Abstract. We consider an initial-boundary value problem for a nonlinear wave equation in one space dimension. The nonlinearity features the damping term |u| m−1 ut and a source term of the form |u| p−1 u, with m, p > 1. We show that whenever m ≥ p, then local weak solutions are global. On the other hand, we prove that whenever p > m and the initial energy is negative, then local weak solutions cannot be global, regardless of the size of the initial data.

In the process of bone remodeling, osteoclasts are responsible for resorption of bone. High levels of intracellular calcium decrease the bone resorbing activity of osteoclasts and increase detachment of osteoclasts from the bone surface. The regulatory role of intracellular calcium in bone resorption is not clearly understood. To understand this phenomenon, we studied the effects of the intracellular calcium modulators ryanodine and ruthenium red on bone resorption using the disaggregated osteoclast pit assay. Changes in intracellular calcium concentrations after treatment with these compounds were detected with the fluoroprobe fura2. With ryanodine, a significant, dose-dependent decrease in bone resorption was detected. This inhibition of bone resorption was reversible upon the removal of ryanodine. Ryanodine increased intracellular calcium concentrations, suggesting that the mechanism of inhibition by ryanodine was via alterations in intracellular stores of calcium. After treatment with ruthenium red, osteoclasts resorbed significantly more bone compared to vehicle-treated cells. This increase in bone resorption correlated with a decrease in intracellular calcium concentrations. The addition of parathyroid hormone or ruthenium red to osteoclast cultures containing ryanodine did not attenuate the decrease in bone resorption caused by ryanodine, suggesting that the mechanism of ryanodine inhibition of bone resorption may involve the "locking" of a calcium channel in an open position.

We study a nonlinear wave equation on the two-dimensional sphere with a blowing-up nonlinearity. The existence and uniqueness of a local regular solution are established. Also, the behavior of the solutions is examined. We show that a large class of solutions to the initial value problem quench in finite time.