24 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
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
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