Related papers: Non-homogeneous wave equation on a cone

We investigate the problem of classification of solutions for the steady Navier-Stokes equations in any cone-like domains. In the form of separated variables, $$u(x,y)=\left( \begin{array}{c} \varphi_1(r)v_1(\theta) \varphi_2(r)v_2(\theta)…

We show wave breaking for the Whitham equation in a range of fractional dispersion, i.e. the solution remains bounded but its slope becomes unbounded in finite time, provided that the initial datum is sufficiently steep.

Criteria for the existence of $T$-periodic solutions of nonautonomous parabolic equation $u_t = \Delta u + f(t,x,u)$, $x\in\mathbb{R}^N$, $t>0$ with asymptotically linear $f$ will be provided. It is expressed in terms of time average…

We look for travelling wave fields $$ E(x,y,z,t)= U(x,y) \cos(kz+\omega t)+ \widetilde U(x,y)\sin(kz+\omega t),\quad (x,y,z)\in\mathbb{R}^3,\, t\in\mathbb{R}, $$ satisfying Maxwell's equations in a nonlinear and cylindrically symmetric…

Some solutions for one class of nonlinear fourth-order partial differential equations \[u_{tt} = ({\kappa u + \gamma u^2})_{xx} + \nu uu_{xxxx} + \mu u_{xxtt} + \alpha u_x u_{xxx} + \beta u_{xx}^2 \] where $\alpha ,\;\beta ,\;\gamma ,\;\mu…

A new class of solitary waves arises in the solution of nonlinear wave equations with constant impedance and no dispersive terms. They depend on a balance between nonlinearity and a dispersion-like effect due to spatial variation in the…

This work is concerned with the equation $ \partial_t \rho = \Delta_x \rho^m $, $ m > 1 $, known as the porous medium equation. It shows stability of the pressure of solutions close to flat travelling wave fronts in the homogeneous…

We show that the parabolic equation $u_t + (-\Delta)^s u = q(x) |u|^{\alpha-1} u$ posed in a time-space cylinder $(0,T) \times \mathbb{R}^N$ and coupled with zero initial condition and zero nonlocal Dirichlet condition in $(0,T) \times…

We consider the wave equation with a damping term on a partially rectangular planar domain, assuming that the damping is concentrated close to the non-rectangular part of the domain. Polynomial decay estimates for the energy of the solution…

Orthogonal polynomials of degree $n$ with respect to the weight function $W_\mu(x) = (1-\|x\|^2)^\mu$ on the unit ball in $\RR^d$ are known to satisfy the partial differential equation $$ [ \Delta - \la x, \nabla \ra^2 - (2 \mu +d) \la x,…

In the present paper we consider global solutions of a class of non-linear wave equations of the form \begin{equation*} \Box u= N(x,t,u)u, \end{equation*} where the nonlinearity~$ N(x,t,u)u$ is assumed to satisfy appropriate boundedness…

This paper is devoted to studying the following two initial-boundary value problems for semilinear wave equations with variable coefficients on exterior domain with subcritical exponent in $n$ space dimensions:…

In this work we consider the defocusing nonlinear wave equation in one-dimensional space. We show that almost all energy is located near the light cone $|x|=|t|$ as time tends to infinity. We also prove that any light cone will eventually…

This paper finds solutions to semilinear wave equations with strongly anomalous propagation of singularities. For very low Sobolev regularity we obtain solutions whose singular support propagates along any ray inside or outside the light…

A numerical explicit method to evaluates transient solutions of linear partial differential inhomogeneous equation with constant coefficients is proposed. A general form of the scheme for a specific linear inhomogeneous equation is shown.…

We consider the linear wave equation $V(x) u_{tt}(x, t) - u_{xx}(x, t) = 0$ on $[0, \infty)\times[0, \infty)$ with initial conditions and a nonlinear Neumann boundary condition $u_x(0, t) = (f(u_t(0,t)))_t$ at $x=0$. This problem is an…

By means of a direct and constructive method based on the theory of semiglobal $C^2$ solution, the local exact boundary observability is shown for nonautonomous 1-D quasilinear wave equations. The essential difference between nonautonomous…

We consider the Cauchy problem for the wave equation in $\Omega\times{\mathbb R}$ with data given on some part of the boundary $\partial\Omega\times{\mathbb R}$. We provide a reconstruction algorithm for this problem based on analytic…

In this paper, we first study the dual fractional parabolic equation \begin{equation*} \partial^\alpha_t u(x,t)+(-\Delta)^s u(x,t) = f(u(x,t))\ \ \mbox{in}\ \ B_1(0)\times\R , \end{equation*} subject to the vanishing exterior condition. We…

For $s \in [1/2, 1)$, let $u$ solve $(\partial_t - \Delta)^s u = Vu$ in $\mathbb R^{n} \times [-T, 0]$ for some $T>0$ where $||V||_{ C^2(\mathbb R^n \times [-T, 0])} < \infty$. We show that if for some $0< c< T$ and $\epsilon>0$…