A Blow-Up Result for a Class Doubly Nonlinear Parabolic Equations with Variable-Exponent Nonlinearities

Abstract

This paper deals with the following doubly nonlinear parabolic equations ($u$ + |$u$|^$r(x)$−2$u$)_$t$ − div(|∇$u$|^$m(x)$−2∇$u$) = |$u$|^$p(x)$−2$u$, where the exponents of nonlinearity $r(x)$, $m(x)$ and $p(x)$ are given functions. Under some appropriate assumptions on the exponents of nonlinearity, and with certain initial data, a blow-up result is established with positive initial energy.