We study the existence and nonexistence of positive (super-) solutions to a singular semilinear elliptic equation $$-\nabla\cdot(|x|^A\nabla u)-B|x|^{A-2}u=C|x|^{A-\sigma}u^p$$ in cone--like domains of $\mathbb R^N$ ($N\ge 2$), for the full range of parameters $A,B,\sigma,p\in\mathbb R$ and $C>0$. We provide a characterization of the set of $(p,\sigma)\in\mathbb R^2$ such that the equation has no positive (super-),solutions, depending on the values of $A,B$ and the principal Dirichlet eigenvalue of the cross--section of the cone. The proofs are based on the explicit construction of appropriate barriers and involve the analysis of asymptotic behavior of super-harmonic functions associated to the Laplace operator with critical potentials, Phragmén-Lindelöf type comparison arguments and an improved version of Hardy's inequality in cone--like domains.

The aim of this paper is to investigate the existence of periodic solutions of a nonlinear planar autonomous system having a limit cycle $x_0$ of least period $T_0>0$ when it is perturbed by a small parameter, $T_1-$periodic perturbation. In the case when $T_0/T_1$ is a rational number $l/k$, with $l, k$ prime numbers, we provide conditions to guarantee, for the parameter perturbation $\varepsilon>0$ sufficiently small, the existence of $klT_0-$ periodic solutions $x_\varepsilon$ of the perturbed system which converge to the trajectory $\tilde x_0$ of the limit cycle as $\varepsilon\to 0$. Moreover, we state conditions under which $T=klT_0$ is the least period of the periodic solutions $x_\varepsilon$. We also suggest a simple criterion which ensures that these conditions are verified. Finally, in the case when $T_0/T_1$ is an irrational number we show the nonexistence, whenever $T>0$ and $\varepsilon>0$, of $T-$periodic solutions $x_\varepsilon$ of the perturbed system converging to $\tilde x_0$. The employed methods are based on the topological degree.

We consider the initial-value problem of the abstract wave equation with dissipation whose coefficient tends to $0$ as $t \to \infty$. In the case that the coefficient of the dissipation is a positive constant, Ikehata-Nishihara and Chill-Haraux obtained the decay estimate of the difference between the solution of this equation and the solution of the corresponding abstract heat equation. The purpose of this paper is to show the decay estimate of the difference between the solution of the above equation and the solution of the corresponding abstract parabolic equation. We also show the decay estimate for the solution of this dissipative wave equation.

In this paper we give new derivations of the heat and wave equation which incorporate the boundary conditions into the formulation of the problems. The principle of least action and the inclusion of a kinetic energy contribution on the boundary are used to derive the wave equation together with kinetic boundary conditions. The methods described for both equations admit all of the standard boundary conditions as well as general Wentzell and dynamic boundary conditions; in addition the boundary conditions arise naturally as part of the formulation of the problems. The physical interpretation for general Wentzell boundary conditions is given for both the heat and wave equations.