This paper establishes oscillation theorems for a class of functional parabolic equations which arises from logistic population models with delays and diffusion.

The oscillation of the solutions of linear parabolic differential equations with deviating arguments are studied and sufficient conditions that all solutions of boundary value problems are oscillatory in a cylindrical domain are given.

DiPerna's and Majda's generalization of Young measures is used to describe oscillations and concentrations in sequences of maps ${\left\{u_k\right\}}_{k\in \mathbb{N}}\subset L^p(\Omega ;{ℝ}^m)$ satisfying a linear differential constraint $𝒜u_k=0$. Applications to sequential weak lower semicontinuity of integral functionals on $𝒜$-free sequences and to weak continuity of determinants are given. In particular, we state necessary and sufficient conditions for weak* convergence of det$\nabla {\varphi }_k\stackrel{*}{\rightharpoonup }\det \nabla \varphi$ in measures on the closure of $\Omega \subset {ℝ}^n$ if ${\varphi }_k\rightharpoonup \varphi$ in $W^{1,n}(\Omega ;{ℝ}^n)$. This convergence holds, for example, under...

We use DiPerna’s and Majda’s generalization of Young measures to describe oscillations and concentrations in sequences of gradients, $\left\{\nabla u_k\right\}$, bounded in $L^p(Ø;{ℝ}^{m\times n})$ if $p\>1$ and $\Omega \subset {ℝ}^n$ is a bounded domain with the extension property in $W^{1,p}$. Our main result is a characterization of those DiPerna-Majda measures which are generated by gradients of Sobolev maps satisfying the same fixed Dirichlet boundary condition. Cases where no boundary conditions nor regularity of $\Omega$ are required and links with lower semicontinuity results...

We use DiPerna's and Majda's generalization of Young measures to describe oscillations and concentrations in sequences of gradients, $\left\{\nabla u_k\right\}$, bounded in $L^p(\Omega ;{ℝ}^{m\times n})$ if p > 1 and $\Omega \subset {ℝ}^n$ is a bounded domain with the extension property in $W^{1,p}$. Our main result is a characterization of those DiPerna-Majda measures which are generated by gradients of Sobolev maps satisfying the same fixed Dirichlet boundary condition. Cases where no boundary conditions nor regularity of Ω are required and links with lower semicontinuity...

It is proved that any weak solution to a nonlinear beam equation is eventually globally oscillatory, i.e., there is a uniform oscillatory interval for large times.

In this paper we have collected some partial results on the sign of u(t,x) where u is a (sufficiently regular) solution of⎧     utt + (-1)m Δmu = 0     (t,x) ∈ R x Ω⎨⎩     u|Γ = ... = Δm-1 u|Γ = 0     t ∈ R.These results rely on the study of a sign of almost periodic functions of a special form restricted to a bounded interval J.

In this paper, sufficient conditions have been obtained for oscillation of solutions of a class of $n$th order linear neutral delay-differential equations. Some of these results have been used to study oscillatory behaviour of solutions of a class of boundary value problems for neutral hyperbolic partial differential equations.

In this paper nonlinear hyperbolic equations of neutral type of a given form are considered, with certain boundary conditions. Under certain constraints on the coefficients of the equation and the boundary conditions, sufficient conditions for oscillation of the solutions of the problems considered are obtained.

In the paper we present an identity of the Picone type for a class of nonlinear differential operators of the second order involving an arbitrary norm $H$ in ${ℝ}^n$ which is continuously differentiable for $x\ne 0$ and such that $H^p$ is strictly convex for some $p>1$. Two important special cases are the $p$-Laplacian and the so-called pseudo $p$-Laplacian. The identity is then used to establish a variety of comparison results concerning nonlinear degenerate elliptic equations which involve such operators. We also get criteria...