Related papers: Non-homogeneous wave equation on a cone

This paper concerns the existence of a nontrivial solution for the following problem \begin{equation} \left\{\begin{aligned} -\Delta u + V(x)u & \in \partial_u F(x,u)\;\;\mbox{a.e. in}\;\;\mathbb{R}^{N},\nonumber u \in…

In this paper, we first consider the Rosenau equation with the quadratic nonlinearity and identify its Lie symmetry algebra. We obtain reductions of the equation to ODEs, and find periodic analytical solutions in terms of elliptic…

Let $G$ be a compact Lie group. In this article, we consider the initial value fractional wave equation with power-type nonlinearity on $G$. Mainly, we investigate some $L^{2}-L^{2}$ estimates of the solutions to the homogeneous fractional…

In this paper we study the semilinear elliptic problem $$ -\Delta u -k^2u=Q|u|^{p-2}u\quad\text{ in }\mathbb{R}^2, $$ where $k>0$, $p\geq 6$ and $Q$ is a bounded function. We prove the existence of real-valued $W^{2,p}$-solutions, both for…

Exact solutions, in terms of special functions, of all wave equations $% u_{xx} - u_{tt} = V(x) u(t,x)$, characterised by eight inequivalent time independent potentials and by variable separation, have been found. The real valueness of the…

Unlike the heat equation or the Laplace equation, solutions of the wave equation on general domains have no known stochastic representation. This short note gives a simple solution to this well known problem in arbitrary dimensions. The…

Let K be a non-Archimedean local field with the normalized absolute value $|\cdot |$. It is shown that a ``plane wave'' $f(t+\omega_1 x_1+... +\omega_nx_n)$, where f is a Bruhat-Schwartz complex-valued test function on K, $(t,x_1,...,…

This paper presents a solution to an initial value problem for the 1-dimensional wave equation on time scales through the application of a Fourier transform and its inverse via contour integrals. The time scale of the spatial dimension is…

For a wave equation with pure delay, we study an inhomogeneous initial-boundary value problem in a bounded 1D domain. Under smoothness assumptions, we prove unique existence of classical solutions for any given finite time horizon and give…

: We establish existence of an infinite family of exponentially-decaying non-radial $C^2$ solutions to the equation $\Delta u + f(u) = 0$ on $R^2$ for a large class of nonlinearities $f$. These solutions have the form $u(r,\theta )=e^{i…

In this paper, we consider the semilinear wave equation involving the nonlinear damping term $g(u_t) $ and nonlinearity $f(u)$. The well-posedness of the weak solution satisfying some additional regularity is achieved under the wider ranges…

In this paper, we study the long-time existence result for small data solutions of quasilinear wave equations exterior to star-shaped regions in two space dimensions. The key novelty is that we establish a Morawetz type energy estimate for…

This paper addresses the inverse problem of simultaneously recovering multiple unknown parameters for semilinear wave equations from boundary measurements. We consider an initial-boundary value problem for a wave equation with a general…

A general class of probability density functions \[u(x,t)=Ct^{-\alpha d}\left (1-\left (\frac{\|x\|}{ct^{\alpha}}\right )^{\beta}\right )_+^{\gamma},\quad x\in \mathbb{R}^d,t>0,\] is considered, containing as particular case the Barenblatt…

In this paper we consider a class of nonlinear wave equation with $x$-dependent coefficients and prove existence of families of time-periodic solutions under the general boundary conditions. Such a model arises from the forced vibrations of…

In this work we consider an energy subcritical semi-linear wave equation ($3 < p < 5$) \[ \partial_t^2 u - \Delta u = \phi(x) |u|^{p-1} u, \qquad (x,t) \in {\mathbb R}^3 \times {\mathbb R} \] with initial data $(u,u_t)|_{t=0} = (u_0,u_1)\in…

We consider the equation $(\partial_t + \rho(\sqrt{-\Delta}))f(t,x) = \mathbf 1_\omega u(t,x)$, $x\in \mathbb R$ or $\mathbb T$. We prove it is not null-controllable if $\rho$ is analytic on a conic neighborhood of $\mathbb R_+$ and…

We consider a nonlinear wave equation with nonconstant coefficients. In particular, the coefficient in front of the second order space derivative is degenerate. We give the blow-up behavior and the regularity of the blow-up set. Partial…

In this article we are interested on the non-homogeneous fractional Schr\"odinger equation \begin{eqnarray}\label{eq00} &(-\Delta)^{\alpha}u(x) + V(x)u(x) = f(u) + h(x) \mbox{ in } \mathbb{R}^{n}. \end{eqnarray} By using mountain pass…

Using nonstandard methods, we show that the time dependent Fourier series of any smooth function F, solving the wave equation, on a finite closed interval, with vanishing boundary conditions, converges uniformly to F.