Volume 84, Nº 3 (2020)

We consider two model non-linear equations describing electric oscillations in systems with distributedparameters on the basis of diodes with non-linear characteristics. We obtain equivalent integral equations forclassical solutions of the Cauchy problem and the first and second initial-boundary value problemsfor the original equations in thehalf-space $x>0$. Using the contraction mapping principle, we prove the local-in-timesolubility of these problems. For one of these equations, we use the Pokhozhaev method of non-linear capacityto deduce a priori bounds giving rise to finite-time blow-up results and obtain upper bounds for the blow-uptime. For the other, we use a modification of Levine's method to obtain sufficient conditions for blow-upin the case of sufficiently large initial data and give a lower bound for the order of growth of a functionalwith the meaning of energy. We also obtain an upper bound for the blow-up time.

Let $k$ be a number field and $S$, $T$ sets of places of $k$. For each prime $p$, we define an invariant $\mathscr{G}=\mathscr{G}_p(k_\infty/k,S,T)$ related to the Galois group of the maximal abelian extension of $k$ which is unramified outside $S$ and splits completely in $T$. In the main theorem we interpret $\mathscr{G}$ in terms of another arithmetic object $\mathscr{U}$ that involves various unit groups and uses genus theory applied to certain modules,which are technically modified from idèle groups. We show that this interpretation is functorial with respect to $S$ and $T$ and thereby providesinteresting connections between $\mathscr{G}$ and $\mathscr{U}$ as $S$ and $T$ vary. The settings and methods are new, and different from the classical genus theoreticmethods for idèle groups. The advantage of the new methods at the finite level not only generalizes but also strengthens certain known results involving the maximal $p$-abelian profinite Galois groupof $k$ that is $S$-ramified and $T$-split in terms of the arithmetic of certain units of $k$. At the infinite level, the method relates the deep arithmeticof special units with those of profinite Galois groups. For example, for special cases of $S$ and $T$, the invariants $\mathscr{G}$ are related to the conjectures of Gross (or Kuz'min–Gross) and Leopoldtand accordingly, in these special cases, the functorial interpretation of $\mathscr{G}$ as $S$ and $T$ vary involves interestingconnections between the conjectures of Gross and Leopoldt in a simpler and more concrete way. As a result, we conjecture that $\mathscr{G}$ is finite for all finite disjoint sets$S$, $T$ over the cyclotomic $\mathbb{Z}_p$-tower of $k$, which includes the conjectures of Gross and Leopoldt as special cases.

We establish an asymptotic formula for the rate of approximation of Fourier series of individual periodic functions by linear averages with an error $\omega_{2m}(f;{1}/{n})$, $m\in\mathbb{N}$. This formula is applicable to the means of Riesz, Gauss–Weierstrass, Picard and others. The result is new even for the arithmetic means of partial Fourier sums. We use the formula to determine the asymptotic behaviour of functions in a certain class. Separately, we consider the case of positive integral convolution operators.