We consider in this paper some class of perturbation for the semilinear wave equation with critical (in the conformal transform sense) power nonlinearity. Working in the framework of similarity variables, we find a Lyapunov functional for the problem. Using a two-step argument based on interpolation and a critical Gagliardo-Nirenberg inequality, we show that the blow-up rate of any sigular solution is given by the solution of the non perturbed associated ODE, namely u ′′ = u p .
In this article, we study the blow-up of the damped wave equation in the scale-invariant case and in the presence of two nonlinearities. More precisely, we consider the following equation: u tt − Δu + 1 + t u t = |u t | p + |u| q , in R N × [0, ∞), with small initial data. For < N(q−1) 2 and ∈ (0, *), where * > 0 is depending on the nonlinearties' powers and the space dimension (* satisfies (q − 1) ((N + 2 * − 1)p − 2) = 4), we prove that the wave equation, in this case, behaves like the one without dissipation (= 0). Our result completes the previous studies in the case where the dissipation is given by (1+t) u t ; > 1, where, contrary to what we obtain in the present work, the effect of the damping is not significant in the dynamics. Interestingly, in our case, the influence of the damping term 1+t u t is important. KEYWORDS blow-up, nonlinear wave equations, scale-invariant damping MSC CLASSIFICATION 35L71; 35B44
The survival outcome of patients with malignant gliomas is still poor, despite advances in surgical techniques, radiation therapy and the development of novel chemotherapeutic agents. The heterogeneity of molecular alterations in signaling pathways involved in the pathogenesis of these tumors contributes significantly to their resistance to treatment. Several molecular targets for therapy have been discovered over the last several years. Therapeutic agents targeting these signaling pathways may provide more effective treatments and may improve survival. This review summarizes the important molecular therapeutic targets and the outcome of published clinical trials involving targeted therapeutic agents in glioma patients.
Bevacizumab discontinuation unrelated to disease progression does not appear to cause rebound recurrence or worsen PFS in patients who benefit from bevacizumab. Additionally, Bev-D patients had an improved response to salvage therapy, findings which provide a strong basis for a prospective study.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.