The Cauchy problem for a fourth order parabolic equation by difference methods

Thesis (Ph.D.)--Boston UniversityThis paper deals with the solution of parabolic partial differential equations by difference methods. It is first concerned with obtaining certain basic results for the nth order equation... This enables one to exhibit a stable difference equation compatible with (5). Once assured of the existence of such an equation, it is employed in proving an existence theorem for a solution of the differential equation. The theorem states that if the coefficients a;(x,t) and the function d(x, t) in (5), and the function f(x) in:(2) possess a sufficient number of uniformly continuous and bounded derivatives in R, and a0(x,t) is negative and bounded away from zero, then there exists a solution of (5), (2) possessing a certain number of uniformly continuous and bounded derivatives. [TRUNCATED

Moment Boundedness of Linear Stochastic Delay Differential Equation with Distributed Delay

This paper studies the moment boundedness of solutions of linear stochastic delay differential equations with distributed delay. For a linear stochastic delay differential equation, the first moment stability is known to be identical to that of the corresponding deterministic delay differential equation. However, boundedness of the second moment is complicated and depends on the stochastic terms. In this paper, the characteristic function of the equation is obtained through techniques of Laplace transform. From the characteristic equation, sufficient conditions for the second moment to be bounded or unbounded are proposed.Comment: 38 pages, 2 figure

Novel Electroweak Symmetry Breaking Conditions From Quantum Effects In The MSSM

We present, in the context of the Minimal Supersymmetric Standard Model, a detailed one-loop analytic study of the minimization conditions of the effective potential in the Higgs sector. Special emphasis is put on the role played by $Str M^4$ in the determination of the electroweak symmetry breaking conditions, where first and second order derivatives of the effective potential are systematically taken into account. Novel, necessary (and sufficient in the Higgs sector) model-independent constraints, are thus obtained analytically, leading to new theoretical lower and upper bounds on $\tan \beta$. Although fully model-independent, these bounds are found to be much more restrictive than the existing model-dependent ones! A first illustration is given in the context of a SUGRA-GUT motivated scenario.Comment: Latex, 45 pages, 5 figure

Bohl-Perron type stability theorems for linear difference equations with infinite delay

Relation between two properties of linear difference equations with infinite delay is investigated: (i) exponential stability, (ii) \l^p-input \l^q-state stability (sometimes is called Perron's property). The latter means that solutions of the non-homogeneous equation with zero initial data belong to \l^q when non-homogeneous terms are in \l^p. It is assumed that at each moment the prehistory (the sequence of preceding states) belongs to some weighted \l^r-space with an exponentially fading weight (the phase space). Our main result states that (i) $\Leftrightarrow$ (ii) whenever $(p,q) \neq (1,\infty)$ and a certain boundedness condition on coefficients is fulfilled. This condition is sharp and ensures that, to some extent, exponential and \l^p-input \l^q-state stabilities does not depend on the choice of a phase space and parameters $p$ and $q$, respectively. \l^1-input \l^\infty-state stability corresponds to uniform stability. We provide some evidence that similar criteria should not be expected for non-fading memory spaces.Comment: To be published in Journal of Difference Equations and Application

Eliminating flutter for clamped von Karman plates immersed in subsonic flows

We address the long-time behavior of a non-rotational von Karman plate in an inviscid potential flow. The model arises in aeroelasticity and models the interaction between a thin, nonlinear panel and a flow of gas in which it is immersed [6, 21, 23]. Recent results in [16, 18] show that the plate component of the dynamics (in the presence of a physical plate nonlinearity) converge to a global compact attracting set of finite dimension; these results were obtained in the absence of mechanical damping of any type. Here we show that, by incorporating mechanical damping the full flow-plate system, full trajectories---both plate and flow---converge strongly to (the set of) stationary states. Weak convergence results require "minimal" interior damping, and strong convergence of the dynamics are shown with sufficiently large damping. We require the existence of a "good" energy balance equation, which is only available when the flows are subsonic. Our proof is based on first showing the convergence properties for regular solutions, which in turn requires propagation of initial regularity on the infinite horizon. Then, we utilize the exponential decay of the difference of two plate trajectories to show that full flow-plate trajectories are uniform-in-time Hadamard continuous. This allows us to pass convergence properties of smooth initial data to finite energy type initial data. Physically, our results imply that flutter (a non-static end behavior) does not occur in subsonic dynamics. While such results were known for rotational (compact/regular) plate dynamics [14] (and references therein), the result presented herein is the first such result obtained for non-regularized---the most physically relevant---models