Related papers: Critical exponent for the wave equation with a tim…
In this paper, we consider the Cauchy problem for semi-linear wave equations with structural damping term $\nu (-\Delta)^2 u_t$, where $\nu >0$ is a constant. As being mentioned in [8,10], the linear principal part brings both the diffusion…
In this paper we find the critical exponent for the global existence (in time) of small data solutions to the Cauchy problem for the semilinear dissipative evolution equations % \[ u_{tt}+(-\Delta)^\delta u_{tt}+(-\Delta)^\alpha…
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 paper we study the existence of global-in-time energy solutions to the Cauchy problem for the Euler-Poisson-Darboux equation, with a power nonlinearity: $$u_{tt}-u_{xx} + \frac\mu{t}\,u_t = |u|^p \,, \quad t>t_0, \…
In this paper, we would like to consider the Cauchy problem for a weakly coupled system of semi-linear damped wave equations with mixed nonlinear terms. Our main objective is to draw conclusions about the critical curve of this problem…
We study the following Cauchy problem for the linear wave equation with both time-dependent friction and time-dependent viscoelastic damping: \begin{equation} \label{EqAbstract}\tag{$\ast$} \begin{cases} u_{tt}- \Delta u + b(t)u_t -…
In this paper, we would like to consider the Cauchy problem for a multi-component weakly coupled system of semi-linear $\sigma$-evolution equations with double dissipation for any $\sigma\ge 1$. The first main purpose is to obtain the…
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 +…
In this paper, we would like to consider the Cauchy problem for a weakly coupled system of semi linear $sigma$ evolution equations with different damping mechanisms for any $\sigma>1$, parabolic like damping and $\sigma$ evolution like…
In this paper, we investigate the critical exponent for a semi-linear damped wave equation involving a Hartree-type nonlinearity of the form $\mathcal{I}_\gamma\left(|u|^{p_1}\right)|u|^{p_2}, p_1, p_2>0, \gamma \in[0, n)$, with initial…
In this paper, we would like to study the critical exponent for semi-linear damped wave equations with power nonlinearity and the initial data belonging to Sobolev spaces of negative order $\dot{H}_m^{-\gamma}$. Precisely, we obtain a new…
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 $a,b\in\mathbb{R}_+$…
In this paper, we consider the Cauchy problem for a semilinear damped wave equation with the nonlinear term $|u|^{1+2/n} \mu(|u|)$, where $\mu$ is a modulus of continuity. In recent papers by Ebert,Girardi,Reissig (Math. Ann. 378 (2020))…
In this note we study the global existence of small data solutions to the Cauchy problem for the semi-linear wave equation with a not effective scale-invariant damping term, namely \[ v_{tt}-\triangle v + \frac2{1+t}\,v_t = |v|^p, \qquad…
We are interested in studying the Cauchy problem for a weakly coupled system of semi-linear $\sigma$-evolution equations with frictional damping. The main purpose of this paper is two-fold. We would like to not only prove the global (in…
In this paper we consider the critical exponent problem for the semilinear wave equation with space-time dependent damping. When the damping is effective, it is expected that the critical exponent agrees with that of only space dependent…
We consider the system of elastic waves with critical space dependent damping $V(x)$. We study the Cauchy problem for this model in the $2$-dimensional Euclidean space ${\bf R}^{2}$, and we obtain faster decay rates of the total energy as…
Let $\mathbb G$ be a graded Lie group with homogeneous dimension $Q$. In this paper, we study the Cauchy problem for a semilinear hypoelliptic damped wave equation involving a positive Rockland operator $\mathcal{R}$ of homogeneous degree…
We study semilinear damped wave equations with power nonlinearity $|u|^p$ and initial data belonging to Sobolev spaces of negative order $\dot{H}^{-\gamma}$. In the present paper, we obtain a new critical exponent…
In this paper, we consider the Cauchy problem for the semilinear damped wave equation on the Heisenberg group with power nonlinearity. We prove that the critical exponent is the Fujita exponent $p_{\mathrm{Fuj}}(\mathscr{Q}) = 1+2 /…