In the present paper, the concepts of module (uniform) approximate amenability and contractibility of Banach algebras that are modules over another Banach algebra, are introduced. The general theory is developed and some hereditary properties are given. In analogy with the Banach algebraic approximate amenability, it is shown that module approximate amenability and contractibility are the same ...

We study a blowup problem for the semilinear heat equation ut = ∆u + u posed in R × (0,∞), where p is supercritical in the Sobolev sense. A solution u(x, t) is said to blow up at a time 0 < T < +∞ if there exists a sequence tn ↗ T as n → ∞ such that |u(·, tn)|∞ → +∞ as n → ∞ with supremum norm | · |∞ in RN . We establish the following result: for given times T1, T2 with 0 < T1 < T2 and any smal...