Related papers: Asymptotics from scaling for nonlinear wave equati…

We survey some of our recent results on inverse problems for evolution equations. The goal is to provide a unified approach to solve various types of evolution equations. The inverse problems we consider consist in determining unknown…

In the analysis of highly-oscillatory evolution problems, it is commonly assumed that a single frequency is present and that it is either constant or, at least, bounded from below by a strictly positive constant uniformly in time. Allowing…

We study the large-time behaviour of the solutions $u$ of the evolution equation involving nonlinear diffusion and gradient absorption $\partial_t u - \Delta_p u + |\nabla u|^q=0$. We consider the problem posed for $x\in {\mathbb R}^N $ and…

Similarity reductions and new exact solutions are obtained for a nonlinear diffusion equation. These are obtained by using the classical symmetry group and reducing the partial differential equation to various ordinary differential…

We study the asymptotic behavior in time of solutions to the one dimensional nonlinear Schr\"odinger equation with a subcritical dissipative nonlinearity $\lambda |u|^\alpha u$, where $0<\alpha<2$, and $\lambda $ is a complex constant…

In this paper, we prove the diffusion phenomenon for the linear wave equation with space-dependent damping. We prove that the asymptotic profile of the solution is given by a solution of the corresponding heat equation in the $L^2$-sense.

We deduce the asymptotic error distribution of the Euler method for the nonlinear filtering problem with continuous-time observations. Previous works by several authors have shown that the error structure of the method is characterized by…

We study the Cauchy problem of the semilinear damped wave equation with polynomial nonlinearity, and establish the local and global existence of the solution for slowly decaying initial data not belonging to $L^2(\mathbb{R}^n)$ in general.…

For small-amplitude semilinear wave equations with power type nonlinearity on the first-order spatial derivative, the expected sharp upper bound on the lifespan of solutions is obtained for both critical cases and subcritical cases, for all…

We present a new method for the solution of the Schrodinger equation applicable to problems of non-perturbative nature. The method works by identifying three different scales in the problem, which then are treated independently: An…

The aim of this paper is to give an account of some applications of pseudodifferential calculus for solving linear wave equations in the limit of high frequency/short wavelength waves. More specifically, on using as a benchmark the case of…

We present a new method for the nonlinear approximation of the solution manifolds of parameterized nonlinear evolution problems, in particular in hyperbolic regimes with moving discontinuities. Given the action of a Lie group on the…

In this paper, the Lie symmetry method is performed for the nonlinear evolution equation for description nonlinear waves in a viscoelastic tube. we will find one and two-dimensional optimal system of Lie subalgebras. Furthermore,…

Variational data assimilation is a technique for combining measured data with dynamical models. It is a key component of Earth system state estimation and is commonly used in weather and ocean forecasting. The approach involves a…

We propose a new approach to the old-standing problem of the anomaly of the scaling exponents of nonlinear models of turbulence. We achieve this by constructing, for any given nonlinear model, a linear model of passive advection of an…

Large-scale simulations of the wave equation in electromagnetism, seismology, and acoustics, can be solved efficiently by finite difference methods. The accuracy of these numerical solutions usually depends on the minimization of…

We improve on recent results that establish the existence of solutions of certain semilinear wave equations possessing an interface that roughly sweeps out a timelike surface of vanishing mean curvature in Minkowski space. Compared to…

In this paper, we prove pointwise decay rates for cubic and higher order nonlinear wave equations, including quasilinear wave equations, on asymptotically flat and time-dependent spacetimes. We assume that the solution to the linear…

We prove a local well-posedness result for an evolution problem consisting of a semilinear wave equation with subcritical nonlinearities posed on a time-dependent compact Riemannian manifold and supplied with a nonlinear dynamical boundary…

We study semilinear wave equations with Ginzburg-Landau type nonlinearities multiplied by a factor $\epsilon^{-2}$, where $\epsilon>0$ is a small parameter. We prove that for suitable initial data, solutions exhibit energy concentration…