共查询到20条相似文献,搜索用时 31 毫秒
1.
We investigate the existence of a global classical solution to the generalized Goursat problem. Under some degenerate assumptions of boundary conditions, we prove that the solution approaches a combination of Lipschitz continuous and a piecewise C1 traveling wave solution. 相似文献
2.
Zhi-Qiang Shao 《Nonlinear Analysis: Theory, Methods & Applications》2010,73(3):600-613
This paper is concerned with the asymptotic behavior of global classical solutions of diagonalizable quasilinear hyperbolic systems with linearly degenerate characteristic fields. On the basis of the existence result for the global classical solution, we prove that when t tends to the infinity, the solution approaches a combination of C1 traveling wave solutions, provided that the C1 norm and the BV norm of the initial data are bounded but possibly large. In contrast to former results obtained by Liu and Zhou [J. Liu, Y. Zhou, Asymptotic behaviour of global classical solutions of diagonalizable quasilinear hyperbolic systems, Math. Methods Appl. Sci. 30 (2007) 479-500], ours do not require their assumption that the system is rich in the sense of Serre. Applications include that to the one-dimensional Born-Infeld system arising in string theory and high energy physics. 相似文献
3.
Jianli Liu 《Nonlinear Analysis: Theory, Methods & Applications》2010,72(1):533-545
In this paper, we investigate the mixed initial-boundary value problem for diagonalizable quasilinear hyperbolic systems with nonlinear boundary conditions on a half-unbounded domain . Under the assumptions that system is strictly hyperbolic and linearly degenerate, we obtain the global existence and uniqueness of C1 solutions with the bounded L1∩L∞ norm of the initial data as well as their derivatives and appropriate boundary condition. Based on the existence results of global classical solutions, we also prove that when t tends to infinity, the solutions approach a combination of C1 travelling wave solutions. Under the appropriate assumptions of initial and boundary data, the results can be applied to the equation of time-like extremal surface in Minkowski space R1+(1+n). 相似文献
4.
Renato Manfrin 《Bulletin des Sciences Mathématiques》2004,128(4):253-263
We prove a result of formation of singularities for the classical solutions of the planar motion of a nonlinear elastic string. In a particular, but physically relevant, case we give a characterization of the global C1 solutions with positive tension. 相似文献
5.
De-Xing Kong 《Journal of Differential Equations》2003,188(1):242-271
In this paper, the author proves the global structure stability of the Lax's Riemann solution , containing only shocks and contact discontinuities, of general n×n quasilinear hyperbolic system of conservation laws. More precisely, the author proves the global existence and uniqueness of the piecewise C1 solution u=u(t,x) of a class of generalized Riemann problem, which can be regarded as a perturbation of the corresponding Riemann problem, for the quasilinear hyperbolic system of conservation laws; moreover, this solution has a global structure similar to that of the solution . Combining the results in Kong (Global structure instability of Riemann solutions of quasilinear hyperbolic systems of conservation laws: rarefaction waves, to appear), the author proves that the Lax's Riemann solution of general n×n quasilinear hyperbolic system of conservation laws is globally structurally stable if and only if it contains only non-degenerate shocks and contact discontinuities, but no rarefaction waves and other weak discontinuities. 相似文献
6.
Cyril Imbert 《Journal of Differential Equations》2005,211(1):218-246
In this paper, we investigate the regularizing effect of a non-local operator on first-order Hamilton-Jacobi equations. We prove that there exists a unique solution that is C2 in space and C1 in time. In order to do so, we combine viscosity solution techniques and Green's function techniques. Viscosity solution theory provides the existence of a W1,∞ solution as well as uniqueness and stability results. A Duhamel's integral representation of the equation involving the Green's function permits to prove further regularity. We also state the existence of C∞ solutions (in space and time) under suitable assumptions on the Hamiltonian. We finally give an error estimate in L∞ norm between the viscosity solution of the pure Hamilton-Jacobi equation and the solution of the integro-differential equation with a vanishing non-local part. 相似文献
7.
Zhi-Qiang Shao 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2014,65(2):241-278
This paper is concerned with the asymptotic behavior of global C 1 solutions of the Goursat problem for quasilinear hyperbolic systems. Based on the existence result on the global classical solution, we prove that when t tends to the infinity, the solution approaches a combination of Lipschitz continuous and piecewise C 1 traveling wave solutions, provided that the C 1 norm of the boundary data is bounded but possibly large, and the BV norm of the boundary data is sufficiently small. Applications include the 1D compressible Euler equations for Chaplygin gases. 相似文献
8.
Przemys?aw Górka 《Nonlinear Analysis: Theory, Methods & Applications》2010,72(6):2852-2858
We study the Cauchy problem for a 1D nonlinear wave equation on R. The nonlinearity can depend on the unknown function and its first order spatial derivative. Using the fixed point theorem we prove the existence of a classical solution. Moreover, the existence of periodic and almost periodic solutions are shown. 相似文献
9.
Peng Qu 《Journal of Differential Equations》2011,251(8):2066-2081
For the Cauchy problem of 1-D first order quasilinear hyperbolic linearly degenerate systems, a new mechanism of singularity formation is given to show that all the W1,p(1<p?+∞) norms of the C1 solution should blow up simultaneously. It gives a way to verify the property of ODE singularity by directly using the energy method in the framework of C1 solution. 相似文献
10.
In this paper, the existence and asymptotic behavior of C1 solutions to the multi-dimensional compressible Euler equations with damping on the framework of Besov space are considered. Comparing with the well-posedness results of Sideris–Thomases–Wang [T. Sideris, B. Thomases, D.H. Wang, Long time behavior of solutions to the three-dimensional compressible Euler with damping, Comm. Partial Differential Equations 28 (2003) 953–978], we weaken the regularity assumptions on the initial data. The global existence lies on a crucial a-priori estimate which is obtained by the spectral localization method. The main analytic tools are the Littlewood–Paley decomposition and Bony’s paraproduct formula. 相似文献
11.
Wladimir Neves 《Journal of Differential Equations》2003,192(2):360-395
We study the initial-boundary-value problems for multidimensional scalar conservation laws in noncylindrical domains with Lipschitz boundary. We show the existence-uniqueness of this problem for initial-boundary data in L∞ and the flux-function in the class C1. In fact, first considering smooth boundary, we obtain the L1-contraction property, discuss the existence problem and prove it by the Young measures theory. In the end we show how to pass the existence-uniqueness results on to some domains with Lipschitz boundary. 相似文献
12.
Éric Dumas 《Journal of Differential Equations》2005,219(2):484-509
Maxwell-Bloch equations describe the propagation of an electromagnetic wave through a quantum medium. For any number of quantum levels, in space dimension 3, we show the global existence of weak (L2) solutions to the initial-value problem. In the case of smoother electromagnetic fields (with curl in L2), the solution is unique. For smooth data (Hs, s?2), the solutions remain smooth for all times. 相似文献
13.
It is proven that the generalized Riemann problem for a class of quasilinear hyperbolic systems of balance laws admits a unique global piecewise C1 solution u=u(t,x) containing only n shock waves with small amplitude on t?0 and this solution possesses a global structure similar to that of the similarity solution u=U(x/t) of the corresponding homogeneous Riemann problem. As an application of our result, we prove the existence of global shock solutions, piecewise continuous and piecewise smooth solution with shock discontinuities, of the flow equations of a model class of fluids with viscosity induced by fading memory with a single jump initial data. 相似文献
14.
Yi Zhou 《Nonlinear Analysis: Theory, Methods & Applications》2010,73(6):1543-1561
In this paper, we consider the mixed initial-boundary value problem for quasilinear hyperbolic systems with nonlinear boundary conditions in the first quadrant . Under the assumptions that the system is strictly hyperbolic and linearly degenerate or weakly linearly degenerate, the global existence and uniqueness of C1 solutions are obtained for small initial and boundary data. We also present two applications for physical models. 相似文献
15.
One of the features of solutions of semilinear wave equations can be found in blow-up results for non-compactly supported data. In spite of finite propagation speed of the linear wave, we have no global in time solution for any power nonlinearity if the spatial decay of the initial data is weak. This was first observed by Asakura (1986) [2] finding out a critical decay to ensure the global existence of the solution. But the blow-up result is available only for zero initial position having positive speed.In this paper the blow-up theorem for non-zero initial position by Uesaka (2009) [22] is extended to higher-dimensional case. And the assumption on the nonlinear term is relaxed to include an example, |u|p−1u. Moreover the critical decay of the initial position is clarified by example. 相似文献
16.
In this article, we give the existence of global L∞ bounded entropy solutions to the Cauchy problem of a generalized n × n hyperbolic system of LeRoux type. The main difficulty lies in establishing some compactness estimates of the viscosity solutions because the system has been generalized from 2 × 2 to n × n and more linearly degenerate characteristic fields emerged, and the emergence of singularity in the region {v1=0} is another difficulty. We obtain the existence of the global weak solutions using the compensated compactness method coupled with the construction of entropy-entropy flux and BV estimates on viscous solutions. 相似文献
17.
Mitsuhiro Nakao 《Nonlinear Analysis: Theory, Methods & Applications》2012,75(4):2236-2248
We prove the global existence of the so-called H2 solutions for a nonlinear wave equation with a nonlinear dissipative term and a derivative type nonlinear perturbation. To show the boundedness of the second order derivatives we need a precise energy decay estimate and for this we employ a ‘loan’ method. 相似文献
18.
Jong Uhn Kim 《Journal of Differential Equations》2011,250(3):1650-1684
We establish the existence and uniqueness of a local smooth solution to the Cauchy problem for a quasi-linear symmetric hyperbolic system with random noise in Rd. When the noise is multiplicative satisfying some nondegenerate conditions and the initial data are sufficiently small, we show that the solution exists globally in time in probability, i.e., the probability of global existence can be made arbitrarily close to one if the initial date are small accordingly. 相似文献
19.
Based on a new a priori estimate method, so-called asymptotic a priori estimate, the existence of a global attractor is proved for the wave equation utt+kg(ut)−Δu+f(u)=0 on a bounded domain Ω⊂R3 with Dirichlet boundary conditions. The nonlinear damping term g is supposed to satisfy the growth condition C1(|s|−C2)?|g(s)|?C3(1+p|s|), where 1?p<5; the damping parameter is arbitrary; the nonlinear term f is supposed to satisfy the growth condition |f′(s)|?C4(1+q|s|), where q?2. It is remarkable that when 2<p<5, we positively answer an open problem in Chueshov and Lasiecka [I. Chueshov, I. Lasiecka, Long-time behavior of second evolution equations with nonlinear damping, Math. Scuola Norm. Sup. (2004)] and improve the corresponding results in Feireisl [E. Feireisl, Global attractors for damped wave equations with supercritical exponent, J. Differential Equations 116 (1995) 431-447]. 相似文献
20.
Wenjun Liu 《Nonlinear Analysis: Theory, Methods & Applications》2010,73(1):244-1904
This paper studies the Cauchy problem for the coupled system of nonlinear Klein-Gordon equations with damping terms. We first state the existence of standing wave with ground state, based on which we prove a sharp criteria for global existence and blow-up of solutions when E(0)<d. We then introduce a family of potential wells and discuss the invariant sets and vacuum isolating behavior of solutions for 0<E(0)<d and E(0)≤0, respectively. Furthermore, we prove the global existence and asymptotic behavior of solutions for the case of potential well family with 0<E(0)<d. Finally, a blow-up result for solutions with arbitrarily positive initial energy is obtained. 相似文献