2025/04/29 by Michael Winkler, Winkler, Michael · 1 citation
Mathematics · Computer Science · Engineering · #Nonlinear Partial Differential Equations #Contact Mechanics and Variational Inequalities #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2504.20473
The model \ utt = (γ(Θ) uxt)x + auxx - (f(Θ))x,
Θt = Θxx + γ(Θ) uxt2 - f(Θ) uxt, . for thermoviscoelastic evolution in one-dimensional Kelvin-Voigt materials is considered. By means of an approach based on maximal Sobolev regularity theory of scalar parabolic equations, it is shown that if γ0>0 is fixed, then there exists δ=δ(γ0)>0 with the property that for suitably regular initial data of arbitrary size an associated initial-boundary value problem posed in an open bounded interval admits a global classical solution whenever γ∈ C2([0,∞)) and f∈ C2([0,∞)) are such that f(0)=0 and |f(ξ)| ≤ Kf ⋅ (ξ+1)α for all ξ≥ 0 and some Kf>0 and α<(3)/(2), and that γ0 ≤ γ(ξ) ≤ γ0 + δ for all ξ≥ 0. This is supplemented by a statement on global existence of certain strong solutions, particularly continuous in both components, under weaker conditions on the initial data.