Related papers: Non-uniqueness for the nonlinear dynamical Lam\'e …
In this paper, we show almost global existence of small solutions to the Cauchy problem for symmetric system of wave equations with quadratic (in 3D) or cubic (in 2D) nonlinear terms and multiple propagation speeds. To measure the size of…
We present a coherent and effective theoretical framework to systematically construct numerically exact nonlinear solitary waves from their respective linear limits. First, all possible linear degenerate sets are classified for a harmonic…
We consider the strictly hyperbolic Cauchy problem \begin{align*} &D_t^m u - \sum\limits_{j = 0}^{m-1} \sum\limits_{|\gamma|+j = m} a_{m-j,\,\gamma}(t,\,x) D_x^\gamma D_t^j u = 0, \newline &D_t^{k-1}u(0,\,x) = g_k(x),\,k = 1,\,\ldots,\,m,…
We deal with the Cauchy problem for a perturbed wave equation in the half-plane with data given on a part of the space-time boundary. The equation in consideration describes a wave process in a laterally inhomogeneous medium. We propose a…
In this paper, we study the precise decay rate in time to solutions of the Cauchy problem for the one-dimensional conservation law with a nonlinearly degenerate viscosity where the far field states are prescribed. Especially, we deal with…
It is shown that the Euler system of hydrodynamic equations for inviscid barotropic fluid for density and velocity is not a complete system of dynamic equations for the inviscicd barotropic fluid. It is only a closed subsystem of four…
We consider the Cauchy problem in ${\bf R}^{n}$ for strongly damped wave equations. We derive asymptotic profiles of these solutions with weighted $L^{1,1}({\bf R}^{n})$ data by using a method introduced in [10].
In this paper, we investigate a system coupled by nonhomogeneous incompressible Navier-Stokes equations and Allen-Cahn equations describing a diffuse interface for two-phase flow of viscous fluids with different densities in a bounded…
We consider a three-dimensional domain occupied by a homogeneous, incompressible, non-Newtonian, heat-conducting fluid with prescribed nonuniform temperature on the boundary and no-slip boundary conditions for the velocity. No external body…
We develop an efficient and convergent numerical method for solving the inverse problem of determining the potential of nonlinear hyperbolic equations from lateral Cauchy data. In our numerical method we construct a sequence of linear…
We study the Cauchy problem in the hyperbolic space for the heat equation with a Fisher-KPP type forcing term. Depending on the relative strength of diffusion, measured by the infimum of the spectrum of the Laplace-Beltrami operator, as…
A combination of some weighted energy estimates is applied for the Cauchy problem of quasilinear wave equations with the standard null conditions in three spatial dimensions. Alternative proofs for global solutions are shown including the…
The Cauchy problem for the Burgers equation with a small dissipation and an initial weak discontinuity and the Cauchy problem with a large initial gradient for a quasilinear parabolic equation and for the Korteweg-de Vries (KdV) equation…
We introduce a new model of the nonlocal wave equations with a logarithmic damping mechanism. We consider the Cauchy poroblem for the new model in the whole space. We study the asymptotic profile and optimal decay and blowup rates of…
We study a nonlocal wave equation with logarithmic damping which is rather weak in the low frequency zone as compared with frequently studied strong damping case. We consider the Cauchy problem for this model in the whole space and we study…
For a class of weakly hyperbolic systems of the form D_t - A(t,x,D_x), where A(t,x,D_x) is a first-order pseudodifferential operator whose principal symbol degenerates like t^{l_*} at time t=0, for some integer l_* \geq 1, well-posedness of…
In this paper we give a meaning to the nonlinear characteristic Cauchy problem for the Wave Equation in base form by replacing it by a family of non-characteristic problems in an appropriate algebra of generalized functions. We prove…
This paper concerns the Cauchy problem of the two-dimensional (2D) nonhomogeneous incompressible Magnetohydrodynamic (MHD) equations with vacuum as far field density. We establish the global existence and uniqueness of strong solutions to…
We derive a linear model of navigation in a two-layer fluid with a variable velocity of the ship. A spectral version of the model including a Rayleigh damping term is analyzed. We prove that the Cauchy problem has a unique solution if the…
We consider the strong and weak solutions to the Cauchy problem of the inhomogeneous incompressible nematic liquid crystal equations in two dimensions. We first establish the local existence and uniqueness of strong solutions by using the…