Quasilinear Hyperbolic–Parabolic Equations of One-Dimensional Viscoelasticity

Quasilinear Hyperbolic{parabolic Equations of One{dimensional Viscoelasticity

We study the global existence of solutions of initial-boundary-value problems for a quasilinear hyperbolic-parabolic equation describing the longitudinal motion of a one-dimensional viscoelastic rod. We treat a variety of nonhomogeneous boundary conditions, requiring separate analyses, because they lead to distinctive physical eeects. We employ a constitutive equation giving the stress as a gen...

Phase Transitions in One-Dimensional Nonlinear Viscoelasticity: Admissibility and Stability

For the motion of a one-dimensional viscoelastic material of rate type with a non-monotonic stress-strain relation, a mixed initial boundary value problem is considered. A simple existence theory is outlined, based on a novel transformation of the equation into the form of a degenerate reaction-diffusion system. This leads to new results characterizing the regularity of weak solutions. It is sh...

One dimensional infinite-horizon variational problems arising in viscoelasticity

Null–controllability of One–dimensional Parabolic Equations

We prove the interior null–controllability of one–dimensional parabolic equations with time independent measurable coefficients.

Quasilinear Parabolic Functional Evolution Equations

Based on our recent work on quasilinear parabolic evolution equations and maximal regularity we prove a general result for quasilinear evolution equations with memory. It is then applied to the study of quasilinear parabolic differential equations in weak settings. We prove that they generate Lipschitz semiflows on natural history spaces. The new feature is that delays can occur in the highest ...