相关论文: On the semilinear damped wave equation with Riesz …
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 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…
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 focus on studying the Cauchy problem for semilinear damped wave equations involving the sub-Laplacian $\mathcal{L}$ on the Heisenberg group $\mathbb{H}^n$ with power type nonlinearity $|u|^p$ and initial data taken from…
We study the Cauchy problem of the damped wave equation \begin{align*} \partial_{t}^2 u - \Delta u + \partial_t u = 0 \end{align*} and give sharp $L^p$-$L^q$ estimates of the solution for $1\le q \le p < \infty\ (p\neq 1)$ with derivative…
In this paper, we study semilinear damped equations $u_{tt}+u_t-\Delta u=|u|^p$ with the initial data in $({\dot{H}^{-\gamma}}\cap H^s)\times({\dot{H}^{-\gamma}}\cap L^2)$. Chen-Reissig (2023) studied the case $0<\gamma<\frac{n}{2}$ and…
In this note, we prove the global existence of solutions to the semilinear damped wave equation in $\mathbb{R}^n$, $n\leq6$, with critical nonlinearity under the assumption that the initial data are small in the energy space $H^1\times L^2$…
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 /…
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…
Our aim in this paper is to discuss the critical exponent in semi-linear structurally damped wave and beam equations with additional dispersion term. The special model we have in mind is $$…
We consider the Cauchy problem in $\mathbb{R}^n,$ $n\geq 1,$ for a semilinear damped wave equation with nonlinear memory. Global existence and asymptotic behavior as $t\rightarrow\infty$ of small data solutions have been established in the…
In this paper we study the existence of global-in-time energy solutions to the Cauchy problem for the Euler-Poisson-Darboux equation, with a power nonlinearity: $$u_{tt}-u_{xx} + \frac\mu{t}\,u_t = |u|^p \,, \quad t>t_0, \…
In the present paper, we study the Cauchy problem for the wave equation with a time-dependent scale invariant damping $\frac{2}{1+t}\partial_t v$ and a cubic convolution $(|x|^{-\gamma}*v^2)v$ with $\gamma\in \left(-\frac{1}{2},3\right)$ in…
We consider the Cauchy problem for semilinear wave equations with variable coefficients and time-dependent scattering damping in $\mathbf{R}^n$, where $n\geq 2$. It is expected that the critical exponent will be Strauss' number $p_0(n)$,…
In this paper we consider the critical exponent problem for the semilinear damped wave equation with time-dependent coefficients. We treat the scale invariant cases. In this case the asymptotic behavior of the solution is very delicate and…
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 investigate in this paper the Cauchy problem of the one-dimensional wave equation with space-dependent damping of the form $\mu_0(1+x^2)^{-1/2}$, where $\mu_0>0$, and time derivative nonlinearity. We establish global existence of mild…
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 consider the Cauchy problem of the semilinear wave equation with a damping term \begin{align*} u_{tt} - \Delta u + c(t,x) u_t = |u|^p, \quad (t,x)\in (0,\infty)\times \mathbb{R}^N,\quad u(0,x) = \varepsilon u_0(x), \ u_t(0,x) =…
The aim of this paper is to determine the critical exponent for the nonlinear wave equations with damping and potential terms of the scale invariant order, by assuming that these terms satisfy a special relation. We underline that our…