Related papers: Cauchy problem for operators with triple effective…
In the present article, a modified Cauchy problem (problem C) for the hyperbolic equation of the third order with the data on the equation's coefficients singularity plane is solved by Riemann method. The special class in which the solution…
A critical point metric is a critical point of the total scalar curvature functional restricted to the space of constant scalar curvature metrics on a closed manifold with unit volume. It was conjectured in 1980's that every critical point…
The theory of weak solutions for nonlinear conservation laws is now well developed in the case of scalar equations [3] and for one-dimensional hyperbolic systems [1, 2]. For systems in several space dimensions, however, even the global…
We consider the Cauchy problem in the Euclidean space for a doubly degenerate parabolic equation with a space-dependent exponential weight, roughly speaking of the type of the exponential of a power of the distance from the origin. We…
We consider the Cauchy problem for a $n\times n$ strictly hyperbolic system of balance laws $$ \{{array}{c} u_t+f(u)_x=g(x,u), x \in \mathbb{R}, t>0 u(0,.)=u_o \in L^1 \cap BV(\mathbb{R}; \mathbb{R}^n), | \lambda_i(u)| \geq c > 0 {for all}…
Results providing conditions on a family of integro-differential operators to determine a formal automorphism are established. Equivalently, the problem can be read in terms of existence and uniqueness of formal solutions of Cauchy problems…
In this paper, we investigate the initial value problem for symmetric hyperbolic systems on globally hyperbolic Lorentzian manifolds with potentials that are both nonlocal in time and space. When the potential is retarded and uniformly…
This paper contributes to the wider study of hyperbolic equations with multiplicities. We focus here on some classes of higher order hyperbolic equations with space dependent coefficients in any space dimension. We prove Sobolev…
In this paper, we consider the well-posedness of the Cauchy problem for a physical model of the extrusion process, which is described by two systems of conservation laws with a free boundary. By suitable change of coordinates and fixed…
We extend the notion of general translation operator to exceptional Laguerre polynomials. To this we investigate the associated singular hyperbolic Cauchy problem. We derive a maximum principle with respect to this Cauchy problem and…
We consider the Cauchy problem and the source problem for normally hyperbolic operators on the Minkowski spacetime, and study the determination of solutions from their integrals along null geodesics. For the Cauchy problem, we give a new…
A 3D coefficient inverse problem for a hyperbolic equation with non-overdetermined data is considered. The forward problem is the Cauchy problems with the initial condition the delta function concentrated at a single plane (i.e. the plane…
We consider the Cauchy problem for weakly hyperbolic $m$-th order partial differential equations with coefficients low-regular in time and smooth in space. It is well-known that in general one has to impose Levi conditions to get $C^\infty$…
We define the notion of critical points at infinity for the charged N-body problem, following the approach of Albouy 1993. We give a characterisation of such points and show how they can be found in the charged 3-body problem. The symmetry…
In this paper we prove an energy estimate with no loss of derivatives for a strictly hyperbolic operator with Zygmund continuous second order coefficients both in time and in space. In particular, this estimate implies the well-posedness…
We consider the Cauchy problem for the hyperbolic-elliptic Ishimori system with general decoupling constant $\kappa \in \mathbb{R}$ and prove global well-posedness in the critical Sobolev space. The proof relies primarily on new bilinear…
Studied in this paper is the well-posedness of the Cauchy problem for the coupled KdV-KdV systems \[ u_t+a_1u_{xxx} = c_{11}uu_x+c_{12}vv_x+d_{11}u_{x}v+d_{12}uv_{x}, \quad u(x,0)= u_0(x) \] \[ v_t+a_2v_{xxx}= c_{21}uu_x+c_{22}vv_x…
We show the short time existence and uniqueness of solutions to the Cauchy problem for fully nonlinear systems of arbitrary even order on closed manifolds which are strongly parabolic at the initial values. The proof uses a linearization…
We revisit the theory of first-order quasilinear systems with diagonalizable principal part and only real eigenvalues, what is commonly referred to as strongly hyperbolic systems. We provide a self-contained and simple proof of local…
We prove some $C^\infty$ and Gevrey well-posedness results for hyperbolic equations whose coefficients lose regularity at one point.