The validity of the weak and strong comparison principles for degenerate parabolic partial differential equations with the p-Laplace operator p is investigated for p > 2. This problem is reduced to the comparison of the trivial solution (≡ 0, by hypothesis) with a nontrivial nonnegative solution u(x, t). The problem is closely related also to the question of uniqueness of a nonnegative solution via the weak comparison principle. In this article, realistic counterexamples to the uniqueness of a nonnegative solution, the weak comparison principle, and the strong maximum principle are constructed with a nonsmooth reaction function that satisfies neither a Lipschitz nor an Osgood standard "uniqueness" condition. Nonnegative multi-bump solutions with spatially disconnected compact supports and zero initial data are constructed between sub-and supersolutions that have supports of the same type.
We study extension ofp-trigonometric functionssinpandcospand ofp-hyperbolic functionssinhpandcoshpto complex domain. Our aim is to answer the question under what conditions onpthese functions satisfy well-known relations for usual trigonometric and hyperbolic functions, such as, for example,sin(z)=-i·sinhi·z. In particular, we prove in the paper that forp=6,10,14,…thep-trigonometric andp-hyperbolic functions satisfy very analogous relations as their classical counterparts. Our methods are based on the theory of differential equations in the complex domain using the Maclaurin series forp-trigonometric andp-hyperbolic functions.
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