Global nonexistence and blow-up results for a quasi-linear evolution equation with variable-exponent nonlinearities

In this paper, we consider a class of quasi-linear parabolic equations with variable exponents, a(x, t)u(t) - Delta(m(.))u = f(p(.)) (u) in which f(p(.)) (u) the source term, a(x, t) > 0 is a nonnegative function, and the exponents of nonlinearity m(x), p(x) are given measurable functions. Under suitable conditions on the given data, a finite-time blow-up result of the solution is shown if the initial datum possesses suitable positive energy, and in this case, we precise estimate for the lifespan T* of the solution. A blow-up of the solution with negative initial energy is also established.