Related papers: The Fourier transform and the wave equation
We consider damped wave equations with a potential and rotational inertia terms. We study the Cauchy problem for this model in the one dimensional Euclidean space and we obtain fast energy decay and L^2-decay of the solution itself as time…
A set of semi-analytical techniques based on Fourier analysis is used to solve wave scattering problems in variously shaped waveguides with varying normal admittance boundary conditions. Key components are newly developed conformal mapping…
In this paper, the one-dimensional time-fractional diffusion-wave equation with the fractional derivative of order $1 \le \alpha \le 2$ is revisited. This equation interpolates between the diffusion and the wave equations that behave quite…
We consider a uniqueness problem concerning the Fourier coefficients of normalized Cauchy transforms. These problems inherently involve proving a simultaneous approximation phenomenon and establishing the existence of cyclic inner functions…
We present an explicit numerical scheme to solve the variable coefficient wave equation in one space dimension with minimal restrictions on the coefficient and initial data.
We prove the local existence for the Water Waves equations with large bathymetric variations on a time interval of size 1/\epsilon, where $\epsilon$ measures the amplitude of the wave. We just need the presence of surface tension.
In physics, phenomena of diffusion and wave propagation have great relevance; these physical processes are governed in the simplest cases by partial differential equations of order 1 and 2 in time, respectively. By replacing the time…
Space-time fractional Zener wave equation, describing viscoelastic materials obeying the time-fractional Zener model and the space-fractional strain measure, is derived and analyzed. This model includes waves with finite speed, as well as…
We consider the Cauchy problem for wave equations with unbounded damping coefficients in the whole space. For a general class of unbounded damping coefficients, we derive uniform total energy decay estimates together with a unique existence…
This paper is concerned with the Cauchy problem for the semilinear wave equation: $u_{tt}-\Delta u=F(u) \ \mbox{in} \ R^n\times[0, \infty)$, where the space dimension $n \ge 2$, $F(u)=|u|^p$ or $F(u)=|u|^{p-1}u$ with $p>1$. Here, the Cauchy…
In this paper, we study the Cauchy problem for a wave equation with general strong damping $-\mu(|D|)\Delta u_t$ motivated by [Tao, Anal. PDE (2009)] and [Ebert-Girardi-Reissig, Math. Ann. (2020)]. By employing energy methods in the Fourier…
We analyse the Gaussian wave packet transform. Based on the Fourier inversion formula and a partition of unity, which is formed by a collection of Gaussian basis functions, a new representation of square-integrable functions is presented.…
We consider the global Cauchy problem for a two-component system of cubic semilinear wave equations in two space dimensions. We give a criterion for large time non-decay of the energy for small amplitude solutions in terms of the radiation…
Six-wave interactions are used for modeling various physical systems, including in optical wave turbulence [16] (where a cascade of photons displays this kind of behavior) and in quantum wave turbulence [31] (for the interaction of Kelvin…
We present a general method of solving the Cauchy problem for multidimensional parabolic (diffusion type) equation with variable coefficients which depend on spatial variable but do not change over time. We assume the existence of the…
We determine the wave front sets of solutions to two special cases of the Cauchy problem for the space-time fractional Zener wave equation, one being fractional in space, the other being fractional in time. For the case of the space…
We focus on the general theory to the Cauchy problem for one dimensional nonlinear wave equations with small initial data. In the general theory, we aim to obtain the lower bound estimate of the lifespan of classical solution. In this…
In this paper, we show for the first time the increasing stability of the inverse source problem for the n-dimensional Helmholtz equation at multiple wave numbers, which is different from the two-or three-dimensional Helmholtz equation. In…
A new approach to the geometrization of the electron theory is proposed. The particle wave function is represented by a geometric entity, i.e., Clifford number, with the translation rules possessing the structure of Dirac equation for any…
We study the Cauchy problem for the semi-linear damped wave equation in any space dimension. We assume that the time-dependent damping term is effective. We prove the global existence of small energy data solutions in the supercritical…