Long-Time Behavior of Quasilinear Thermoelastic Kirchhoff-Love Plates with Second Sound

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 well-posedness 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.Comment: 46 page

Nonlinear Evolution Problems

In this workshop three types of nonlinear evolution problems— geometric evolution equations (essentially of parabolic type), nonlinear hyperbolic equations, and dispersive equations— were the subject of 22 talks

ANALYSIS OF STEADY-STATE AND DYNAMICAL RADIALLY-SYMMETRIC PROBLEMS OF NONLINEAR VISCOELASTICITY

This thesis treats radially symmetric steady states and radially symmetric motions of nonlinearly elastic and viscoelastic plates and shells subject to dead-load and hydrostatic pressures on their boundaries and with the plate subject to centrifugal force. The plates and shells are described by specializations of the exact (nonlinear) equations of three-dimensional continuum mechanics. The treatment in every case is very general and encompasses large classes of constitutive functions (characterizing the material response). We first treat the radially symmetric steady states of plates and shells and the radially symmetric steady rotations of plates. We show that the existence, multiplicity, and qualitative behavior of solutions for problems accounting for the live loads due to hydrostatic pressure and centrifugal force depend critically on the material properties of the bodies, physically reasonable refined descriptions of which are given and examined here with great care, and on the nature of boundary conditions.The treatment here, giving new and sharp results, employs several different mathematical tools, ranging from phase-plane analysis to the mathematically more sophisticated direct methods of the Calculus of Variations, fixed-point theorems, and global continuation methods, each of which has different strengths and weaknesses for handling intrinsic difficulties in the mechanics. We then treat the initial-boundary-value problems for the radially symmetric motions of annular plates and spherical shells that consist of a nonlinearly viscoelastic material of strain-rate type. We discuss a range of physically natural constitutive equations. We first show that when the material is strong in a suitable sense relative to externally applied loads, solutions exist for all time, depend continuously on the data, and consequently are unique. We study the role of the constitutive restrictions and that of the regularity of the data in ensuring the preclusion of a total compression and of an infinite extension for finite time. We then show that when the material is not sufficiently strong then under certain conditions on the (hydrostatic) pressure terms there are globally defined unbounded solutions and there are solutions that blow up in finite time. The practical importance of these results is that for each problem involving live loads they furnish thresholds in material response delimiting materials for which solutions are ill behaved. A mathematical or numerical study limited to a particular class of materials may dangerously indicate well-behaved solutions when there are other realistic materials for which solutions are ill behaved. Moreover this work furnishes so-called trivial solutions for the subsequent study (not given here) of bifurcation of stable equilibrium configurations from these trivial solutions

Decay estimates for the time-fractional evolution equations with time-dependent coefficients

In this paper, the initial-boundary value problems for the time-fractional degenerate evolution equations are considered. Firstly, in the linear case, we obtain the optimal rates of decay estimates of the solutions. The decay estimates are also established for the time-fractional evolution equations with nonlinear operators such as: p-Laplacian, the porous medium operator, degenerate operator, mean curvature operator, and Kirchhoff operator. At the end, some applications of the obtained results are given to derive the decay estimates of global solutions for the time-fractional Fisher-KPP-type equation and the time-fractional porous medium equation with the nonlinear source.Comment: 23 pages. The previous version of the paper has been edited according to the comments of the reviewer