Related papers: On the degenerate Cauchy problem for a nonlinear v…
In this paper, we establish the existence of spatially inhomogeneous classical self-similar solutions to a non-Lipschitz semi-linear parabolic Cauchy problem with trivial initial data. Specifically we consider bounded solutions to an…
We study one-dimensional viscoelastic phase transitions modeled by a Ginzburg--Landau energy with a non-convex cubic stress-strain law. Extending the isothermal model, we couple the momentum equation to a heat equation for the temperature…
We study inhomogeneous non-strictly hyperbolic systems of two equations, which are a formal generalization of the transformed one-dimensional Euler-Poisson equations. For such systems, a complete classification of the behavior of the…
In this paper, we study the Cauchy problem for a quasilinear degenerate parabolic stochastic partial differential equation driven by a cylindrical Wiener process. In particular, we adapt the notion of kinetic formulation and kinetic…
We consider solutions of a scalar reaction-diffusion equation of the ignition type with a random, stationary and ergodic reaction rate. We show that solutions of the Cauchy problem spread with a deterministic rate in the long time limit. We…
In this paper we establish the optimal regularity estimates for the Cauchy problem of stochastic kinetic equations with random coefficients in anisotropic Besov spaces. As applications, we study the nonlinear filtering problem for a…
We consider the Cauchy problem for one-dimensional p-system with damping of space-dependent coefficient. This system models the compressible flow through porous media in the Lagrangean coordinate. Our concern is an asymptotic behavior of…
We discuss the Cauchy problem for anisotropic wave equations. Precisely, we address the question to know which kind of Cauchy data on the lateral boundary are necessary to guarantee uniqueness of solutions of an anisotropic wave equation.…
We study the Cauchy problem for the defocusing modified Korteweg-de Vries (mKdV) equation with step-like initial data approaching nonzero constants $c_l$ and $c_r$ as $x \to -\infty$ and $x\to+\infty$, respectively. Assuming $c_l>c_r>0$,…
This paper is devoted to study the wave propagation and its stability for a class of two-component discrete diffusive systems. We first establish the existence of positive monotone monostable traveling wave fronts. Then, applying the…
We consider the following Cauchy problem for weakly coupled systems of semi-linear damped elastic waves with a power source non-linearity in three-dimensions: \begin{equation*} U_{tt}-a^2\Delta U-\big(b^2-a^2\big)\nabla\text{div }…
I describe a new algorithm for solving nonlinear wave equations. In this approach, evolution takes place on characteristic hypersurfaces. The algorithm is directly applicable to electromagnetic, Yang-Mills and gravitational fields and other…
This article is focused on a multidimensional nonlinear variational wave equation which is the Euler-Lagrange equation of a variational principle arising form the theory of nematic liquid crystals. By using the method of characteristics, we…
In this paper, we investigate the asymptotic behavior of solutions toward a multiwave pattern of the Cauchy problem for the scalar viscous conservation law where the far field states are prescribed. Especially, we deal with the case when…
We study the scattering problem for the nonlinear wave equation with potential. In the absence of the potential, one has sharp existence results for the Cauchy problem with small initial data; those require the data to decay at a rate…
We consider the Cauchy problem for wave equations with unbounded damping coefficients in the whole space. For a general class of unbounded damping coefficients, we derive uniform total energy decay estimates together with a unique existence…
Local structures, beyond the well-known `frozen-in' to the barotropic flows of the generalized vorticities, of the two-fluid model of plasma flows are presented. More general non-barotropic situations are also considered. A modified Euler…
We study the Cauchy problem for systems of cubic nonlinear Klein-Gordon equations with different masses in one space dimension. Under a suitable structural condition on the nonlinearity, we will show that the solution exists globally and…
We prove that solutions to Cauchy problems related to the $p$-parabolic equations are stable with respect to the nonlinearity exponent $p$. More specifically, solutions with a fixed initial trace converge in an $L^q$-space to a solution of…
This paper investigates solitary water waves propagating along the surface of a two-dimensional dielectric fluid with constant vorticity in the presence of an external electric field. We formulate the system as a nonlinear free boundary…