Daniel Toundykov

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"Well-posedness and stability of a semilinear Mindlin-Timoshenko plate model" I will discuss well-posedness and long-time behavior of Mindlin-Timoshenko plate equations that describe vibrations of thin plates. This system of partial differential equations was derived by R. Mindlin in 1951 (though E. Reissner also considered an analogous model earlier in(More)
Advisers: Petronela Radu and Daniel Toundykov This thesis explores several models in continuum mechanics from both local and nonlo-cal perspectives. The first portion settles a conjecture proposed by Filippo Gazzola and his collaborators on the finite-time blow-up for a class of fourth-order differential equations modeling suspension bridges. Under suitable(More)
Weak solutions and blow-up for wave equations of p-Laplacian type with supercritical sources" (2015). This paper investigates a quasilinear wave equation with Kelvin-Voigt damping, u t t − ∆ p u − ∆u t = f (u), in a bounded domain Ω ⊂ R 3 and subject to Dirichlét boundary conditions. The operator ∆ p , 2 < p < 3, denotes the classical p-Laplacian. The(More)
This paper settles a conjecture by Gazzola and Pavani [10] regarding solutions to the fourth order ODE w (4) + kw + f (w) = 0 which arises in models of traveling waves in suspension bridges when k > 0. Under suitable assumptions on the nonlinearity f and initial data, we demonstrate blow-up in finite time. The case k ≤ 0 was first investigated by Gazzola et(More)
We study regular solutions to wave equations with super-critical source terms, e.g., of exponent p > 5 in 3D. Such high-order sources have been a major challenge in the investigation of finite-energy (H 1 × L 2) solutions to wave PDEs for many years. The well-posedness question has been answered in part, but even the local existence, for instance, in 3(More)
We derive global in time a priori bounds on higher-level energy norms of strong solutions to a semilinear wave equation: in particular, we prove that despite the influence of a nonlinear source, the evolution of a smooth initial state is globally bounded in the strong topology ∼ H 2 × H 1. And the bound is uniform with respect to the corresponding norm of(More)
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