具C^∞系数的非主型方程的Goursat问题

本文以逼近和正则化方法，对一类具Ｃ＾∞系数和数据的非主型方程的Ｇｏｕｒｓａｔ问题证明了Ｃ＾∞解的存在唯一性。     

MULTIDIMENSIONAL GOURSAT PSOBLEM FOR SEMILINEAR WAVE EQUATIONS

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MULTIDIMENSIONAL GOURSAT PROBLEM FOR SEMILINEAR HYPERBOLIC EQUATIONS

In this paper we study the Goursat problem for semilinear hyperbolicequations with zero boundary condition where the boundary is the characteristic conefor hyperbolic operator. Our result shows that the solution is Lipschitz and is smoothaway from the characteristic cone.     

Riemann and Goursat step data problems for extensible strings

Riemann and Goursat step data problems for extensible nonlinear elastic strings are solved in the class of regulated functions. In the first paragraph, the solution to the simplest initial and boundary value problem, i.e., Goursat problem in strains, is constructed. This solution points out four vector-valued functions of a vector variable, which are the tools used in solving the Goursat problem in velocity and the Riemann problem.     

On existence and nonexistence of global solutions of Cauchy–Goursat problem for nonlinear wave equations

We consider the Cauchy–Goursat initial characteristic problem for nonlinear wave equations with power nonlinearity. Depending on the power of nonlinearity and the parameter in an equation we investigate the problem on existence and nonexistence of global solutions of the Cauchy–Goursat problem. The question on local solvability of the problem is also considered.     

Numerical solution of the Goursat problem on a triangular domain with mixed boundary conditions

For the Goursat problem, we consider a triangular domain with mixed Dirichlet and impedance boundary conditions imposed on it. We develop an algorithm for its numerical solution mainly based on Runge-Kutta method and trapezoidal formula. Iterative techniques are constructed to compute some data for the nonlinear part of the differential equation and the impedance boundary condition. Error estimates are derived. Examples are presented to illustrate the effectiveness of the method.     

Decompositions of functions and Dirichlet problems in the unit disc

In this paper, we establish a decomposition theorem for polyharmonic functions and consider its applications to some Dirichlet problems in the unit disc. By the decomposition, we get the unique solution of the Dirichlet problem for polyharmonic functions (PHD problem) and give a unified expression for a class of kernel functions associated with the solution in the case of the unit disc introduced by Begehr, Du and Wang. In addition, we also discuss some quasi-Dirichlet problems for homogeneous mixed-partial differential equations of higher order. It is worthy to note that the decomposition theorem in the present paper is a natural extension of the Goursat decomposition theorem for biharmonic functions.     

Characteristic boundary-value problem for a system of first-order partial differential equations with shifted arguments of the desired function

For a system of first-order partial differential equations of a form not studied earlier we consider a variant of Goursat problem and prove the existence and uniqueness of the solution to the problem.     

具线性退化特征场对角型方程组Goursat问题的奇性形成机制

For an inhomogeneous quasilinear hyperbolic system of diagonal form, under the assumptions that the system is linearly degenerate and the C^1 norm of the boundary data is bounded, we show that the mechanism of the formation of singularities of C^1 classical solution to the Goursat problem with C^1 compatibility conditions at the origin must be an ODE type. The similar result is also obtained for the weakly discontinuous solution with C^0 compatibility conditions at the origin.     

One Goursat problem in a Sobolev space

In this paper we consider a hyperbolic-type differential equation with L _p -coefficients in a three-dimensional space. For this equation we study the Goursat problem with nonclassical boundary constraints not requiringmatched conditions. We prove the equivalence of these boundary conditions to classical ones in the case when one seeks for a solution to the stated problem in an anisotropic space introduced by S. L. Sobolev. In addition, we prove the correct solvability of the Goursat problem by the method of integral equations.     

Characteristic boundary value problem for a third-order functional-differential equation with the Bianchi operator

We consider a version of the Goursat problem for a previously unstudied third-order equation and prove the existence and uniqueness of the solution of this problem.     

Two-dimensional Riemann problem of the Euler equations to the Van der Waals gas around a sharp corner

In this paper, we study the Riemann problem of the two-dimensional (2D) pseudo-steady supersonic flow with Van der Waals gas around a sharp corner expanding into vacuum. The essence of this problem is the interaction of the centered simple wave with the planar rarefaction wave, which can be solved by a Goursat problem or a mixed characteristic boundary value and slip boundary value problem for the 2D self-similar Euler equations. We establish the hyperbolicity and a priori C¹ estimates of the solution through the methods of characteristic decompositions and invariant regions. Moreover, we construct the pentagon invariant region in order to obtain the global solution. In addition, based on the generality of the Van der Waals gas, we construct the subinvariant regions and get the hyperbolicity of the solution according to the continuity of the subinvariant region. At last, the global existence of solution to the gas expansion problem is obtained constructively.     

The Goursat problem for the fractional telegraph equation with Caputo derivatives

The Goursat problem for the fractional telegraph equation with Caputo derivatives is studied. An existence and uniqueness theorem for the solution of the problem is proved.     

Global existence of classical solutions of Goursat problem for quasilinear hyperbolic systems with BV data

We investigate the existence of a global classical solution to the Goursat problem for linearly degenerate quasilinear hyperbolic systems. As the result in [A. Bressan, Contractive metrics for nonlinear hyperbolic systems, Indiana Univ. Math. J. 37 (1988) 409–421] suggests that one may achieve global smoothness even if the C¹ norm of the initial data is large, we prove that, if the C¹ norm of the boundary data is bounded but possibly large, and the BV norm of the boundary data is sufficiently small, then the solution remains C¹ globally in time. Applications include the equation of time‐like extremal surfaces in Minkowski space R^{1 + (1 + n)} and the one‐dimensional Chaplygin gas equations. Copyright © 2013 John Wiley & Sons, Ltd.     

Numerical solution of the Goursat problem by a nonlinear trapezoidal formula

In this paper the solution of the Goursat problem is obtained by the use of a nonlinear Trapezoidal formula based on geometric means. The numerical results indicate the new strategy to be superior.     

Solvability of the Goursat problem in quadratures

We find new variants of Goursat problem solution in quadratures on the basis of the combination of the Riemann method and the cascade integration. The results are applied to two Volterra equations with particular integrals.     

Application of the fundamental principle to complex Cauchy problem

In this paper we give an explicit formula for the solution of the non-homogeneous complex Cauchy problem with Cauchy data given on a bounded smooth strictly convex domain in a non-characteristic hyperplane. These formulas are obtained using the explicit version of the fundamental principle given in terms of residue currents; moreover, we characterize the domain of definition of the solution and we generalize these techniques to the non-homogeneous Goursat problem.     

Global exact boundary observability for first‐order quasilinear hyperbolic systems of diagonal form

In this paper, by means of a constructive method based on the theory of the existence and the uniqueness of the C¹ solution to the Cauchy problem and the Goursat problem, the global exact boundary observability for the first‐order quasilinear hyperbolic systems of diagonal form with linearly degenerate characteristics is obtained. In the case that the system has no zero characteristics, we realize the two‐sided and one‐sided global exact boundary observability by the boundary observed values and obtain the observability inequality. Copyright © 2012 John Wiley & Sons, Ltd.     

Influence of Lower Terms on the Well-Posedness of Characteristics Problems for Third-Order Hyperbolic Equations

In this paper, we consider a general problem of Goursat type for third-order hyperbolic equations of general form with dominated lower terms. We study the influence of the lower terms contained in the equation, as well as those in the boundary conditions, on the well-posedness of the problem under consideration.     

Goursat problem for a fourth-order equation with the biwave operator

We consider the Goursat problem for a fourth-order linear partial differential equation with the biwave operator. By using the energy inequality method and averaging operators with variable increment, we prove the existence and uniqueness of a strong solution.