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 are interested in entire solutions for the semilinear biharmonic equation ∆u = f(u) in R , where f(u) = e or −u−p (p > 0). For the exponential case, we prove that for the polyharmonic problem ∆u = e with positive integer m, any classical entire solution verifies ∆2m−1u < 0, this completes the results in [6, 14]; we obtain also a refined asymptotic expansion of radial separatrix solution to ∆...

The blow-up rate estimate for the solution to a semilinear parabolic equation ut = ∆u+V (x)|u|p−1u in Ω×(0, T ) with 0-Dirichlet boundary condition is obtained. As an application, it is shown that the asymptotic behavior of blow-up time and blow-up set of the problem with nonnegative initial data u(x, 0) = Mφ(x) as M goes to infinity, which have been found in [5], are improved under some reason...

In this paper we study the existence of nontrivial solutions for the following system of coupled semilinear Poisson equations:    −∆u = v , in Ω, −∆v = f(u) , in Ω, u = 0 and v = 0 , on ∂Ω, where Ω is a bounded domain in R . We assume that 0 < p < 2 N−2 , and the function f is superlinear and with no growth restriction (for example f(s) = s e); then the system has a nontrivial (strong) sol...

In this paper, we study a semilinear p -Laplacian problem −Δpu+h(x)|∇u| = b(x)g(u), u > 0, x ∈ R , lim |x|→∞ = 0, where q ∈ (p− 1, p], b, h ∈Cα loc(R) for some α ∈ (0,1), h(x) 0, b(x) > 0,∀x ∈ RN , and g ∈ C1((0,∞),(0,∞)) which may be singular at 0 . Using a sub-supersolution argument and a perturbed argument, we obtain the existence of entire solutions to the problem. No monotonicity condition...

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...