共查询到20条相似文献,搜索用时 15 毫秒
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Le Xuan Truong Le Thi Phuong Ngoc Alain Pham Ngoc Dinh Nguyen Thanh Long 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(18):6933-6949
This paper is devoted to studying a nonlinear wave equation with boundary conditions of two-point type. First, we state two local existence theorems and under the suitable conditions, we prove that any weak solutions with negative initial energy will blow up in finite time. Next, we give a sufficient condition to guarantee the global existence and exponential decay of weak solutions. Finally, we present numerical results. 相似文献
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HuangWenhua ShenZuhe 《高校应用数学学报(英文版)》1999,14(2):131-136
An existence and uniqueness result concerned with the Diriehlet boundary value prob-lem u“ cu‘ g(t,u)=e(t),u(0)=u(π)=0 is offered. 相似文献
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Aldo T. Lourêdo M.A. Ferreira de Araújo M. Milla Miranda 《Mathematical Methods in the Applied Sciences》2014,37(9):1278-1302
This paper is concerned with the existence and decay of solutions of the mixed problem for the nonlinear wave equation with boundary conditions Here, Ω is an open bounded set of with boundary Γ of class C2; Γ is constituted of two disjoint closed parts Γ0 and Γ1 both with positive measure; the functions μ(t), f(s), g(s) satisfy the conditions μ(t) ≥ μ0 > 0, f(s) ≥ 0, g(s) ≥ 0 for t ≥ 0, s ≥ 0 and h(x,s) is a real function where x ∈ Γ1, ν(x) is the unit outward normal vector at x ∈ Γ1 and α, β are non‐negative real constants. Assuming that h(x,s) is strongly monotone in s for each x ∈ Γ1, it is proved the global existence of solutions for the previous mixed problem. For that, it is used in the Galerkin method with a special basis, the compactness approach, the Strauss approximation for real functions and the trace theorem for nonsmooth functions. The exponential decay of the energy is derived by two methods: by using a Lyapunov functional and by Nakao's method. Copyright © 2013 John Wiley & Sons, Ltd. 相似文献
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P. Jameson Graber 《Journal of Evolution Equations》2012,12(1):141-164
We consider a wave equation with nonlinear acoustic boundary conditions. This is a nonlinearly coupled system of hyperbolic
equations modeling an acoustic/structure interaction, with an additional boundary damping term to induce both existence of
solutions as well as stability. Using the methods of Lasiecka and Tataru for a wave equation with nonlinear boundary damping,
we demonstrate well-posedness and uniform decay rates for solutions in the finite energy space, with the results depending
on the relationship between (i) the mass of the structure, (ii) the nonlinear coupling term, and (iii) the size of the nonlinear
damping. We also show that solutions (in the linear case) depend continuously on the mass of the structure as it tends to
zero, which provides rigorous justification for studying the case where the mass is equal to zero. 相似文献
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Jong Uhn Kim 《Transactions of the American Mathematical Society》2008,360(2):575-607
We prove the existence and uniqueness of solutions to the initial boundary value problem for a one-dimensional wave equation with unilateral boundary conditions and random noise. We also establish the existence of an invariant measure.
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Andrzej Nowakowski 《Nonlinear Analysis: Theory, Methods & Applications》2010,73(6):1495-1514
We discuss solvability for the semilinear equation of the vibrating string xtt(t,y)−Δx(t,y)+f(t,y,x(t,y))=0 in a bounded domain, and certain type of nonlinearity on the boundary. To this effect we derive a new dual variational method. Next we discuss stability of solutions with respect to initial conditions. 相似文献
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We consider a variable-coefficient wave equation with nonlinear damped acoustic boundary conditions. Well-posedness in the Hadamard sense for strong and weak solutions is proved by using the theory of nonlinear semigroups. 相似文献
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Le Thi Phuong Ngoc Nguyen Anh Triet Nguyen Thanh Long 《Nonlinear Analysis: Theory, Methods & Applications》2010,72(3-4):1865-1885
The paper is devoted to the study of a linear wave equation with a nonlinear boundary condition of viscoelastic type. The existence of a weak solution is proved by using the Faedo–Galerkin method. The uniqueness, regularity and asymptotic expansion of the solution are also discussed. 相似文献
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George J. Bautista Juan Límaco Leyter Potenciano-Machado 《Mathematical Methods in the Applied Sciences》2024,47(2):1132-1152
In this paper, we prove the well-posedness of a nonlinear wave equation coupled with boundary conditions of Dirichlet and acoustic type imposed on disjoints open boundary subsets. The proposed nonlinear equation models small vertical vibrations of an elastic medium with weak internal damping and a general nonlinear term. We also prove the exponential decay of the energy associated with the problem. Our results extend the ones obtained in previous results to allow weak internal dampings and removing the dimensional restriction . The method we use is based on a finite-dimensional approach by combining the Faedo-Galerkin method with suitable energy estimates and multiplier techniques. 相似文献
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Le Xuan Truong Le Thi Phuong Ngoc Alain Pham Ngoc Dinh Nguyen Thanh Long 《Nonlinear Analysis: Real World Applications》2010,11(3):1289-1303
This paper is concerned with the existence and the regularity of global solutions to the linear wave equation associated the boundary conditions of two-point type. We also investigate the decay properties of the global solutions to this problem by the construction of a suitable Lyapunov functional. Finally, we present some numerical results. 相似文献
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Numerical methods for weak solution of wave equation with van der Pol type nonlinear boundary conditions
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Jun Liu Yu Huang Haiwei Sun Mingqing Xiao 《Numerical Methods for Partial Differential Equations》2016,32(2):373-398
We develop computational methods for solving wave equation with van der Pol type nonlinear boundary conditions under the framework of weak solutions. Based on the wave reflection on the boundaries, we first solve the Riemann invariants by constructing two iteration mappings, and then show that the weak solution can be obtained by the integration of the Riemann invariants on the boundaries. If the compatible conditions are not satisfied or only hold with a low degree, a high‐order integration method is developed for the numerical solution. When the initial condition is sufficiently smooth and compatible conditions hold with a sufficient degree, we establish a sixth‐order finite difference scheme, which only needs to solve a linear system at any given time instance. Numerical experiments are provided to demonstrate the effectiveness of the proposed approaches. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 373–398, 2016 相似文献
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M.M. Cavalcanti V.N. Domingos Cavalcanti J.A. Soriano 《Journal of Mathematical Analysis and Applications》2003,281(1):108-124
We study the global existence of solutions of the nonlinear degenerate wave equation (ρ?0)
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The goal of this work is to study a model of the wave equation with semilinear porous acoustic boundary conditions with nonlinear boundary/interior sources and a nonlinear boundary/interior damping. First, applying the nonlinear semigroup theory, we show the existence and uniqueness of local in time solutions. The main difficulty in proving the local existence result is that the Neumann boundary conditions experience loss of regularity due to boundary sources. Using an approximation method involving truncated sources and adapting the ideas in Lasiecka and Tataru (1993) [28], we show that the existence of solutions can still be obtained. Second, we prove that under some restrictions on the source terms, then the local solution can be extended to be global in time. In addition, it has been shown that the decay rates of the solution are given implicitly as solutions to a first order ODE and depends on the behavior of the damping terms. In several situations, the obtained ODE can be easily solved and the decay rates can be given explicitly. Third, we show that under some restrictions on the initial data and if the interior source dominates the interior damping term and if the boundary source dominates the boundary damping, then the solution ceases to exists and blows up in finite time. Moreover, in either the absence of the interior source or the boundary source, then we prove that the solution is unbounded and grows as an exponential function. 相似文献
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S. A. Brykalov 《Georgian Mathematical Journal》1994,1(3):243-249
The paper gives sufficient conditions for the existence and nonuniqueness of monotone solutions of a nonlinear ordinary differential equation of the second order subject to two nonlinear boundary conditions one of which is two-point and the other is integral. The proof is based on an existence result for a problem with functional boundary conditions obtained by the author in [6]. 相似文献
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Ludmila S. Pulkina 《Proceedings of the Steklov Institute of Mathematics》2012,278(1):199-207
Two problems with nonlinear boundary conditions are studied. Existence and uniqueness theorems are proved for generalized solutions to each problem. 相似文献
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Jean C. Cortissoz 《Archiv der Mathematik》2011,97(1):69-78
Using ideas arising in the works of LeJan and Sznitman and Mattingly and Sinai on their study of the Navier–Stokes equations,
we investigate the blow-up behavior of a nonlinear parabolic equation subject to periodic boundary conditions. 相似文献