One-dimensional solutions of non-local Allen-Cahn-type equations with rough kernels

We are interested in the study of local and global minimizers for an energy functional of the type where W is a smooth, even double-well potential and K is a non-negative symmetric kernel in a general class, which contains as a particular case the choice K(z)=|z|-N-2s, with s 08(0, 1), related to the fractional Laplacian. We show the existence and uniqueness (up to translations) of one-dimensional minimizers in the full space RN and obtain sharp estimates for some quantities associated to it. In particular, we deduce the existence of solutions of the non-local Allen-Cahn equation which possess one-dimensional symmetry.The results presented here were proved in [9,10,36] for the model case K(z)=|z|-N-2s. In our work, we consider instead general kernels which may be possibly non-homogeneous and truncated at infinity

Oxaliplatin plus fluoropyrimidines as adjuvant therapy for colon cancer in older patients: A subgroup analysis from the TOSCA trial

Background: Previous studies on oxaliplatin and fluoropyrimidines as adjuvant therapy in older patients with stage III colon cancer (CC) produced conflicting results. Patients and methods: We assessed the impact of age on time to tumour recurrence (TTR), disease-free survival (DFS), cancer-specific survival (CSS), and overall survival (OS) in 2360 patients with stage III CC (1667 aged <70 years and 693 ≥ 70 years) randomised to receive 3 or 6 months of FOLFOX or CAPOX within the frame of the phase III, TOSCA study. Results: Older patients compared with younger ones presented more frequently an Eastern Cooperative Oncology Group performance status equal to 1 (10.5% vs 3.3%, p < 0.001), a greater number of right-sided tumours (40.9% vs 26.6%, p < 0.001), and were at higher clinical risk (37.2% vs 33.2%, p = 0.062). The treatments were almost identical in the two cohorts (p = 0.965). We found a greater proportion of dose reductions (46.7% vs 41.4%, p = 0.018), treatment interruptions (26.1% vs 19.3%, p < 0.001) and a higher proportion of recurrences (24.2% vs 20.3%, p = 0.033) in the older patients. The multivariable analysis of the TTR did not indicate a statistically significant effect of age (hazard ratio [HR]: 1.19; 95% confidence interval [CI]: 0.98–1.44; p = 0.082). The HR comparing older with younger patients was 1.34 (95% CI: 1.12–1.59; p = 0.001) for DFS, 1.58 (95% CI: 1.26–1.99; p < 0.001) for OS, and 1.28 (95% CI: 0.96–1.70; p = 0.089) for CSS. Conclusions: Worse prognostic factors and reduced treatment compliance have a negative impact on the efficacy of oxaliplatin-based adjuvant therapy in older patients