Sharp lifespan estimates for semilinear fractional evolution equations with critical nonlinearity

In this paper we consider semilinear wave equation and semilinear second order σ-evolution equations with different (effective or non-effective) damping mechanisms driven by fractional Laplace operators; in particular, the nonlinear term is the product of a power nonlinearity $|u{|}^p$ with the critical exponent $p=p_c\left(n\right)$ and a modulus of continuity $\mu \left(\right|u\left|\right)$. We derive a critical condition on the nonlinearity by proving a global in time existence result under the Dini condition on μ and a blow-up result when μ does not satisfy the Dini condition. Especially, in this latter case we determine new sharp estimates for the lifespan of local solutions, obtaining coincident upper and lower bounds of the lifespan. In particular, we derive a new sharp estimate for the wave equation with structural damping and classical power nonlinearity $|u{|}^p$ in the critical case $p=p_c\left(n\right)$, not yet determined in previous literature. The proof of the blow-up results and the upper bound estimates of the lifespan require the introduction of new test functions which allows to overcome some new difficulties due to the presence of both non-local differential operators and general nonlinearities.