Global classical solutions to a kind of quasilinear hyperbolic systems in several space variables

For a kind of quasilinear hyperbolic systems in several space variables whose coefficient matrices commute each other, by means of normalized coordinates, formulas of wave decomposition and pointwise decay estimates, the global existence of classical solution to the Cauchy problem for small and decaying initial data is obtained, under hypotheses of weak linear degeneracy and weakly strict hyperbolicity.     

The method of plane waves for a hyperbolic system with several spatial variables

In the 80s of the XXth century the Riemann method for hyperbolic equations of the second order was extended to hyperbolic systems in general form with one spatial variable. In this paper, this result is extended to hyperbolic systems in general form with several spatial variables with time dependent coefficients.     

On a class of strongly hyperbolic systems

In this paper we prove a well-posedness result for the Cauchy problem. We study a class of first order hyperbolic differential [2] operators of rank zero on an involutive submanifold ofT ^* R ⁿ⁺¹-{0} and prove that under suitable assumptions on the symmetrizability of the lifting of the principal symbol to a natural blow up of the “singular part” of the characteristic set, the operator is strongly hyperbolic.     

On the asymptotics of the solution to a singularly perturbed hyperbolic system of equations with several spatial variables in the critical case

A complete asymptotic expansion of the solution to an initial value problem for a singularly perturbed hyperbolic system of equations in several spatial variables is constructed and justified. A specific feature of the problem is that its solution has a spike zone in a neighborhood of which the asymptotics is described by a parabolic equation.     

Global classical solutions for a class of quasilinear hyperbolic systems

In this paper we investigate the mechanism of singularity formation to the Cauchy problem of quasilinear hyperbolic system.Moreover, we present some examples to show some significant problems.     

On some hyperbolic systems of temple class

The aim of this paper is the statement of a general class of Temple systems of conservation laws that includes both the chromatography/electrophoresis like systems and the 2×2 LeRoux system that we generalize to any dimension. We show that this class actually belongs to the Temple type, and compute a complete set of strict Riemann invariants in a generic situation. As the property “the integral curve of this eigenvector is a straight line”is essentially a linear algebra property, we aim to deduce the results with simple (linear algebra) hypothesis and arguments, from the structure of the jacobian matrix.     

A stability theorem for entropy solutions of initial value problems for first order quasilinear hyperbolic systems in several space variables

In this paper we prove an uniqueness and stability theorem for the solutions of Cauchy problem for the systems $$\frac{\partial }{{\partial t}}u + \sum\limits_{i = 1}^n { \frac{\partial }{{\partial x_i }} } f^i (x,t,u) = g(x,t,u),$$ whereu is a vector function (u ₁ (x, t),..., u _r (x, t)),f ⁱ =(a ₁ ⁱ (x, t, u),..., a _r ⁱ (x, t, u)), i=1,...,n, g=(g ₁ (x, t, u),...,g _r (x, t, u),i G ? ⁿ and t≥0. We use the concept of entropy solution introduced by Kruskov and improved by Lax, Dafermos and others autors. We assume that the Jacobian matricesf _u ⁱ are symmetric and the Hessian(a _j ⁱ ) _uu (i=1,...,n; j=1,...,r) are positive. We obtain uniqueness and stability inL _loc ² within the class of those entropy solutions which satisfy $$\frac{{u_j (---,x_i ,---,t)---u_j (---,y_i ,---,t)}}{{x_i - y_i }} \geqslant - K(t),$$ (i=1,...,n; j=1,...,r) for (?,x _i ,?,t), (?,y _i ,?,t) on a compact setD ? ? ⁿ x (0, ∞) and a functionK(t)∈L _loc ¹ ([0, ∞)) depending onD. Here we denote by (?,x _i ,?,t) and (?,y _i ,?,t) two points whose coordinates only differ in thei-th space variable. At the end we relax the hypotheses of symmetry and convexity on the system and give a theorem of uniqueness and stability for entropy solutions which are locally Lipschitz continuous on a strip ? ⁿ x [0,T].     

On the fundamental solutions of a class of hyperbolic quartic operators in dimension 3

The fundamental solutions of the linear hyperbolic partial differential operators with constant coefficients of the form are represented by elliptic integrals of the first kind. Mathematics Subject Classification (2000) 35A08, 35A20, 35E05     

Full regularity for a class of degenerated parabolic systems in two spatial variables

The authors consider quasilinear parabolic systems