Related papers: Method for solving hyperbolic systems with initial…
A new approach using a hyperbolic-equation system (HES) is proposed to solve for the electron fluids in quasi-neutral plasmas. The HES approach avoids treatments of cross-diffusion terms which cause numerical instabilities in conventional…
We investigate the finite time stability property of one-dimensional nonautonomous initial boundary value problems for linear decoupled hyperbolic systems with nonlinear boundary conditions. We establish sufficient and necessary conditions…
An averaging method for getting uniformly valid asymptotic approximations of the solution of hyperbolic systems of equations is presented. The averaged system of equations disintegrates into independent equations for non-resonance systems.…
Two new approaches to solving first-order quasilinear elliptic systems of PDEs in many dimensions are proposed. The first method is based on an analysis of multimode solutions expressible in terms of Riemann invariants, based on links…
We consider the initial-boundary value problem for systems of quasilinear wave equations on domains of the form $[0,T] \times \Sigma$, where $\Sigma$ is a compact manifold with smooth boundaries $\partial\Sigma$. By using an appropriate…
We study hyperbolic systems of one-dimensional partial differential equations under general, possibly non-local boundary conditions. A large class of evolution equations, either on individual 1-dimensional intervals or on general networks,…
It was shown recently that the constraints on the initial data for Einstein's equations may be posed as an evolutionary problem [9]. In one of the proposed two methods the constraints can be replaced by a first order symmetrizable…
We give sufficient conditions on the initial data so that a semilinear wave inequality blows-up in finite time. Our method is based on the study of an associated second order differential inequality. The same method is applied to some…
Coupled wave equations are popular tool for investigating longitudinal dynamical effects in semiconductor lasers, for example, sensitivity to delayed optical feedback. We study a model that consists of a hyperbolic linear system of partial…
In this paper, we investigate the speed of convergence and higher-order asymptotics of solutions to the porous medium equation posed in $\mathbf{R}^N$. Applying a nonlinear change of variables, we rewrite the equation as a diffusion on a…
We prove the global classical solvability of initial-boundary problems for semilinear first-order hyperbolic systems subjected to local and nonlocal nonlinear boundary conditions. We also establish lower bounds for the order of nonlinearity…
A method is presented for solving the characteristic initial value problem for the collision and subsequent nonlinear interaction of plane gravitational or gravitational and electromagnetic waves in a Minkowski background. This method…
We assign some kind of invariant manifolds to a given integrable PDE (its discrete or semi-discrete variant). First, we linearize the equation around its arbitrary solution $u$. Then we construct a differential (respectively, difference)…
Well-posedness of certain semilinear parabolic problems with nonlocal initial conditions is shown in time-weighted spaces. The result is applied to recover the initial states in semilinear parabolic problems with nonlinearities of…
This paper investigates solutions of hyperbolic diffusion equations in $\mathbb{R}^3$ with random initial conditions. The solutions are given as spatial-temporal random fields. Their restrictions to the unit sphere $S^2$ are studied. All…
We derive and analyse well-posed boundary conditions for the linear shallow water wave equation. The analysis is based on the energy method and it identifies the number, location and form of the boundary conditions so that the initial…
We discuss a purely variational approach to the study of a wide class of second order nonhomogeneous dissipative hyperbolic PDEs. Precisely, we focus on the wave-like equations that present also a nonzero source term and a…
The paper deals with initial-boundary value problems for linear non-autonomous first order hyperbolic systems whose solutions stabilize to zero in a finite time. We prove that problems in this class remain exponentially stable in $L^2$ as…
In classical continuum physics, a wave is a mechanical disturbance. Whether the disturbance is stationary or traveling and whether it is caused by the motion of atoms and molecules or the vibration of a lattice structure, a wave can be…
We discuss solution concepts for linear hyperbolic equations with coefficients of regularity below Lipschitz continuity. Thereby our focus is on theories which are based either on a generalization of the method of characteristics or on…