English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Uniqueness and stability theorems are established for coupled systems of parabolic differential equations which may involve a Volterra-type dependence on the past history of the process. We allow retarded or deviating arguments, convolution-type memory terms, and strong coupling. (This means that all the space derivatives up to a given order can occur in all the equations.) Our results for strong coupling depend on the concept of "admissible monomial" which is here introduced for the first time and has no counterpart in the linear case. It is possible for uniqueness to fail in general, but to be restored (relative to a tolerably large class of functions of (x, t)) if a single solution independent of x exists. Another curious feature of these theorems, depending again on the concept of admissible monomial, is that conditions for uniqueness can involve derivatives of order much higher than those occurring in the equation. Examples given elsewhere show that the results are, in various respects, sharp. Thus, the seemingly peculiar hypotheses do not arise from deficient technique, but from the actual behavior of strongly coupled systems. The paper concludes with a new method of dealing with unbounded regions for the difficult case in which the functional occurs in the boundary operator as well as in the differential equation. 0. Historical introduction. As the reader will recall, a powerful method of getting inequalities of the form u(x, t) < p(x, t) for solutions u of parabolic inequalities was introduced by Nagumo in 1939 and rediscovered by Westphal in 1949. The essence of the method is to choose p so that p(x, 0) > u(x, 0) and so that a contradiction is obtained at the first value t such that u(x, t) = p(x, t) for some x. Such a point (x, t) is called a Nagumo point. The subject of this paper has its origin in certain extensions of this basic procedure. (a) Inspired by early work of Max Müller (1926), Szarski and Mlak extended the method to weakly coupled systems in 1955, 1957 and 1959. Here one considers the first value t such that the kth component uk(x, t) = p(x, t) for some x and some k. (b) It was observed by Redheffer in 1963 that the Nagumo procedure applies to equations containing functionals u —* Fu, provided the functional is monotone and of Volterra type; that is, provided (Fu)(x, t) can be estimated from above by use of sup4 supT<( u(i, r). (c) In 1975 Nickel introduced a novel approach to the theory of strongly coupled systems depending on the fact that, in a suitable function class, the space derivatives Du can be estimated at (x, t) by means of sup£|t/(£, t)\. (The possibility of such estimation in one variable follows from inequalities of Landau, Bernstein and Kolmogorov.) Nickel's ingenious method [3]-[5] is, first, to substitute Received by the editors March 6, 1979. 1980 Mathematics Subject Classification. Primary 35K55, 35R10, 35R45. 'Supported by NSF Grant No. MCS 77-03570. © 1980 American Mathematical Society 0002-9947/80/0000-05 10/$05.50

Stability of the null solution of parabolic functional inequalities