On the asymptotic behaviour of solutions of stochastic differential equations

Summary In this paper we study the asymptotic behaviour of the solution of the stochastic differential equation dX _t=g(X _t)dt+(X _t)dW _t, where and g are positive functions and W _tis a Wiener process. We clarify, under which conditions X _tmay be approximated on {X _t} by means of a deterministic function. Further the question is treated, whether X _tconverges in distribution on {X _t. We deal with the Ito-solution as well as the Stratonovitch-solution and compare both.Partially supported by the SFB 123 Stochastische Mathematische Modelle, Heidelberg     

On the asymptotic behavior of solutions of nonlinear second-order parabolic equations

We study the asymptotic behavior as t → +∞ of solutions to a semilinear second-order parabolic equation in a cylindrical domain bounded in the spatial variable. We find the leading term of the asymptotic expansion of a solution as t → +∞ and show that each solution of the problem under consideration is asymptotically equivalent to a solution of some nonlinear ordinary differential equation.     

On the behaviour of solutions of degenerated nonlinear parabolic equations

The aim of this work is to study the behaviour of solutions of the initial boundary problem for degenerated nonlinear parabolic equations of the second order. The conditions of existence and non-existence solutions are established. Moreover, the behaviour of the solution is studied. We obtain the estimations in terms of characterizing the initial and weight functions on infinity, without a lower bound on the initial function.     

On the asymptotic integration of nonlinear differential equations

The aim of the present paper is twofold. Firstly, the paper surveys the literature concerning a specific topic in asymptotic integration theory of ordinary differential equations: the class of second order equations with Bihari-like nonlinearity. Secondly, some general existence results are established with regard to a condition that has been found recently to be of significant use in the theory of elliptic partial differential equations.     

On the boundedness and the asymptotic behaviour of the non-negative solutions of Volterra-Hammerstein integral equations

We study the Volterra-Hammerstein integral equation $$U(t,x) = U_O (t,x) + \mathop \smallint \limits_O^t \mathop \smallint \limits_D f(y, U (t - s,y)) h (x,y,s)dsdy,$$ t≥0, x∈D. We derive sufficient conditions for the boundedness of all non-negative solutions U. We show that, for bounded non-negative solutions U, U(t,.) is positive on D for sufficiently large t>0, if we impose appropriate positivity assumptions on f and h. If we additionally assume that, for y∈D, rf(y,r) strictly monotone increases and f(y,r)/r strictly monotone decreases as r>0 increases, the following alternative holds for any bounded non-negative solution U: Either U(t,.) converges toward zero for t→∞, pointwise on D, or U(t,.) converges, for t→∞, toward the unique bounded positive solution of the corresponding Hammerstein integral equation, uniformly on D. We indicate conditions for the occurrence of each of the two cases.     

On asymptotic behaviour of neutral integrodifferential equations

The necessary and sufficient conditions for all nontrivial solutions of $$\tfrac{d}{{dt}}\left[ {x\left( t \right) - c\int_0^\infty {H\left( s \right)x\left( {t - s} \right)ds} } \right] + a\int_0^\infty {k\left( s \right)x\left( {t - s} \right)ds + bx\left( t \right) = 0} $$ to have ‘zero crossing’ are given. Also, an application to level crossing is given.     

Verification of asymptotic solutions for one-dimensional nonlinear parabolic equations

The present article proves a result that is new for partial differential equations. According to this result, the solution of the Cauchy problem for a nonlinear parabolic equation with variable, slowly changing coefficients will turn into (asymptotically approach) a special asymptotic solution, either a solution or a kink, for high values of t.Translated from Matematicheskie Zametki, Vol. 52, No. 3, pp. 10–16, September, 1992.