The nonlocal differential equation\begin{equation}-A \big ( \big (b*u^{p(\cdot)} \big )(1) \big )u''(t)=\lambda f\big(t,u(t)\big)\text{, }t\in(0,1)\notag\end{equation}is considered, where $$\displaystyle(a*b)(t):=\int_0^ta(t-s)b(s)\ ds , \ \ \ t\ge0 ,$$ is a finite convolution. The existence of at least one positive solution is considered when the equation is subject to boundary conditions. A model case of the above problem occurs when $b(t)\equiv1$, as in this case the problem reduces to\begin{equation}-A \Big (\int_0^1\big(u(s)\big)^{p(s)}\ ds \Big )u''(t)=\lambda f\big(t,u(t)\big)\text{, }t\in(0,1).\notag\end{equation}As a consequence nonlocal coefficients possessing $p(x)$-growth are considered. By using a specialized order cone together with topological fixed point theory we are able to provide existence theorems with a minimum of assumptions on the coefficient function $A$. In particular, $A$ can vanish or change sign on intervals of infinite measure.

We study the local well-posedness in the framework of the Sobolev space $H^1(\mathbb R^N)$, $N \geq 3$, for a semilinear parabolic equation with asymptotically polynomial nonlinearity up to the critical Sobolev growth. Then we establish the dichotomy between blow-up and global existence for solutions with small energy by means of variational methods and the so-called potential well argument.

In this paper, we study the local well-posedness of the initial-boundary value problem for the 1D quasilinear wave equation $$\partial^2 _t u = \partial_x ( u^{2a} \partial_x u) + F(u)u_x $$ with the zero Dirichlet data on boundaries and the Levi condition on $F$. The boundary degeneracy causes losses in the principal part of the equation, which results in previous results only obtaining the estimates with loss of regularity. To overcome the main difficulty, we introduce singular weight functions to derive the estimates of preserving the initial regularity for the solution and its derivatives. The local existence and uniqueness for the degenerate initial-boundary value problem are established by applyingthe method of characteristic, the contraction mapping principle, and $L^\infty$ estimate with singular weight functions.

We prove estimates for the maximal and minimal predator and prey populations on the unique limit cycle in a standard predator-prey system.Our estimates are valid when the cycle exhibits small predator and prey abundances and large amplitudes.The proofs consist of constructions of several Lyapunov-type functions and derivation of a large number of non-trivial estimates,and should be of independent interest.This study generalizes results proved by the authors in [11] to a wider class of systems and, in addition, it gives simpler proofs of some already known estimates.