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On the strong maximum principle for parabolic differential equations

  • Wolfgang Walter (a1)

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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.

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References

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1.Besala, P., An extension of the strong maximum principle for parabolic equations, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astr. Phys. 19 (1971), 10031006.
2.Colton, D., The strong maximum principle for the heat equation, Proc. Edinburgh Math. Soc. 27 (1984), 297299.
3.W., Walter, Differential- und Integral-Ungleichungen (Springer Tracts in Natural Philosophy, Springer-Verlag, 1964).
4.Walter, W., Differential and Integral Inequalities (Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 55, Springer-Varlag, 1970) (enlarged translation of [3]).
5.Watson, N. A., The weak maximum principle for parabolic differential inequalities, Rend. Circ. Mat. Palermo, Serie II 33 (1984), 421425.

On the strong maximum principle for parabolic differential equations

  • Wolfgang Walter (a1)

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