Decay rates of Volterra equations on ℝ N

This note is concerned with the linear Volterra equation of hyperbolic type

Pullback attractors for non-autonomous reaction-diffusion equations on ℝ n

We study the long time behavior of solutions of the non-autonomous reaction-diffusion equation defined on the entire space ℝ ⁿ when external terms are unbounded in a phase space. The existence of a pullback global attractor for the equation is established in L ²(ℝ ⁿ ) and H ¹(ℝ ⁿ ), respectively. The pullback asymptotic compactness of solutions is proved by using uniform a priori estimates on the tails of solutions outside bounded domains.      

Solvability of forward–backward stochastic partial differential equations

In this paper we study the solvability of a class of fully-coupled forward–backward stochastic partial differential equations (FBSPDEs). These FBSPDEs cannot be put into the framework of stochastic evolution equations in general, and the usual decoupling methods for the Markovian forward–backward SDEs are difficult to apply. We prove the well-posedness of the FBSPDEs, under various conditions on the coefficients, by using either the method of contraction mapping or the method of continuation. These conditions, especially in the higher dimensional case, are novel in the literature.     

Some Notes on Solvability of LPDO

The purpose of this paper is to discuss how the lower order terms of a LPDO with multiple characteristics influence the solvability of the operators. We begin with the followiog operator: L(x,D)=D_1~2+x_1~2D_2~2+pD_2+qD_1+c where p,q,c are assumed be complex constants. Theorem 1 The operator L will be locally solvable for any q,c, if either |Re p|<1 or Im p=0.     

The Jacobi-Dunkl transform on ℝ and the convolution product on new spaces of distributions

In this work, we consider the Jacobi-Dunkl operator Λ _α,β , a 3 b 3 \frac-12\alpha\geq\beta\geq\frac{-1}{2} , a 1 \frac-12\alpha\neq\frac{-1}{2} , on ℝ. The eigenfunction Y_l^a,b\Psi_{\lambda}^{\alpha,\beta} of this operator permits to define the Jacobi-Dunkl transform. The main idea in this paper is to introduce and study the Jacobi-Dunkl transform and the Jacobi-Dunkl convolution product on new spaces of distributions     

On the positive radial solutions of some semilinear elliptic equations on ℝ n

We study the existence and behavior of positive radial solutions of the equationu + f(u) = 0 in ⁿ . This equation arises in various problems in applied mathematics, e.g. in the study of phase transitions, nuclear cores and more recently in population genetics and solitary waves. The important model casef(u) = – u + u ^p, p>1, describes for instance the pressure distribution in a van der Waals fluid. In this case, we obtain fairly complete knowledge of all positive radial solutions.Supported in part by an NSF grant and a research grant from the Graduate School of the University of Minnesota.     

Long-term analysis of degenerate parabolic equations in ℝ N

Longtime behavior of degenerate equations with the nonlinearity of polynomial growth of arbitrary order on the whole space R^N is considered. By using -trajectories methods, we proved that weak solutions generated by degenerate equations possess an (L_U² (R^N), L_loc² (R^N))-global attractor. Moreover, the upper bounds of the Kolmogorov ε-entropy for such global attractor are also obtained.     

Solvability of fully nonlinear functional equations involving Erdélyi-Kober fractional integrals on the unbounded interval

In this paper, we study the existence of solutions to fully nonlinear functional equations involving Erdélyi-Kober fractional integrals on the unbounded interval. By constructing a closed and convex subset of a Fréchet space, sufficient conditions are presented by using a fixed point theorem via the Kuratowski measure. Finally, an example is given to illustrate the obtained result.     

Stability analysis of Runge-Kutta methods for nonlinear neutral delay integro-differential equations

The sufficient conditions for the stability and asymptotic stability of Runge-Kutta methods for nonlinear neutral delay integro-differential equations are derived. A numerical test that confirms the theoretical results is given in the end.     

He’s homotopy perturbation method for systems of integro-differential equations

In this article, the homotopy perturbation method [He JH. Homotopy perturbation technique. Comput Meth Appl Mech Eng 1999;178:257–62; He JH. A coupling method of homotopy technique and perturbation technique for nonlinear problems. Int J Non-Linear Mech 2000;35(1):37–43; He JH. Comparison of homotopy perturbation method and homotopy analysis method. Appl Math Comput 2004;156:527–39; He JH. Homotopy perturbation method: a new nonlinear analytical technique. Appl Math Comput 2003;135:73–79; He JH. The homotopy perturbation method for nonlinear oscillators with discontinuities. Appl Math Comput 2004;151:287–92; He JH. Application of homotopy perturbation method to nonlinear wave equations Chaos, Solitons & Fractals 2005;26:695–700] is applied to solve linear and nonlinear systems of integro-differential equations. Some nonlinear examples are presented to illustrate the ability of the method for such system. Examples for linear system are so easy that has been ignored.     

Inverse problem for a nonlinear Benney–Luke type integro-differential equations with degenerate kernel

We consider the problem on the unique solvability of the inverse problem for a nonlinear partial Benney–Luke type integro-differential equation of the fourth order with a degenerate kernel. We modify the degenerate kernelmethod which has been designed for Fredholm integral equations of the second kind to apply to the case of the above-mentioned equation. We exploit the Fouriermethod of separation of variables. By means of designations, the Benney–Luke type integro-differential equation is reduced to a system of algebraic equations. Using an additional condition, we obtain the countable system of nonlinear integral equations with respect to the main unknown function. We employ the method of successive approximations together with the contraction mapping principle. Finally, the restore function is defined.     

Effect of noise on front propagation in reaction-diffusion equations of KPP type

We consider reaction-diffusion equations of KPP type in one spatial dimension, perturbed by a Fisher-Wright white noise, under the assumption of uniqueness in distribution. Examples include the randomly perturbed Fisher-KPP equations

$\partial_tu=\partial_x^2u+u(1-u)+\epsilon\sqrt{u(1-u)}\dot{W},$

and

$\partial_tu=\partial_x^2u+u(1-u)+\epsilon\sqrt{u}\dot{W},$

where \(\dot{W}=\dot{W}(t,x)\) is a space-time white noise. We prove the Brunet-Derrida conjecture that the speed of traveling fronts for small ε is

$2-\pi^2|{\log}\,\epsilon^2|^{-2}+O((\log|{\log}\,\epsilon|)|{\log}\,\epsilon|^{-3}).$

     

Solvability of a class of elastic beam equations with strong Carathéodory nonlinearity

We study the existence of a solution to the nonlinear fourth-order elastic beam equation with nonhomogeneous boundary conditions

An efficient numerical approximation for the linear class of Fredholm integro-differential equations based on Cattani’s method

This paper provides with a generalization of the work by Cattani (Math. Probl. Eng. (2008) 1–24), who has introduced the connection coefficients of the Shannon wavelets. We apply the Shannon wavelets approximation based on Cattani’s connection coefficients together the collocation points for solving the linear Fredholm integro-differential equations. Finally, numerical results are included to demonstrate the validity and applicability of the method and some comparisons are made with existing results.     

Error estimates and stability of itô-type systems of nonlinear stochastic integro-differential equations

By developing an integral representation of a function of solution process of a system of Ito-type stochastic nonlinear integro-differential equations, the error estimates on absolute p-th deviation of a solution process with the solution of the mean of the corresponding deterministic system of integro-differential equations is analyzed. Furthermore, the p-th moment stability properties of the steady state of the system is studied. The obtained results would provide a partial solution to one of the fundamental problems in modelling of dynamic systems, namely, to what extent incorporating randomness in the system causes the change in behavior of the system relative to its deterministic version     

Phragmen-Lindelöf type results for second order quasilinear parabolic equations in ℝ2

In this paper Phragmen-Lindelöf type growth-decay estimates are derived for solutions of initial-boundary value problems associated with a class of quasilinear parabolic equations defined on a semi-infinite strip in ². The particular problems considered are ones in which homogeneous initial data and homogeneous Dirichlet conditions on the long sides of the strip are prescribed.This research was carried out while the first author held a visiting appointment at Cornell University and was partially supported by NSF grant #DMS-9100876.