Fite and Kamenev type oscillation criteria for the second order nonlinear damped elliptic differential equation ${\sum }_{i,j=1}^N{D}_i\left[a_{ij}\left(x\right)D_{j}y\right]+{\sum }_{i=1}^N{b}_i\left(x\right)D_{i}y+p\left(x\right)f\left(y\right)=0$ are obtained. Our results are extensions of those for ordinary differential equations and improve some known oscillation criteria in the literature. Several examples are given to show the significance of the results.

Certain hyperbolic equations with continuous distributed deviating arguments are studied, and sufficient conditions are obtained for every solution of some boundary value problems to be oscillatory in a cylindrical domain. Our approach is to reduce the multi-dimensional oscillation problems to one-dimensional oscillation problems for functional differential inequalities by using some integral means of solutions.

We consider a quasilinear elliptic problem whose left-hand side is a Leray-Lions operator of $p$-Laplacian type. If $p<\gamma <N$ and the right-hand side is a Radon measure with singularity of order $\gamma$ at $x_0\in \Omega$, then any supersolution in $W_{\mathrm{l}oc}^{1,p}\left(\Omega \right)$ has singularity of order at least $\frac{(\gamma -p)}{(p-1)}$ at $x_0$. In the proof we exploit a pointwise estimate of $𝒜$-superharmonic solutions, due to Kilpeläinen and Malý, which involves Wolff’s potential of Radon’s measure.

The problems of Gevrey hypoellipticity for a class of degenerated quasi-elliptic operators are studied by several authors (see [1]–[5]). In this paper we obtain the Gevrey hypoellipticity for a degenerated quasi-elliptic operator in ${ℝ}^2$, without any restriction on the characteristic polyhedron.

In this paper, we discuss the special diffusive hematopoiesis model $$\frac{\partial P(t,x)}{\partial t}=\Delta P(t,x)-\gamma P(t,x)+\frac{\beta P(t-\tau ,x)}{1+P^n(t-\tau ,x)}$$ with Neumann boundary condition. Sufficient conditions are provided for the global attractivity and oscillation of the equilibrium for Eq. (*), by using a new theorem we stated and proved. When P(t, χ) does not depend on a spatial variable χ ∈ Ω, these results are also true and extend or complement existing results. Finally, existence and stability of the Hopf bifurcation for Eq. (*) are studied.

We examine the parabolic system of three equations $u_t$ - Δu = $v^p$, $v_t$ - Δv = $w^q$, $w_t$ - Δw = $u^r$, x ∈ ${ℝ}^N$, t > 0 with p, q, r positive numbers, N ≥ 1, and nonnegative, bounded continuous initial values. We obtain global existence and blow up unconditionally (that is, for any initial data). We prove that if pqr ≤ 1 then any solution is global; when pqr > 1 and max(α,β,γ) ≥ N/2 (α, β, γ are defined in terms of p, q, r) then every nontrivial solution exhibits a finite blow up time.