This paper is concerned with the existence, nonexistence, and uniqueness of convex monotone positive solutions of elastic beam equations with a parameter λ. The boundary conditions mean that the beam is fixed at one end and attached to a bearing device or freed at the other end. By using fixed point theorem of cone expansion, we show that there exists λ * ≥ λ * > 0 such that the beam equation has at least two, one, and no positive solutions for 0 < λ ≤ λ * , λ * < λ ≤ λ * and λ > λ * , respectively; furthermore, by using cone theory we establish some uniqueness criteria for positive solutions for the beam and show that such solution x λ depends continuously on the parameter λ. In particular, we give an estimate for critical value of parameter λ.

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In this paper, a strongly damped semilinear wave equation with a general nonlinearity is considered. With the help of a newly constructed auxiliary functional and the concavity argument, a general finite time blow-up criterion is established for this problem. Furthermore, the lifespan of the weak solution is estimated from both above and below. This partially extends some results obtained in recent literatures and sheds some light on the similar effect of power type nonlinearity and logarithmic nonlinearity on finite time blow-up of solutions to such problems.

In this paper, the authors consider the reaction-diffusion equation with nonlinear absorption and nonlinear nonlocal Neumann boundary condition. They prove that the solution either exists globally or blows up in finite time depending on the initial data, the weighting function on the border, and nonlinear indexes in the equation by using the comparison principle.