Stability of the null solution of parabolic functional inequalities

Authors:
Ray Redheffer and Wolfgang Walter

Journal:
Trans. Amer. Math. Soc. **262** (1980), 285-302

MSC:
Primary 35R10; Secondary 35K60

DOI:
https://doi.org/10.1090/S0002-9947-1980-0583856-3

MathSciNet review:
583856

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Abstract: 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 ) 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.

**[1]**A. Kolmogoroff,*On inequalities between upper bounds of consecutive derivatives of an arbitrary function defined on an infinite interval*, Učen. Zap. Moskov. Gos. Univ. Mat.**30**(1939), 3-16. (Russian) MR**0001787 (1:298c)****[2]**Karl Nickel,*The Crocco-transformation for the three-dimensional Prandtl boundary layer equations*, MRC Report 1594, University of Wisconsin, 1975.**[3]**-,*Error bounds and uniqueness for the solutions of nonlinear strongly coupled parabolic systems of differential equations*, MRC Report 1596, University of Wisconsin, 1975.**[4]**-,*Fehlerschranken und Eindeutigkeitsaussagan für die Lösungen nichtlinearer, stark gekoppelter parabolischer Differentialgleichungen*. Math. Z.**152**(1976), 33-45.**[5]**-,*New results on strongly coupled systems of parabolic differential equations*. Proc. Fourth Dundee Conf. on Ordinary and Partial Differential Equations (March 30-April 2, 1976), Lecture Notes in Math., vol. 564, Springer-Verlag, Berlin and New York, 1976. MR**0605666 (58:29279)****[6]**-,*The lemma of Max Müller-Nagumo-Westphal for strongly coupled systems of parabolic functional differential equations*, MRC Report 1800, 1977. (German transl, available from the University of Freiburg.)**[7]**R. M. Redheffer,*Die Collatzsche Monotonie bei Anfangswertproblemen*, Arch. Rational Mech. Anal.**14**(1963), 196-212. MR**0157102 (28:342)****[8]**R. M. Redheffer and W. Walter,*Existence theorems for strongly coupled systems of partial differential equations over Bernstein classes*, Bull. Amer. Math. Soc.**82**(1976), 899-902. MR**0457925 (56:16129)****[9]**R. M. Redheffer,*Uniqueness, stability and error estimation for parabolic functional-differential equations*, Ber. 9, Univ. Karlsruhe (1976). Accepted by the Soviet Academy of Sciences in October, 1976 for the Jubilee Volume dedicated to the 70th anniversary of Academician I. N. Vekua.**[10]**-,*Das Maximumprinzip in unbeschränkten Gebieten für parabolische Ungleichungen mit Funktionalen*, Math. Ann.**226**(1977), 155-170. MR**0450786 (56:9079)****[11]**-,*Inequalities involving derivatives*, Pacific J. Math.**85**(1979), 165-178. MR**571634 (81f:26012)****[12]**-,*Comparison theorems for parabolic functional inequalities*, Pacific J. Math.**85**(1979), 447-470. MR**574929 (81i:35164)****[13]**-,*Nonuniqueness of the null solution for strongly coupled systems of parabolic differential equations*, Math. Z.**171**(1980), 83-90.**[14]**J. Szarski,*Differential inequalities*, Monogr. Mat., Tom 43, Warsaw, 1965. MR**0190409 (32:7822)****[15]**W. Walter,*Differential and integral inequalities*, Ergebnisse der Mathematik und ihrer Grenzgebiete, Bd. 55, Springer-Verlag, Berlin and New York, 1970. (Further references to the early history can be found here.) MR**0271508 (42:6391)**

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DOI:
https://doi.org/10.1090/S0002-9947-1980-0583856-3

Article copyright:
© Copyright 1980
American Mathematical Society