We consider a Cauchy problem $$y^{\prime }\left(x\right)+y^2\left(x\right)=q\left(x\right),{\left.y\left(x\right)\right|}_{x=x_0}=y_0$$ where $x_0$ , $y_0\in \mathbb{R}$ and $q\left(x\right)\in L_1^{\text{loc}}\left(R\right)$ is a non-negative function satisfying the condition: $${\int }_{-\mathrm{\infty }}^{x}q\left(t\right)dt>0,{\int }_x^{\mathrm{\infty }}q\left(t\right)dt>0\text{ for }x\in \mathbb{R}.$$ We obtain the conditions under which $y\left(x\right)$ can be continued to all of $\mathbb{R}$. This depends on $x_0$ , $y_0$ and the properties of $q\left(x\right)$.

We prove a characterisation of sets with finite perimeter and $BV$ functions in terms of the short time behaviour of the heat semigroup in ${\mathbf{R}}^N$. For sets with smooth boundary a more precise result is shown.

In this paper, we study the relationship between the long time behavior of a solution $u(t,x)$ of the nonlinear heat equation $u_t-\Delta u+{\left|u\right|}^{\alpha }u=0$ on ${ℝ}^N$ (where $\alpha \>0$) and the asymptotic behavior as $\left|x\right|\to \infty$ of its initial value $u_0$. In particular, we show that if the sequence of dilations ${\lambda }_n^{2/\alpha }{u}_0({\lambda }_n·)$ converges weakly to $z(·)$ as ${\lambda }_n\to \infty$, then the rescaled solution $t^{1/\alpha }u(t,·\sqrt{t})$ converges uniformly on ${ℝ}^N$ to $𝒰\left(1\right)z$ along the subsequence $t_n={\lambda }_n^2$, where $𝒰\left(t\right)$ is an appropriate flow. Moreover, we show there exists an initial value $U_0$ such that the set of all possible $z$ attainable...

Let G be a group generated by r elements $g_1,\dots ,g_r$. Among the reduced words in $g_1,\dots ,g_r$ of length n some, say ${\gamma }_n$, represent the identity element of the group G. It has been shown in a combinatorial way that the 2nth root of ${\gamma }_{2n}$ has a limit, called the cogrowth exponent with respect to the generators $g_1,\dots ,g_r$. We show by analytic methods that the numbers ${\gamma }_n$ vary regularly, i.e. the ratio ${\gamma }_{2n+2}/{\gamma }_{2n}$ is also convergent. Moreover, we derive new precise information on the domain of holomorphy of γ(z), the generating function...

Let $R\left(\lambda \right)$ be the error term in Weyl’s law for a 3-dimensional Riemannian Heisenberg manifold. We prove that ${\int }_1^T{\left|R\left(t\right)\right|}^{2}dt=c{T}^{\frac{5}{2}}+O_{\delta }\left(T^{\frac{9}{4}+\delta }\right)$, where $c$ is a specific nonzero constant and $\delta$ is an arbitrary small positive number. This is consistent with the conjecture of Petridis and Toth stating that $R\left(t\right)=O_{\delta }\left(t^{\frac{3}{4}+\delta }\right)$.The idea of the proof is to use the Poisson summation formula to write the error term in a form which can be estimated by the method of the stationary phase. The similar result will be also proven in the $2n+1$-dimensional case. ...