2(ω)=δ, then ‖(u−v)(⋅,T)‖L2(Ω)?Cδα where constants C>0 and α∈(0,1) can be independent of u and v in certain cases; (ii) if two solutions of the above equation hold the same value over ω×{T}, then they coincide over Ω×[0,Tm). Tm indicates the maximum number such that these two solutions exist on [0,Tm). 相似文献
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We consider the initial-value problem for the bidirectional Whitham equation, a system which combines the full two-way dispersion relation from the incompressible Euler equations with a canonical shallow-water nonlinearity. We prove local well-posedness in classical Sobolev spaces, using a square-root type transformation to symmetrise the system. 相似文献
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We prove that the flow map of the Kadomtsev–Petviashvili-I (KP-I) equation is not uniformly continuous on bounded sets of
the natural energy space.
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In this paper we study the case of equalities in some comparison results for L
1-norm or L
-norm of the solutions of Dirichlet elliptic problem or Hamilton-Jacobi equations. We show that equalities are achieved only in spherically symmetric situations.Work partially supported by MURST (40%). 相似文献
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Guoguang LIN Department of Mathematics Yunnan University Kunming China e-mail: gglin@ynu. edu. cn 《Communications in Nonlinear Science & Numerical Simulation》2000,5(1)
1 FormulationLet fi be a bounded set of R", n 2 2, tv'ith a sufficiently smooth boundary off. ac'e considerthe motion of an incompressible, viscous fluid in fi, which is described by the following systemll]with the initial conditionand the Dirichlet boundary conditionwhere v = (yi, .". u.) is the velocity, p is the pressure, the tensor r = (n j) is defined asp is a. positive constant. Notice that when p = 2 the s}l'steln (1), (2), (5) turns to the N*avierStokes system.A scalar potential to t… 相似文献
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We prove a sharp three sphere inequality for solutions to third order perturbations of a product of two second order elliptic operators with real coefficients. Then we derive various kinds of quantitative estimates of unique continuation for the anisotropic plate equation. Among these, we prove a stability estimate for the Cauchy problem for such an equation and we illustrate some applications to the size estimates of an unknown inclusion made of different material that might be present in the plate. The paper is self-contained and the Carleman estimate, from which the sharp three sphere inequality is derived, is proved in an elementary and direct way based on standard integration by parts. 相似文献
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Quantitative estimates of unique continuation for parabolic equations and inverse initial-boundary value problems with unknown boundaries 总被引:4,自引:0,他引:4
B. Canuto E. Rosset S. Vessella 《Transactions of the American Mathematical Society》2002,354(2):491-535
In this paper we obtain quantitative estimates of strong unique continuation for solutions to parabolic equations. We apply these results to prove stability estimates of logarithmic type for an inverse problem consisting in the determination of unknown portions of the boundary of a domain in , from the knowledge of overdetermined boundary data for parabolic boundary value problems.
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Let u be the solutions to some elliptic equations of the form
in some Lipschitz domains, a direct and elementary proof of the doubling property for u2 over balls near the boundary is presented, and the unique continuation at the boundary for Dini domains is founded.AMS Subject Classification (1991): 42B20, 35J10 相似文献