Self-similar extinction for a diffusive Hamilton-Jacobi equation with critical absorption

International audienceThe behavior near the extinction time is identified for non-negative solutions to the diffusive Hamilton-Jacobi equation with critical gradient absorption ∂_t u − ∆_p u + |∇u|^{p−1} = 0 in (0, ∞) × R^N , and fast diffusion 2N/(N + 1) < p < 2. Given a non-negative and radially symmetric initial condition with a non-increasing profile which decays sufficiently fast as |x| → ∞, it is shown that the corresponding solution u to the above equation approaches a uniquely determined separate variable solution of the form U (t, x) = (T_e − t)^{1/(2−p)} f_* (|x|), (t, x) ∈ (0, T_e) × R^N , as t → T_e , where T_e denotes the finite extinction time of u. A cornerstone of the convergence proof is an underlying variational structure of the equation. Also, the selected profile f_* is the unique non-negative solution to a second order ordinary differential equation which decays exponentially at infinity. A complete classification of solutions to this equation is provided, thereby describing all separate variable solutions of the original equation. One important difficulty in the uniqueness proof is that no monotonicity argument seems to be available and it is overcome by the construction of an appropriate Pohozaev functional

Mating alters gene expression patterns in Drosophila melanogaster male heads

<p>Abstract</p> <p>Background</p> <p>Behavior is a complex process resulting from the integration of genetic and environmental information. <it>Drosophila melanogaster </it>rely on multiple sensory modalities for reproductive success, and mating causes physiological changes in both sexes that affect reproductive output or behavior. Some of these effects are likely mediated by changes in gene expression. Courtship and mating alter female transcript profiles, but it is not known how mating affects male gene expression.</p> <p>Results</p> <p>We used <it>Drosophila </it>genome arrays to identify changes in gene expression profiles that occur in mated male heads. Forty-seven genes differed between mated and control heads 2 hrs post mating. Many mating-responsive genes are highly expressed in non-neural head tissues, including an adipose tissue called the fat body. One fat body-enriched gene, <it>female-specific independent of transformer </it>(<it>fit</it>), is a downstream target of the somatic sex-determination hierarchy, a genetic pathway that regulates <it>Drosophila</it> reproductive behaviors as well as expression of some fat-expressed genes; three other mating-responsive loci are also downstream components of this pathway. Another mating-responsive gene expressed in fat, <it>Juvenile hormone esterase </it>(<it>Jhe</it>), is necessary for robust male courtship behavior and mating success.</p> <p>Conclusions</p> <p>Our study demonstrates that mating causes changes in male head gene expression profiles and supports an increasing body of work implicating adipose signaling in behavior modulation. Since several mating-induced genes are sex-determination hierarchy target genes, additional mating-responsive loci may be downstream components of this pathway as well.</p