Polynomial Growth Condition Research Articles

Long-time behavior of reaction–diffusion equations with dynamical boundary condition

In this paper, we study the long-time behavior of the reaction–diffusion equation with dynamical boundary condition, where the nonlinear terms f and g satisfy the polynomial growth condition of arbitrary order. Some asymptotic regularity of the solution has been proved. As an application of the asymptotic regularity results, we can not only obtain the existence of a global attractor A in ( H 1 ( Ω ) ∩ L p ( Ω ) ) × L q ( Γ ) immediately, but also can show further that A attracts every L 2 ( Ω ) × L 2 ( Γ ) -bounded subset with ( H 1 ( Ω ) ∩ L p + δ ( Ω ) ) × L q + κ ( Γ ) -norm for any δ , κ ∈ [ 0 , ∞ ) .

Stochastic suppression and stabilization of delay differential systems

AbstractIn this paper, we investigate stochastic suppression and stabilization for nonlinear delay differential system \documentclass{article}\usepackage{amssymb}\usepackage[mathscr]{euscript}\footskip=0pc\pagestyle{empty}\begin{document}${\dot{x}}(t)=f(x(t),x(t-\delta(t)),t)$\end{document}, where δ(t) is the variable delay and f satisfies the one‐sided polynomial growth condition. Since f may defy the linear growth condition or the one‐sided linear growth condition, this system may explode in a finite time. To stabilize this system by Brownian noises, we stochastically perturb this system into the nonlinear stochastic differential system dx(t)=f(x(t), x(t−δ(t)), t)dt+qx(t)dw1(t)+σ|x(t)|βx(t)dw2(t) by introducing two independent Brownian motions w1(t) and w2(t). This paper shows that the Brownian motion w2(t) may suppress the potential explosion of the solution of this stochastic system for appropriate choice of β under the condition σ≠0. Moreover, for sufficiently large q, the Brownian motion w1(t) may exponentially stabilize this system. Copyright © 2010 John Wiley & Sons, Ltd.

Analysis of Noise under Regime Switching

In this paper we consider a stochastic nonlinear system under regime switching. Given a system x(t)=f(x(t),r(t),t) in which f satisfies so-called one-side polynomial growth condition. We introduce two Brownian noise feedbacks and stochastically perturb this system into dx(t)=(x(t),r(t),t)dt+ σ (r(t))|x(t)|βx(t)dW1(t)+q(r(t))x(t)dW2(t) . It can be proved that appropriate noise intensity may suppress the potentially explode in a finite time and ensure that this system is almost surely exponentially stable although the corresponding system without Brownian noise perturbation may be unstable system.

Spatiotemporal dynamics of two generic predator–prey models

We present the analysis of two reaction–diffusion systems modelling predator–prey interactions, where the predator displays the Holling type II functional response, and in the absence of predators, the prey growth is logistic. The local analysis is based on the application of qualitative theory for ordinary differential equations and dynamical systems, while the global well-posedness depends on invariant sets and differential inequalities. The key result is an L ∞-stability estimate, which depends on a polynomial growth condition for the kinetics. The existence of an a priori L p -estimate, uniform in time, for all p≥1, implies L ∞-uniform bounds, given any nonnegative L ∞-initial data. The applicability of the L ∞-estimate to general reaction–diffusion systems is discussed, and how the continuous results can be mimicked in the discrete case, leading to stability estimates for a Galerkin finite-element method with piecewise linear continuous basis functions. In order to verify the biological wave phenomena of solutions, numerical results are presented in two-space dimensions, which have interesting ecological implications as they demonstrate that solutions can be ‘trapped’ in an invariant region of phase space.

Asymptotic regularity for p-Laplacian equation

This paper is devoted to proving some asymptotic regularity of the solutions of the p-Laplacian equation ut−div(|∇u|p−2∇u)+f(u)=g(x) (p∊(2,N)) considered on a bounded domain Ω⊂RN(N≥3). The nonlinear term f satisfies the polynomial growth condition of arbitrary order c1|u|q−k≤f(u)u≤c2|u|q+k, where q≥2 is arbitrary. As an application of the asymptotic regularity results, we not only can obtain the existence of a (L2(Ω),W01,p(Ω)∩Lq(Ω))-global attractor A immediately but also can show further that A attracts every bounded subsets of L2(Ω) under the W01,p∩Lq+δ-norm for any δ∊[0,∞). Furthermore, the fractal dimension of A is finite in Lq+δ(Ω) for any δ∊[0,∞).

On removability of the isolated singularity for solutions of high-order elliptic equations

A Serrin-like condition is found for removability of the isolated singularity for high-order elliptic equations in divergent form whose coefficients satisfy a polynomial growth conditions and an ellipticity condition stronger than one that usually is considered for high-order elliptic equations.

Weighted estimates for commutators of multilinear Calderón-Zygmund operators with non-doubling measures

Let μ be a nonnegative Radon measure on ℝ d, which only satisfies the polynomial growth condition. Under this assumption, the authors obtain some weighted weak-type estimates for the commutators generated by the multilinear Calderón-Zygmund operators and RBMO(μ)functions.

Suppression and stabilisation of noise

In this article, we investigate the stochastic suppression and stabilisation of nonlinear systems. Given an unstable differential equation , in which f satisfies the one-sided polynomial growth condition, we introduce two Brownian noise feedbacks and therefore stochastically perturb this system into the nonlinear stochastic differential equation . This article shows that appropriate β may guarantee that this stochastic system exists as a unique global solution although the corresponding deterministic may explode in a finite time. Then sufficiently large q may ensure that this stochastic system is almost surely exponentially stable.

Multiple Unbounded Positive Solutions for Three-Point BVPs with Sign-Changing Nonlinearities on the Positive Half-Line

This work is concerned with the existence of unbounded positive solutions for a second-order nonlinear three-point boundary value problem on the positive half-line. The interesting points of the results are that the nonlinearity depends on the solution and its derivative and may change sign. Moreover, it satisfies general polynomial growth conditions. New existence results of nontrivial single and multiple positive solutions are proved using recent fixed point theorems on cones in a special Banach space.

On the singular generalized Fisher-like equation with derivative depending nonlinearity

This work is devoted to the existence of positive nontrivial solutions for a second-order nonlinear boundary value problem posed on the positive half-line. The problem stems from epidemiology and combustion theory. The positive nonlinearity depends on the solution and its derivative and may exhibit a singularity at the origin. It satisfies general polynomial growth conditions. New existence results of multiple positive solutions are proved using recent fixed point theorems on cones in Banach spaces. Examples with numerical computations illustrate the applicability of the existence theorems. Comparisons with existing results are provided.

Robust control of uncertain systems with polynomial nonlinearity by output feedback

AbstractThe problem of global robust stabilization by output feedback is investigated for two classes of uncertain systems with polynomial nonlinearity—one is with controllable/observable linearization and the other is not. The uncertainties in the systems are assumed to be dominated by both lower‐ and higher‐order nonlinearities multiplying by an output‐dependent growth rate. There are two ingredients in this study. One is to exploit the idea of how to handle polynomial growth conditions via homogeneity and domination without introducing an observer gain updated law. The other is the development of a recursive design algorithm for the construction of reduced‐order observers, which is not only interesting in its own right but also has a valid counterpart, capable of dealing with strongly nonlinear systems, even lack of uniform observability and smooth stabilizability. Copyright © 2008 John Wiley & Sons, Ltd.

Far field asymptotics of solutions to convection equation with anomalous diffusion

The initial value problem for the conservation law $$\partial_t u + (-\Delta)^{\alpha/2}u + \nabla \cdot f(u) = 0$$ is studied for $$\alpha \in (1, 2)$$ and under natural polynomial growth conditions imposed on the nonlinearity. We find the asymptotic expansion as $$\mid x \mid \rightarrow \infty$$ of solutions to this equation corresponding to initial conditions, decaying sufficiently fast at infinity.

Long-time behavior of solutions of some doubly nonlinear equation

The subject of this investigation is the qualitative properties of solutions of the equation where nonlinear term g is supposed to satisfy the upper polynomial growth condition of arbitrary order and g′(s) > −l. The existence of a , -global attractor for this equation with Dirichlet boundary condition is proved.

Proof of existence of global solutions for m-component reaction–diffusion systems with mixed boundary conditions via the Lyapunov functional method

To prove global existence for solutions of m-component reaction–diffusion systems presents fundamental difficulties in the case in which some components of the system satisfy Neumann boundary conditions while others satisfy nonhomogeneous Dirichlet boundary conditions and nonhomogeneous Robin boundary conditions. The purpose of this paper is to prove the existence of a global solution using a single inequality for the polynomial growth condition of the reaction terms. Our technique is based on the construction of polynomial functionals. This result generalizes those obtained recently by Kouachi et al (at press), Kouachi (2002 Electron. J. Diff. Eqns 2002 1), Kouachi (2001 Electron. J. Diff. Eqns 2001 1) and independently by Malham and Xin (1998 Commun. Math. Phys. 193 287).

Singular control with state constraints on unbounded domain

We study a class of stochastic control problems where a cost of the form $$\mathbb{E}\int_{[0,\infty)}e^{-\beta s}[\ell(X_{s})\,ds+h(Y^{\circ}_{s})\,d|Y|_{s}]$$ is to be minimized over control processes Y whose increments take values in a cone $\mathbb{Y}$ of ℝp, keeping the state process X=x+B+GY in a cone $\mathbb{X}$ of ℝk, k≤p. Here, $x\in\mathbb{X}$, B is a Brownian motion with drift b and covariance Σ, G is a fixed matrix, and Y○ is the Radon–Nikodym derivative dY/d|Y|. Let ℒ=−(1/2)trace(ΣD2)−b⋅D where D denotes the gradient. Solutions to the corresponding dynamic programming PDE, $$[(\mathscr{L}+\beta)f-\ell]\vee\sup_{y\in\mathbb{Y}:|Gy|=1}[-Gy\cdot Df-h(y)]=0,$$ on $\mathbb{X}^{o}$ are considered with a polynomial growth condition and are required to be supersolution up to the boundary (corresponding to a “state constraint” boundary condition on $\partial\mathbb{X}$). Under suitable conditions on the problem data, including continuity and nonnegativity of ℓ and h, and polynomial growth of ℓ, our main result is the unique viscosity-sense solvability of the PDE by the control problem’s value function in appropriate classes of functions. In some cases where uniqueness generally fails to hold in the class of functions that grow at most polynomially (e.g., when h=0), our methods provide uniqueness within the class of functions that, in addition, have compact level sets. The results are new even in the following special cases: (1) The one-dimensional case k=p=1, $\mathbb{X}=\mathbb{Y}=\mathbb{R}_{+}$; (2) The first-order case Σ=0; (3) The case where ℓ and h are linear. The proofs combine probabilistic arguments and viscosity solution methods. Our framework covers a wide range of diffusion control problems that arise from queueing networks in heavy traffic.

Strong convergence rates for backward Euler on a class of nonlinear jump-diffusion problems

We generalise the current theory of optimal strong convergence rates for implicit Euler-based methods by allowing for Poisson-driven jumps in a stochastic differential equation (SDE). More precisely, we show that under one-sided Lipschitz and polynomial growth conditions on the drift coefficient and global Lipschitz conditions on the diffusion and jump coefficients, three variants of backward Euler converge with strong order of one half. The analysis exploits a relation between the backward and explicit Euler methods.

Error Feedback and a Servomechanism Problem for Uncertain Nonlinear Systems

The robust servomechanism problem, also known as the problem of robust output regulation, is considered for a family of uncertain triangular systems. In contrast to the previous work where the uncertain vector field of the controlled plant was assumed to be bounded by a linear growth of the unmeasurable states multiplied by a polynomial function of the system output, we show that the polynomial growth condition can be relaxed and robust output regulation by error feedback is still possible, as long as the uncertain system is dominated by a continuous function of the system output multiplied by a linear growth of the unmeasurable states.

Global attractors for p-Laplacian equation

The existence of a ( L 2 ( Ω ) , W 0 1 , p ( Ω ) ∩ L q ( Ω ) ) -global attractor is proved for the p-Laplacian equation u t − div ( | ∇ u | p − 2 ∇ u ) + f ( u ) = g on a bounded domain Ω ⊂ R n ( n ⩾ 3 ) with Dirichlet boundary condition, where p ⩾ 2 . The nonlinear term f is supposed to satisfy the polynomial growth condition of arbitrary order c 1 | u | q − k ⩽ f ( u ) u ⩽ c 2 | u | q + k and f ′ ( u ) ⩾ − l , where q ⩾ 2 is arbitrary. There is no other restriction on p and q. The asymptotic compactness of the corresponding semigroup is proved by using a new a priori estimate method, called asymptotic a priori estimate.

Optimal regularity conditions for elliptic problems via $L^p_\delta$-spaces

We consider positive solutions of the Dirichlet problem for nonlinear elliptic equations, where the nonlinearity is assumed to satisfy the polynomial upper growth condition f(x,u) ≤ C(1 + up). It is known from a classical work of Brezis and Turner that the condition p pBT, for a suitable nonlinearity f(x,u) = a(x)up, with a(x) ≥ 0 and a ∈ L∞, we prove the existence of an unbounded weak solution. We next consider the case of systems and show that the polynomial growth conditions for a priori estimates recently obtained by Quittner and the author are also optimal. Our results are strongly connected with the regularity theory of the Laplace operator in the spaces $L^p_\delta(\Omega)$, the Lebesgue spaces weighted by the distance to the boundary. As a by-product, we in turn establish the optimality of the known linear $L^p_\delta$-regularity estimates. Our proofs are based on the construction of a solution of the Laplace equation with a suitable boundary singularity, with conical support, and we use recent results on the boundary behavior of heat kernels.

Asymptotic hyperfunctions, tempered hyperfunctions, and asymptoticexpansions

We introduce new subclasses of Fourier hyperfunctions of mixed type, satisfying polynomial growth conditions at infinity, and develop their sheaf and duality theory. We use Fourier transformation and duality to examine relations of these asymptotic and tempered hyperfunctions to known classes of test functions and distributions, especially the Gel′fand‐Shilov spaces. Further it is shown that the asymptotic hyperfunctions, which decay faster than any negative power, are precisely the class that allows asymptotic expansions at infinity. These asymptotic expansions are carried over to the higher‐dimensional case by applying the Radon transformation for hyperfunctions.