Long-time behavior of quasilinear thermoelastic Kirchhoff–Love plates with second sound

Abstract

We consider an initial-boundary-value problem for a thermoelastic Kirchhoff & Love plate, thermally insulated and simply supported on the boundary, incorporating rotational inertia and a quasilinear hypoelastic response, while the heat effects are modeled using the hyperbolic Maxwell-Cattaneo-Vernotte law giving rise to a 'second sound' effect. We study the local wellposedness of the resulting quasilinear mixed-order hyperbolic system in a suitable solution class of smooth functions mapping into Sobolev H k -spaces. Exploiting the sole source of energy dissipation entering the system through the hyperbolic heat flux moment, provided the initial data are small in a lower topology (basic energy level corresponding to weak solutions), we prove a nonlinear stabilizability estimate furnishing global existence & uniqueness and exponential decay of classical solutions.

Large deflections of inextensible cantilevers: modeling, theory, and simulation

“…First, we note that our well-posedness results are consistent with what is to be expected for quasilinear beams and plates (see, e.g., [34,35]), owing to the fact that the [NL Stiffness] is quasilinear in nature. Moreover, the uniqueness in Theorem 4.7 is nontrivial, based on exploiting the particular polynomial structure of the [NL Stiffness] term.…”

“…In line with [35] and other papers on quasilinear beam and plate equations, we seek global solutions when damping is present. Indeed, since damping is -rather oddly -necessary for us to obtain existence for the full system (σ = ι = 1) when inertia is present, we may ask what sort of stability for the system is gained as a byproduct.…”

Theory of Solutions for an Inextensible Cantilever

“…We proceed as in [20,25] to obtain global existence indirectly via the Barrier method, which exploits the superlinearity in the problem. Using the damping, we will employ stabilization type multipliers at every energy level to obtain an inequality of the form in the theorem below, which we take from [20]:…”

Boundary stabilization of quasilinear Maxwell equations

“…Proof of Theorem 2.2. Fixing δ = min{δ 1 , 1/2C}, estimate (4.2) furnishes the enhanced linear stabilizability inequality If we have here T * = ∞, the result follows as in the linear case, see[30, Remark 4.2] or the proofs of[32, Corollary 5.7] or[6, Theorem A.1].Suppose that T * < ∞. Equation (2.14) then yields z(T * ) = δ 2 .…”