相关论文: On the defocusing semilinear wave equations in thr…
We study a semilinear wave inequality with double damping on a complete noncompact Riemannian manifold. The considered problem involves a potential function $V$ depending on the space variable in front of the power nonlinearity and an…
The commuting vector fields approach, devised for strichartz estimates in [13], was developed for proving the local well-posedness in the Sobolev spaces $H^s$ with $s>2+\frac{2-\sqrt{3}}{2}$ for general quasi-linear wave equation in…
A nonlinear generalisation of Schrodinger's equation had previously been obtained using information-theoretic arguments. The nonlinearities in that equation were of a nonpolynomial form, equivalent to the occurence of higher-derivative…
This paper is concerned with long-time dynamics of semilinear wave equations defined on bounded domains of $\mathbb{R}^3$ with cubic nonlinear terms and locally distributed damping. The existence of regular finite-dimensional global…
Explicit solutions are obtained for a class of semilinear radial Schrodinger equations with power nonlinearities in multi-dimensions. These solutions include new similarity solutions and other new group-invariant solutions, as well as new…
This paper studies the upper bound of the lifespan of classical solutions of the initial value problems for one dimensional wave equations with quasilinear terms of space-, or time-derivatives of the unknown function. The results are same…
The nonlinear wave and Schrodinger equations on Euclidean space of any dimension, with general power nonlinearity and with both the focusing and defocusing signs, are proved to be ill-posed in the Sobolev space of index s whenever the…
We consider the semilinear equation $$ \epsilon^{2s} (-\Delta)^s u + V(x)u - u^p = 0, \quad u>0, \quad u\in H^{2s}(\R^N) $$ where $0<s<1,\ 1<p<\frac{N+2s}{N-2s}$, $ V(x)$ is a sufficiently smooth potential with $\inf_\R V(x)> 0$, and…
We consider the quadratically semilinear wave equation on R^d, d>=3, equipped with a Riemannian metric. This metric is non-trapping and approaches the Euclidean metric polynomially at infinity. Using Mourre estimates and the Kato theory of…
We study the existence and concentration of positive and nodal solutions to a Schr\"odinger equation in the presence of a shrinking self-focusing core of arbitrary shape. Via a suitable rescaling, the concentration gives rise to a limiting…
In the present paper, we are concerned with the semilinear viscoelastic wave equation subject to a locally distributed dissipative effect of Kelvin-Voigt type, posed on a bounded domain with smooth boundary. We begin with an auxiliary…
In this paper, we study a semilinear weakly coupled system of wave equations with power nonlinearities. More precisely, we couple (through the nonlinear terms) a wave equation and a damped wave equation with a time-dependent coefficient for…
This study investigates a semilinear wave equation characterized by nonlinear damping $g(u_t) $ and nonlinearity $f(u)$. First, the well-posedness of weak solutions across broader exponent ranges for $g$ and $f$ is established, by utilizing…
We consider a semilinear elliptic equation with Dirichlet boundary conditions in a smooth, possibly unbounded, domain. Under suitable assumptions, we deduce a condition on the size of the domain that implies the existence of a positive…
In this note, we study the semilinear wave equation with power nonlinearity $|u|^p$ on compact Lie groups. First, we prove a local in time existence result in the energy space via Fourier analysis on compact Lie groups. Then, we prove a…
This paper studies for large frequency number $k>0$ the existence and multiplicity of solutions of the semilinear problem $$ -\Delta u -k^2 u=Q(x)|u|^{p-2}u\quad\text{ in }\mathbb{R}^N, \quad N\geq 2. $$ The exponent $p$ is subcritical and…
We present and analyse a new conforming space-time Galerkin discretisation of a semi-linear wave equation, based on a variational formulation derived from De Giorgi's elliptic regularisation viewpoint of the wave equation in second-order…
We consider the pointwise decay of solutions to wave-type equations in two model singular settings. Our main result is a form of Price's law for solutions of the massless Dirac-Coulomb system in (3+1)-dimensions. Using identical techniques,…
We consider the wave equation with focusing power nonlinearity. The associated ODE in time gives rise to a self-similar solution known as the ODE blowup. We prove the nonlinear asymptotic stability of this blowup mechanism outside of radial…
In this paper we study the long time behavior for a semilinear wave equation with space-dependent and nonlinear damping term. After rewriting the equation as a first order system, we define a class of approximate solutions that employ…