Blow-up at arbitrary energy levels for a strongly damped viscoelastic wave equation with a variable-exponent logarithmic source

Abstract

We study a strongly damped viscoelastic wave equation with a logarithmic source whose measurable exponent satisfies $2<p_{-}\le p(x)\le p_{+}<2^{\ast }$. An optimal pointwise correction to the source--potential inequality defines a \emph{shifted energy}. Under suitable kernel conditions, we prove finite-time blow-up for negative shifted energy and for a class of nonnegative shifted energies with no fixed upper energy threshold. Explicit upper lifespan bounds follow from an estimate for reaching negative shifted energy and a concavity argument. We also construct smooth blow-up data at every prescribed real energy level and show that our lifespan estimate covers additional data in the constant-exponent case. Explicit examples are given to illustrate the blow-up results.