Two-sided estimates of the blow-up time for a semilinear wave equation with fractional structural damping

Abstract

We consider the initial--boundary value problem for the semilinear wave equation with fractional structural damping $$ u_{tt}+(-Δ)^θu_{t}-Δu=|u|^{p-1}u $$ in a bounded domain, where the exponent $θ\in[0,1]$ interpolates between external frictional damping ($θ=0$) and internal Kelvin--Voigt viscoelastic damping ($θ=1$), and therefore parametrises the frequency dependence of the dissipation mechanism. For initial data with energy below the depth of the potential well and negative Nehari functional we prove finite-time blow-up together with an explicit upper bound for the blow-up time. The bound comes from a single concavity functional in which the fractional dissipation cancels identically; it therefore has the same form for every $θ\in[0,1]$, and it admits a variant that is uniform in $θ$. Conversely, for $1<p\le\frac{n+2θ}{n-2}$ when $n\ge3$, we establish an explicit lower bound for the blow-up time. Mechanically, the two bounds delimit a guaranteed interval of existence and a guaranteed failure time for the model. The admissible range of exponents in the lower bound widens linearly with $θ$, which quantifies how the strength of the internal damping enlarges the class of nonlinear loads for which such a guarantee can be computed; we do not claim monotonicity in $θ$ of the numerical value of the bound. Both theorems are proved for a class of energy solutions specified by a short list of requirements, so that they are independent of any particular local existence theorem, and they carry over unchanged to the elasticity and plate operators used in structural models. The two endpoint cases recover known results for frictional and strong damping.