Semilinear nonautonomous parabolic equations with unbounded coefficients in the linear part

We study the Cauchy problem for the semilinear nonautonomous parabolic equation $u_t=\mathcal{A}(t)u+\psi(t,u)$ in $[s,\tau]\times {{\mathbb R}^d}$, $\tau> s$, in the spaces $C_b([s, \tau]\times{{\mathbb R}^d})$ and in $L^p((s, \tau)\times{{\mathbb R}^d}, \nu)$. Here $\nu$ is a Borel measure defined via a tight evolution system of measures for the evolution operator $G(t,s)$ associated to the family of time depending second order uniformly elliptic operators $\mathcal{A}(t)$. Sufficient conditions for existence in the large and stability of the null solution are also given in both $C_b$ and $L^p$ contexts. The novelty with respect to the literature is that the coefficients of the operators $\mathcal{A}(t)$ are allowed to be unbounded

Remarks on Bessel beams, signals and superluminality

We address the question about the velocity of signals carried by Bessel beams wave packets propagating in vacuum and having well defined wavefronts in time. We find that this problem is analogous to that of propagation of usual plane wave packets within dispersive media and conclude that the signal velocity can not be superluminal.Comment: LaTeX, 16 pages, no figures. Completely revised version, accepted for publication in Physics Letters

Schauder theorems for Ornstein-Uhlenbeck equations in infinite dimension

We prove Schauder type estimates for stationary and evolution equations driven by the classical Ornstein-Uhlenbeck operator in a separable Banach space, endowed with a centered Gaussian measure

Traces of Sobolev functions on regular surfaces in infinite dimensions

In a Banach space $X$ endowed with a nondegenerate Gaussian measure, we consider Sobolev spaces of real functions defined in a sublevel set $O= \{x\in X:\;G(x) <0\}$ of a Sobolev nondegenerate function $G:X\mapsto \R$. We define the traces at $G^{-1}(0)$ of the elements of $W^{1,p}(O, \mu)$ for $p>1$, as elements of $L^1(G^{-1}(0), \rho)$ where $\rho$ is the surface measure of Feyel and de La Pradelle. The range of the trace operator is contained in $L^q(G^{-1}(0), \rho)$ for $1\leq q<p$ and even in $L^p(G^{-1}(0), \rho)$ under further assumptions. If $O$ is a suitable halfspace, the range is characterized as a sort of fractional Sobolev space at the boundary. An important consequence of the general theory is an integration by parts formula for Sobolev functions, which involves their traces at $G^{-1}(0)$

Differential regulation of expression of the multiple ADP/ATP translocase genes in human cells

The expression of the genes encoding the three isoforms of the human ADP/ATP translocase (T1, T2, and T3) has been investigated in cultured cell systems under different experimental conditions, using isoform- specific probes. In several human cell lines tested, i.e. HeLa, Hep3B, 143B, HL60, the T3 gene is expressed as a single 1300-nucleotide mRNA, whereas the T2 gene produces two species of mRNA, 1450 and 1600 nucleotides in size. These two species, which are present in HeLa cells in approximately equivalent amounts, were shown to derive from the use of two different polyadenylation signals. The gene for the muscle-specific isoform of ADP/ATP translocase, T1, was not found to be expressed in any of the cell lines investigated. The levels of T2 and T3 mRNAs in HeLa cells are differentially affected by the growth conditions. In fact, the T2 mRNA level remains relatively constant throughout the exponential and stationary phases, whereas the T3 mRNA level decreases progressively in the second half of the exponential phase and in the stationary phase down to less than 50%. This difference in quantitative behavior of the two mRNAs must reflect changes in their rates of synthesis, since their half-lives are very similar (t^1/2 = 5-6 h), with no significant growth-related differences. Treatment of HL60 cells with 12-O-tetradecanoylphorbol-13-acetate or retinoic acid, two agents which induce cessation of cell proliferation and cell differentiation, resulted in a marked decrease in both T2 and T3 mRNA levels. Exposure of HeLa cells to chloramphenicol produced a pronounced decrease in the levels of both T2 and T3 mRNAs after 48 to 72 h of treatment. Half-life time measurements strongly suggested that this decrease reflected a reduction in the rate of synthesis of the two transcripts. Treatment of HeLa cells with dinitrophenol also produced a dramatic decrease in the steady state levels of both T2 and T3 mRNA, which, however, in contrast to the just mentioned situation, could be accounted for by a decrease in their metabolic stability. Control experiments indicated that the chloramphenicol- and dinitrophenol-induced changes were not a nonspecific consequence of mitochondrial dysfunction. The observations reported here clearly demonstrate that the expression of the multiple ADP/ATP translocase genes in human cells is sensitive to the cell physiological conditions, responding to the varying cellular demands by changes in the rate of synthesis or stability of their mRNAs

Maximal L2L^2 regularity for Dirichlet problems in Hilbert spaces

We consider the Dirichlet problem $\lambda U - {\mathcal{L}}U= F$ in \mathcal{O}, U=0 on $\partial \mathcal{O}$. Here $F\in L^2(\mathcal{O}, \mu)$ where $\mu$ is a nondegenerate centered Gaussian measure in a Hilbert space $X$, $\mathcal{L}$ is an Ornstein-Uhlenbeck operator, and $\mathcal{O}$ is an open set in $X$ with good boundary. We address the problem whether the weak solution $U$ belongs to the Sobolev space $W^{2,2}(\mathcal{O}, \mu)$. It is well known that the question has positive answer if $\mathcal{O} = X$; if $\mathcal{O} \neq X$ we give a sufficient condition in terms of geometric properties of the boundary $\partial \mathcal{O}$. The results are quite different with respect to the finite dimensional case, for instance if \mathcal{O} is the ball centered at the origin with radius $r$ we prove that $U\in W^{2,2}(\mathcal{O}, \mu)$ only for small $r$

A note on Duffin-Kemmer-Petiau equation in (1+1) space-time dimensions

In the last years several papers addressed the supposed spin-1 sector of the massive Duffin-Kemmer-Petiau (DKP) equation restricted to (1+1) space-time dimensions. In this note we show explicitly that this is a misleading approach, since the DKP algebra in (1+1) dimensions admits only a spin-0 representation. Our result also is useful to understand why several recent papers found coincident results for both spin-0 and spin-1 sectors of the DKP theory in (3+1) dimensions when the dynamics is restricted to one space dimension.Comment: 3 pages, no figure