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.
The weak and strong comparison principles (WCP and SCP, respectively) are investigated for quasilinear elliptic boundary value problems with the p-Laplacian in one space dimension, ∆ p (u) def = d dx |u | p−2 u . We treat the "degenerate" case of 2 < p < ∞ and allow also for the nontrivial convection velocity b : [−1, 1] → R in the underlying domain Ω = (−1, 1). We establish the WCP under a rather general, "natural sufficient condition" on the convection velocity, b(x), and the reaction function, ϕ(x, u). Furthermore, we establish also the SCP under a number of various additional hypotheses. In contrast, with these hypotheses being violated, we construct also a few rather natural counterexamples to the SCP and discuss their applications to an interesting classical problem of fluid flow in porous medium, "seepage flow of fluids in inclined bed". Our methods are based on a mixture of classical and new techniques.
We study extension of p-trigonometric functions sinp and cosp to complex domain. For p = 4, 6, 8,. . ., the function sinp satisfies the initial value problem which is equivalent to (*) −(u ′) p−2 u ′′ − u p−1 = 0, u(0) = 0, u ′ (0) = 1 in R. In our recent paper, Girg, Kotrla (2014), we showed that sinp(x) is a real analytic function for p = 4, 6, 8,. .. on (−πp/2, πp/2), where πp/2 = 1 0 (1 − s p) −1/p. This allows us to extend sinp to complex domain by its Maclaurin series convergent on the disc {z ∈ C : |z| < πp/2}. The question is whether this extensions sinp(z) satisfies (*) in the sense of differential equations in complex domain. This interesting question was posed by Došlý and we show that the answer is affirmative. We also discuss the difficulties concerning the extension of sinp to complex domain for p = 3, 5, 7,. .. Moreover, we show that the structure of the complex valued initial value problem (*) does not allow entire solutions for any p ∈ N, p > 2. Finally, we provide some graphs of real and imaginary parts of sinp(z) and suggest some new conjectures.
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