Kneser's property for a parabolic partial differential equation

Abstract

au/at = Au + 1~1”~ for t > 0 and x E R”, (1) with the initial condition u(Gx)l,=, = 0 for x E R”. (2) The problem has a smooth solution u(t, x) = t*/4 other than the trivial solution. The purpose of this paper is to investigate properties of the set of solutions for problem (1) with condition (2). In Section 2 we shall present the main result: the cross-section of the solutions is a continuum (i.e. a compact and connected set) in a given function space, whose fact is known as Kneser’s property [l] in the theory of ordinary differential equations with nonuniqueness of solutions. In ordinary differential equations, Hukuhara [2, 31 made more detailed investigations on Kneser’s property and found Hukuhara’s phenomenon. For partial differential equations of parabolic type, we have tried to discuss the same problem stated above [4], regarding which Goldstein [5] pointed out that our assertion is valid only for equations which generate a compact C,-semigroup. On the other hand, Ballotti [6] has shown Aronszajn’s theorem [7] and hence Kneser’s property for a partial differential equation generating a compact C,-semigroup. In Ballotti’s paper [6] such an example has been treated: