29 research outputs found
The maximum principle and sign changing solutions of the hyperbolic equation with the Higgs potential
In this article we discuss the maximum principle for the linear equation and
the sign changing solutions of the semilinear equation with the Higgs
potential. Numerical simulations indicate that the bubbles for the semilinear
Klein-Gordon equation in the de Sitter spacetime are created and apparently
exist for all times
Fundamental Solutions for the Klein-Gordon Equation in de Sitter Spacetime
In this article we construct the fundamental solutions for the Klein-Gordon
equation in de Sitter spacetime. We use these fundamental solutions to
represent solutions of the Cauchy problem and to prove estimates for
the solutions of the equation with and without a source term
Huygens' Principle for the Klein-Gordon equation in the de Sitter spacetime
In this article we prove that the Klein-Gordon equation in the de Sitter
spacetime obeys the Huygens' principle only if the physical mass of the
scalar field and the dimension of the spatial variable are tied by
the equation . Moreover, we define the incomplete Huygens'
principle, which is the Huygens' principle restricted to the vanishing second
initial datum, and then reveal that the massless scalar field in the de Sitter
spacetime obeys the incomplete Huygens' principle and does not obey the
Huygens' principle, for the dimensions , only. Thus, in the de Sitter
spacetime the existence of two different scalar fields (in fact, with m=0 and
), which obey incomplete Huygens' principle, is equivalent to
the condition (in fact, the spatial dimension of the physical world). For
these two values of the mass are the endpoints of the so-called in
quantum field theory the Higuchi bound. The value of the
physical mass allows us also to obtain complete asymptotic expansion of the
solution for the large time. Keywords: Huygens' Principle; Klein-Gordon
Equation; de Sitter spacetime; Higuchi Boun
The influence of oscillations on energy estimates for damped wave models with time-dependent propagation speed and dissipation
The aim of this paper is to derive higher order energy estimates for
solutions to the Cauchy problem for damped wave models with time-dependent
propagation speed and dissipation. The model of interest is \begin{equation*}
u_{tt}-\lambda^2(t)\omega^2(t)\Delta u +\rho(t)\omega(t)u_t=0, \quad
u(0,x)=u_0(x), \,\, u_t(0,x)=u_1(x). \end{equation*} The coefficients
and are shape functions and
is an oscillating function. If and
is an "effective" dissipation term, then energy
estimates are proved in [2]. In contrast, the main goal of the present paper is
to generalize the previous results to coefficients including an oscillating
function in the time-dependent coefficients. We will explain how the interplay
between the shape functions and oscillating behavior of the coefficient will
influence energy estimates.Comment: 37 pages, 2 figure
An interesting connection between hypoellipticity and branching phenomena for certain differential operators with degeneracy of infinite order
In the present paper the influence of lower order term is studied on the qualitative properties of some infinitely degenerate elliptic operators. Using different methods one can prove a n interesting connection between the non-hypollipticity for infinitely degenerate elliptic operators and branching of singularities for corresponding weaklyhyperbolic operators. The questionfor local and nonlocal solvability is considered, too. The results show, that the fulfilment of C°-type Levi conditions is not sufficient to characterize the qualitative properties of degenerate elliptic operators