Related papers: On nonlinear damped wave equations for positive op…
We consider the Cauchy problem for the weakly dissipative wave equation $$ \square u+\frac\mu{1+t} u_t=0 $$ with parameter $\mu\ge2$. Based on the explicit representations of solutions provided in [Math. Meth. Appl. Sci. 2004; {\bf…
In this paper we consider energy decay estimates for the Cauchy problems of dissipative wave equations with time dependent coefficients, in particular, the coefficients consisting of weak dissipation and very fast oscillating terms. For…
In this paper we address the problem of wave dynamics in presence of concentrated nonlinearities. Given a vector field $V$ on an open subset of $\CO^n$ and a discrete set $Y\subset\RE^3$ with $n$ elements, we define a nonlinear operator…
We consider the Cauchy problem in R^n for some types of damped wave equations. We derive asymptotic profiles of solutions with weighted L^{1,1}(R^n) initial data by employing a simple method introduced by the first author. The obtained…
This paper investigates the Cauchy problem for the semilinear damped wave equation $u_{tt}+\mathcal{L}_{a,b}u+u_t=|u|^p$ with the mixed local-nonlocal operator $\mathcal{L}_{a,b}:=-a\Delta+b(-\Delta)^{\sigma}$, where $a,b\in\mathbb{R}_+$…
We consider the total energy decay of the Cauchy problem for wave equations with a potential and an effective damping. We treat it in the whole one-dimensional Euclidean space. Fast energy decay is established with the help of potential.…
We study the Cauchy problem for the cubic nonlinear Schroedinger equation, perturbed by (higher order) dissipative nonlinearities. We prove global in-time existence of solutions for general initial data in the energy space. In particular we…
We consider the Cauchy problem for the defocusing power type nonlinear wave equation in $(1+3)$-dimensions for energy subcritical powers $p$ in the range $3 < p< 5$. We prove that any solution is global-in-time and scatters to free waves in…
This paper is concerned with the semilinear damped wave equation on a measure space with a self-adjoint operator, instead of the standard Laplace operator. Under a certain decay estimate on the corresponding heat semigroup, we establish the…
In this paper we consider the Cauchy problem for the semilinear damped wave equation $u_{tt}-\Delta u + u_t = h(u);\qquad u(0;x) = f(x); \quad u_t(0;x) = g(x);$ where $h(s) = |s|^{1+2/n}\mu(|s|)$. Here n is the space dimension and $\mu$ is…
We study the Cauchy problem for the nonlinear wave equations (NLW) with random data and/or stochastic forcing on a two-dimensional compact Riemannian manifold without boundary. (i) We first study the defocusing stochastic damped NLW driven…
We establish sharp energy decay rates for a large class of nonlinearly first-order damped systems, and we design discretization schemes that inherit of the same energy decay rates, uniformly with respect to the space and/or time…
In this paper, we study the following Cauchy problem for linear visco-elastic damped wave models with a general time-dependent coefficient $g=g(t)$: \begin{equation} \label{EqAbstract} \tag{$\star$} \begin{cases} u_{tt}- \Delta u +…
First, using the uniform decomposition in both physical and frequency spaces, we obtain an equivalent norm on modulation spaces. Secondly, we consider the Cauchy problem for the dissipative evolutionary pseudo-differential equation…
In this paper we consider the Cauchy problem for linear dissipative generalized Klein-Gordon equations with nonlinear memory in the right hand side. Our goal is to study the effect of this nonlinearity on both the decay estimates of global…
In this paper, we consider the Cauchy problem for a non-homogeneous wave equation generated by the fractional Laplacian and involving different kinds of lower order terms. We allow the equation coefficients and data to be of distributional…
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 by using an iteration argument some blow-up results for a semilinear damped wave equation in generalized Einstein-de Sitter spacetime with a time-dependent coefficient for the damping term and power nonlinearity. Then, we…
We consider the Cauchy problem of a dissipative nonlinear Schr\"odinger equation with a time dependent harmonic potential. We find a critical situation that the $L^2$-norm of dissipative solutions decays or not and which is decided by a…
We introduce a new model of the logarithmic type of wave like plate equation with a nonlocal logarithmic damping mechanism. We consider the Cauchy problem for this new model in the whole space, and study the asymptotic profile and optimal…