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On the strong maximum principle for parabolic differential equations

Published online by Cambridge University Press:  20 January 2009

Wolfgang Walter
Wolfgang Walter
Affiliation:
Mathematisches Institut, Universitāt Karlsruhe
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In a recent paper [2], D. Colton has given a new proof for the strong maximum principle with regard to the heat equation ut = Δu. His proof depends on the analyticity (in x) of solutions. For this reason it does not carry over to the equation

or to more general equations. But in order to tread mildly nonlinear equations such asut = Δu + f(u) which are important in many applications, it is essential to have the strong maximum principle at least for equation (*). It should also be said that this proof uses nontrivial facts about the heat equation.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 29 , Issue 1 , February 1986 , pp. 93 - 96
Copyright
Copyright © Edinburgh Mathematical Society 1986

References

REFERENCES

1.Besala, P., An extension of the strong maximum principle for parabolic equations, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astr. Phys. 19 (1971), 1003–1006.Google Scholar
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3.W., Walter, Differential- und Integral-Ungleichungen (Springer Tracts in Natural Philosophy, Springer-Verlag, 1964).Google Scholar
4.Walter, W., Differential and Integral Inequalities (Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 55, Springer-Varlag, 1970) (enlarged translation of [3]).CrossRefGoogle Scholar
5.Watson, N. A., The weak maximum principle for parabolic differential inequalities, Rend. Circ. Mat. Palermo, Serie II 33 (1984), 421–425.Google Scholar