to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

Differential equations of infinite order are an increasingly important class of equations in theoretical physics. Such equations are ubiquitous in string field theory and have recently attracted considerable interest also from cosmologists. Though these equations have been studied in the classical mathematical literature, it appears that the physics community is largely unaware of the relevant formalism. Of particular importance is the fate of the initial value problem. Under what circumstances do infinite order differential equations possess a well-defined initial value problem and how many initial data are required? In this paper we study the initial value problem for infinite order differential equations in the mathematical framework of the formal operator calculus, with analytic initial data. This formalism allows us to handle simultaneously a wide array of different nonlocal equations within a single framework and also admits a transparent physical interpretation. We show that differential equations of infinite order do not generically admit infinitely many initial data. Rather, each pole of the propagator contributes two initial data to the final solution. Though it is possible to find differential equations of infinite order which admit well-defined initial value problem with only two initial data, neither the dynamical equations of p-adic string theory nor string field theory seem to belong to this class. However, both theories can be rendered ghost-free by suitable definition of the action of the formal pseudo-differential operator. This prescription restricts the theory to frequencies within some contour in the complex plane and hence may be thought of as a sort of ultra-violet cut-off. Our results place certain recent attempts to study inflation in the context of nonlocal field theories on a much firmer mathematical footing.

Dynamics with infinitely many derivatives: The Initial value problem

Differential equations of infinite order are an increasingly important class of equations in theoretical physics. Such equations are ubiquitous in string field theory and have recently attracted considerable interest also from cosmologists. Though these equations have been studied in the classical mathematical literature, it appears that the physics community is largely unaware of the relevant formalism. Of particular importance is the fate of the initial value problem. Under what circumstances do infinite order differential equations possess a well-defined initial value problem and how many initial data are required? In this paper we study the initial value problem for infinite order differential equations in the mathematical framework of the formal operator calculus, with analytic initial data. This formalism allows us to handle simultaneously a wide array of different nonlocal equations within a single framework and also admits a transparent physical interpretation. We show that differential equations of infinite order do not generically admit infinitely many initial data. Rather, each pole of the propagator contributes two initial data to the final solution. Though it is possible to find differential equations of infinite order which admit well-defined initial value problem with only two initial data, neither the dynamical equations of p-adic string theory nor string field theory seem to belong to this class. However, both theories can be rendered ghost-free by suitable definition of the action of the formal pseudo-differential operator. This prescription restricts the theory to frequencies within some contour in the complex plane and hence may be thought of as a sort of ultra-violet cut-off.

Dynamics with Infinitely Many Derivatives: The Initial Value Problem

The study of the control and stabilization of the KdV equation began with the work of Russell and Zhang in late 1980s. Both exact control and stabilization problems have been intensively studied since then and significant progresses have been made due to many people's hard work and contributions. In this article, the authors intend to give an overall review of the results obtained so far in the study but with an emphasis on its recent progresses. A list of open problems is also provided for further investigation.

Control and stabilization of the Korteweg-de Vries equation: recent progresses

Wave equations with strong damping in Hilbert spaces

We study global existence in time of small solutions to the Cauchy problem for the nonlinear damped wave equation \begin{equation} \left\{ \begin{array}{c} \partial _{t}^{2}u+\partial _{t}u-\Delta u=\mathcal{N}\left( u\right) ,\quad x\in \mathbf{R}^{n},\text{ }t>0, \\ u(0,x)=\varepsilon u_{0}\left( x\right) ,\partial _{t}u(0,x)=\varepsilon u_{1}\left( x\right) , \quad x\in \mathbf{R}^{n}, \end{array} \right. \tag*{(0.1)} \end{equation} where $\varepsilon >0$. The nonlinearity $\mathcal{N}\left( u\right) \in \mathbf{C}^{k}\left( \mathbf{R}\right) $ satisfies the estimate \begin{equation*} \Big | \frac{d^{j}}{du^{j}}\mathcal{N}\left( u\right) \Big | \leq C\big | u\big | ^{\rho -j}, \quad 0\leq j\leq k\leq \rho . \end{equation*} The power $\rho >1+\frac{2}{n}$ is considered as super critical for large time. We assume that the initial data \begin{equation*} u_{0}\in \mathbf{H}^{\alpha ,0}\cap \mathbf{H}^{0,\delta },\text{ }u_{1}\in \mathbf{H}^{\alpha -1,0}\cap \mathbf{H}^{0,\delta }, \end{equation*} where $\delta >\frac{n}{2},$ $\left[ \alpha \right] \leq \rho ,\alpha \geq \frac{n}{2}-\frac{1}{\rho -1}$ for $n\geq 2$, and $\alpha \geq \frac{1}{2}- \frac{1}{2\left( \rho -1\right) }$ for $n=1.$ Weighted Sobolev spaces are \begin{equation*} \mathbf{H}^{l,m}= \Big \{ \phi \in \mathbf{L}^{2}; \big \| \left\langle x\right\rangle ^{m}\left\langle i\partial _{x}\right\rangle ^{l}\phi \left( x\right) \big \| _{\mathbf{L}^{2}}s 0$ such that for any $\varepsilon \in \left( 0,\varepsilon _{0}\right] $ there exists a unique global solution $ u\in \mathbf{C}\left( \left[ 0,\infty \right) ;\mathbf{H}^{\alpha ,0}\cap \mathbf{H}^{0,\delta }\right) $ for the Cauchy problem (0.1) and solutions satisfy the time decay property \begin{equation*} \big \| u ( t ) \big \| _{\mathbf{L}^{p}}\leq Ct^{-\frac{n}{2 } ( 1-\frac{1}{p} ) } \end{equation*} for all $t>0$, where $2\leq p\leq $ $\frac{2n}{n-2\alpha }$ if $\alpha s < \frac{n}{2},$ $2\leq ps <$ $\infty $ if $\alpha =\frac{n}{2},$ and $2\leq p\leq $ $\infty $ if $\alpha \gt\frac{n}{2}.$

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Damped wave equation with super critical nonlinearities

Introduction 1 Preliminaries 2 Sobolev spaces 3 General Theory 4 Nonlinear Schrodinger Type Equations 5 Whitham Equation 6 Korteweg-de Vries-Burgers Equation 7 Large Initial Data 8 KdV-B Type Equation 9 Dirichlet Problem for KdV Equation 10 Neumann Problem for KdV Equation 11 Landau-Ginzburg Equations 12 Burgers Equation with Pumping 13 KdVB Equation on a Segment 14 NLS Equation on Segment 15 Periodic Problem Bibliography Index

Nonlinear Theory of Pseudodifferential Equations on a Half-Line

We study large time asymptotics of small solutions to the Cauchy problem for nonlinear damped wave equations with a critical nonlinearity { ∂ 2 t u + ∂ t u - Δu + λu 1+2 n = 0, x ∈ R n , t > 0, u(0, x) = eu 0 (x), ∂ t u(0, x) = eu 1 (x), x ∈ R n , where e > 0, and space dimensions n = 1, 2,3. Assume that the initial data uo ∈ H δ,0 n H 0,δ , u 1 e H δ-1,0 n H -1,δ , where δ > n 2, weighted Sobolev spaces are H l,m = {Φ e L2; m l Φ(x)∥ L 2 = √1 + x 2 . Also we suppose that λθ 2/n > 0, ∫u 0 (x) dx > 0, where Then we prove that there exists a positive e 0 such that the Cauchy problem above has a unique global solution u ∈ C ([0, oo); H δ,0 ) satisfying the time decay property ∥u(t)-eθG(t,x)e -φ(t) ∥ Lp ≤ Ce 1+2 n g -1-n 2 (t) -n 2(1-1/p) for all t > 0, 1 ≤ p ≤ ∞, where e ∈ (0, e 0 ].

/pdf/damped-wave-equation-with-a-critical-nonlinearity-5feoity113.pdf

Damped wave equation with a critical nonlinearity

We study the Cauchy problem for the nonlinear Schrodinger equation with dissipation where L is a linear pseudodifferential operator with dissipative symbol ReL(ξ) ≥ C1|ξ|2/(1 + ξ2) and |L′(ξ)| ≤ C2(|ξ|+ |ξ|n) for all ξ ∈ R. Here, C1, C2 > 0, n ≥ 1. Moreover, we assume that L(ξ) = αξ2 + O(|ξ|2+γ) for all |ξ| 0, Re α > 0, Im α ≥ 0. When L(ξ) = αξ2, equation (A) is the nonlinear Schrodinger equation with dissipation ut − αuxx + i|u|2u = 0. Our purpose is to prove that solutions of (A) satisfy the time decay estimate under the conditions that u0 ∈ Hn,0 ∩ H0,1 have the mean value and the norm ‖u0‖Hn,0 + ‖u0‖H0,1 = e is sufficiently small, where σ = 1 if Im α > 0 and σ = 2 if Im α = 0, and Therefore, equation (A) is considered as a critical case for the large-time asymptotic behaviour because the solutions of the Cauchy problem for the equation ut − αuxx + i|u|p−1u = 0, with p > 3 have the same time decay estimate ‖u‖L∞ = O(t−½) as that of solutions to the linear equation. On the other hand, note that solutions of the Cauchy problem (A) have an additional logarithmic time decay. Our strategy of the proof of the large-time asymptotics of solutions is to translate (A) to another nonlinear equation in which the mean value of the nonlinearity is zero for all time.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0308210500000561

Elena I. Kaikina

Papers

Damped wave equation with super critical nonlinearities

Nonlinear Theory of Pseudodifferential Equations on a Half-Line

Damped wave equation with a critical nonlinearity

Large-time behaviour of solutions to the dissipative nonlinear Schrödinger equation

On the critical nonlinear damped wave equation with large initial data