On behavior at infinity of solutions of parabolic differential equations in a Banach space
Abstract
For a differential equation of the form $y'(t) + Ay(t) = 0, \ t \in (0, \infty)$, where $A$ is the generating operator of a $C_{0}$-semigroup of linear operators on a Banach space $\mathfrak{B}$, we give conditions on the operator $A$, under which this equation is uniformly (uniformly exponentially) stable, that is, every its weak solution defined on the open semiaxis $(0, \infty)$ tends (tends exponentially) to 0 as $t \to \infty$. As distinguished from the previous works dealing only with solutions continuous at 0, in this paper no conditions on the behavior of a solution near 0 are imposed. In the case where the equation is parabolic, there always exist weak solutions which have singularities of any order. The criterions below not only generalize, but make more precise a number of earlier results in this direction.
Key words: Differential equation in a Banach space, uniformly and uniformly exponentially stable equation, weak solution, weak Cauchy problem, C0-semigroup of linear operators, bounded analytic C0-semigroup, infinitely differentiable, entire, entire of exponential
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