In this paper we show the small data solvability of suitable semilinear wave and Klein-Gordon equations on geometric classes of spaces, which include so-called asymptotically de Sitter and Kerr-de Sitter spaces, as well as asymptotically Minkowski spaces. These spaces allow general infinities, called conformal infinity in the asymptotically de Sitter setting; the Minkowski type setting is that ...

A semilinear partial differential equation of hyperbolic type with a convolution term describing simple viscoelastic materials with fading memory is considered. Ž . Regarding the past history memory of the displacement as a new variable, the equation is transformed into a dynamical system in a suitable Hilbert space. The dissipation is extremely weak, and it is all contained in the memory term....

We consider the following superlinear elliptic equation on Sn  ε∆Snu− u+ f(u) = 0 in D; u > 0 in D and u = 0 on ∂D, where D is a geodesic ball on Sn with geodesic radius θ1, and ∆Sn is the Laplace-Beltrami operator on Sn. We prove that for any θ ∈ ( 2 , π) and for any positive integer N ≥ 1, there exist at least 2N + 1 solutions to the above problem for ε sufficiently small. Moreover, the asym...

Under suitable conditions we are able to solve the semilinear wave equation in any dimension. We are also able to compute the essential spectrum of the linear wave operator for the rotationally invariant periodic case.

We consider the semilinear elliptic equation 4u = p(x)uα + q(x)uβ on a domain Ω ⊆ Rn, n ≥ 3, where p and q are nonnegative continuous functions with the property that each of their zeroes is contained in a bounded domain Ωp or Ωq, respectively in Ω such that p is positive on the boundary of Ωp and q is positive on the boundary of Ωq. For Ω bounded, we show that there exists a nonnegative soluti...