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

Abstract

"In this paper, we consider a class of quasi-linear parabolic equations with variable exponents, $$a\left( x,t\right) u_{t}-\Delta _{m\left( .\right) }u=f_{p\left( .\right)}\left( u\right)$$ in which $f_{p\left( .\right)}\left( u\right)$ 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 $T^{\ast }$ of the solution. A blow-up of the solution with negative initial energy is also established."