Abstract
This paper deals with the behavior of solutions of non‐linear ordinary differntial equations in a Hilbert space with applications to non‐linear partial differential equations.
Highlights
This paper deals with the behavior of solutions of non-linear ordinary differntial equations in a Hilbert space with applications to non-linear partial dif,fereDtial equations
Recall that we do net have this situation ip a linear case because this par of our theerems have n, analog in the linear case
We study two cases of non-linear differential equations: the case of deqenerate equation in bounded interval of time ar.d the case of non-degenerate equation iF, unbounded interval of time
Summary
Bx here x(>-, s is compact set of R n or smooth compact n-manifold In this special case, it is possible to obtain more complete results and more simple form of condition (I.19). Let u(t) be a solution of equation (I.I) under Condition A such that condition (1.24) is satisfied for some non-negative bounded functions a(t), b(t) in the interval I. where constant v _> 0 depending on a(t),b(t) and u(t) itself, and constant >_ 0 depending on a(t), b(t) only. Let A(t,u) for each u(t) c DA with flat norm lu(t)l satisfy the following conditionsi) (A(t,u(t))v,) 0 in some interval (0,) with depending on u(t) for each Ilv H, llvll I. d--[ ii) (a(t,u(t))v,v)] is bounded in interval (0,) for each v H, lvll (this condition follows from Conoition A,). 0 f.or solutions of equation (I.I) under those conditions
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