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We construct global weak solutions of the Euler equations in an infinite cylinder $\Pi=\{x\in \mathbb{R}^{3}\ |\ x_h=(x_1,x_2),\ r=|x_h|<1\}$ for axisymmetric initial data without swirl when initial vorticity…
In this paper we extend analysis of the WaveHoltz iteration -- a time-domain iterative method for the solution of the Helmholtz equation. We expand the previous analysis of energy conserving problems and prove convergence of the WaveHoltz…
We present a preliminary study of a new phenomena associated with the Euler-Poisson equations -- the so called critical threshold phenomena, where the answer to questions of global smoothness vs. finite time breakdown depends on whether the…
We consider the asymptotic behavior of the soltion to the wave equation with time-dependent damping and analytic nonlinearity. Our main goal is to prove the convergence of a global solution to an equilibrium as time goes to infinity by…
Motivated by numerically modeling surface waves for inviscid Euler equations, we analyze linear models for damped water waves and establish decay properties for the energy for sufficiently regular initial configurations. Our findings give…
In this paper, we study the Cauchy problem to the linear damped $\sigma$-evolution equation with time-dependent damping in the effective cases \begin{equation*} u_{t t}+(-\Delta)^\sigma u+b(t)(-\Delta)^\delta u_t=0, \end{equation*} and…
In this article, we have studied the convergence behavior of the Dirichlet-Neumann and Neumann- Neumann waveform relaxation algorithms for time-fractional sub-diffusion and diffusion-wave equations in 1D & 2D for regular domains, where the…
This paper is concerned with the initial-boundary value problem on the full Euler-Poisson system for ions over a half line. We establish the existence of stationary solutions under the Bohm criterion similar to the isentropic case and…
We investigate the homogeneous Dirichlet problem for the Fast Diffusion Equation $u_t=\Delta u^m$, posed in a smooth bounded domain $\Omega\subset \mathbb{R}^N$, in the exponent range $m_s=(N-2)_+/(N+2)<m<1$. It is known that bounded…
We study the one-dimensional isentropic compressible Euler equations with linear (frictional) damping, subject to multiplicative, white-in-time stochastic forcing. The system is posed on a bounded interval with $L^\infty$ initial data and…
We consider the asymptotic behaviour of finite energy solutions to the one-dimensional defocusing nonlinear wave equation $-u_{tt} + u_{xx} = |u|^{p-1} u$, where $p > 1$. Standard energy methods guarantee global existence, but do not…
We consider an initial-boundary value problem for $\partial_tu-\partial_t^{-\alpha}\nabla^2u=f(t)$, that is, for a fractional diffusion ($-1<\alpha<0$) or wave ($0<\alpha<1$) equation. A numerical solution is found by applying a…
In this paper, we investigate the well-posedness and the long-time asymptotic behavior for the initial-boundary value problem for multi-term time-fractional diffusion equations, where the time differentiation consists of a finite summation…
In this paper we obtain higher order asymptotic profilles of solutions to the Cauchy problem of the linear damped wave equation in $\textbf{R}^n$ \begin{equation*} u_{tt}-\Delta u+u_t=0, \qquad u(0,x)=u_0(x), \quad u_t(0,x)=u_1(x),…
The large time $t$ asymptotics for scalar, constant coefficient,linear, third order, dispersive equations are obtained for asymptotically time-periodic Dirichlet boundary data and zero initial data on the half-line modeling a wavemaker…
We consider a class of porous medium type of equations with Caputo time derivative. The prototype problem reads as $\Dc u=-\A u^m$ and is posed on a bounded Euclidean domain $\Omega\subset\mathbb{R}^N$ with zero Dirichlet boundary…
A class of semi-bounded solutions of the two-dimensional incompressible Euler equations satisfying either periodic or Dirichlet boundary conditions is examined. For smooth initial data, new blowup criteria in terms of the initial concavity…
In this article we are concerned with an inverse boundary value problem for a non-linear wave equation of divergence form with space dimension $n\geq 3$. In particular the so called the interior determination problem. This non-linear wave…
In this paper we study a semilinear wave equation with nonlinear, time-dependent damping in one space dimension. For this problem, we prove a well-posedness result in $W^{1,\infty}$ in the space-time domain $(0,1)\times [0,+\infty)$. Then…
We study the large time behavior of solutions to the Cauchy problem for the quasilinear absorption-diffusion equation $$ \partial_tu=\Delta u^m-|x|^{\sigma}u^p, \quad (x,t)\in\real^N\times(0,\infty), $$ with exponents $p>m>1$ and $\sigma>0$…