关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

Special Functions and Their Applications. By N. N. Lebedev, translated by R. A. Silverman. Pp. xii, 308. 96s. 1965. (Prentice-Hall)

Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping–source interaction

Given $\alpha \in [0,2)$ and $f \in L^2 ((0,T)\times(0,1))$, we derive new Carleman estimates for the degenerate parabolic problem $w_t + (x^\alpha w_x) _x =f$, where $(t,x) \in (0,T) \times (0,1)$, associated to the boundary conditions $w(t,1)=0$ and $w(t,0)=0$ if $0 \leq \alpha <1$ or $(x^\alpha w_x)(t,0)=0$ if $1\leq \alpha <2$ The proof is based on the choice of suitable weighted functions and Hardy-type inequalities As a consequence, for all $0 \leq \alpha <2$ and $\omega\subset\subset(0,1)$, we deduce null controllability results for the degenerate one-dimensional heat equation $u_t - (x^\alpha u_x)_x = h \chi _\omega$ with the same boundary conditions as above

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Carleman Estimates for a Class of Degenerate Parabolic Operators

these plants can be readily propa gated from seed; some exhibit complex seed dormancies and are more easily propagated by root division or stem cuttings. This article will focus on propagation of wildflowers and grasses using seed, as this is the commonly used, but often misunderstood, method of producing native herbaceous perennials. The methods described herein are based upon our 30 years of experience at Prairie Nursery in propagating a wide variety of native plants from seed. Most native perennials require that their seed be pre-treated to break dormancy prior to seeding. There are four basic types of seed treatments or planting methods that we use to overcome seed dormancy: Dry Stratification: Seed is exposed to freezing temperatures for 30 or more days. Moist Stratification: Seed is mixed with a damp inert substrate and stored in a refrigerated environment at 34-36 degrees Fahrenheit (1-2 degrees Celsius). The seed should not be frozen, as this may damage the cell walls and destroy the seed. Scarification: Seed with hard seed-coats are scratched with sandpaper to allow moisture to penetrate into the seed and initiate the germination process. Hot Water: Seeds that are stimulated to germinate by wildfires are treated with nearboiling water.

전자파 Propagation 개요

We prove an estimate of Carleman type for the one dimensional heat equation
$$ u_t - \left( {a\left( x \right)u_x } \right)_x + c\left( {t,x} \right)u = h\left( {t,x} \right),\quad \left( {t,x} \right) \in \left( {0,T} \right) \times \left( {0,1} \right), $$ where a(·) is degenerate at 0. Such an estimate is derived for a special pseudo-convex weight function related to the degeneracy rate of a(·). Then, we study the null controllability on [0, 1] of the semilinear degenerate parabolic equation
 $$ u_t - \left( {a\left( x \right)u_x } \right)_x + f\left( {t,x,u} \right) = h\left( {t,x} \right)\chi _\omega \left( x \right), $$ where (t, x) ∈(0, T) × (0, 1), ω=(α, β) ⊂⊂ [0, 1], and f is locally Lipschitz with respect to u.

Carleman estimates for degenerate parabolic operators with applications to null controllability

This work is concerned with the stabilization of hyperbolic systems by a nonlinear feedback which can be localized on a part of the boundary or locally distributed. We show that general weighted integral inequalities together with convexity arguments allow us to produce a general semi-explicit formula which leads to decay rates of the energy in terms of the behavior of the nonlinear feedback close to the origin. This formula allows us to unify for instance the cases where the feedback has a polynomial growth at the origin, with the cases where it goes exponentially fast to zero at the origin. We also give three other significant examples of nonpolynomial growth at the origin. Our work completes the work of [15] and improves the results of [21] and [22] (see also [23] and [10]). We also prove the optimality of our results for the one-dimensional wave equation with nonlinear boundary dissipation. The key property for obtaining our general energy decay formula is the understanding between convexity properties of an explicit function connected to the feedback and the dissipation of energy.

Convexity and Weighted Integral Inequalities for Energy Decay Rates of Nonlinear Dissipative Hyperbolic Systems

Decay estimates for second order evolution equations with memory

A general method for proving sharp energy decay rates for memory-dissipative evolution equations

We consider systems of Timoshenko type in a one-dimensional bounded domain. The physical system is damped by a single feedback force, only in the equation for the rotation angle, no direct damping is applied on the equation for the transverse displacement of the beam. Moreover the damping is assumed to be nonlinear with no growth assumption at the origin, which allows very weak damping. We establish a general semi-explicit formula for the decay rate of the energy at infinity in the case of the same speed of propagation in the two equations of the system. We prove polynomial decay in the case of different speed of propagation for both linear and nonlinear globally Lipschitz feedbacks.

Fatiha Alabau-Boussouira

Papers

Carleman estimates for degenerate parabolic operators with applications to null controllability

Convexity and Weighted Integral Inequalities for Energy Decay Rates of Nonlinear Dissipative Hyperbolic Systems

Decay estimates for second order evolution equations with memory

A general method for proving sharp energy decay rates for memory-dissipative evolution equations

Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control