Characterization of Turing diffusion-driven instability on evolving domains

In this paper we establish a general theoretical framework for Turing diffusion-driven instability for reaction-diffusion systems on time-dependent evolving domains. The main result is that Turing diffusion-driven instability for reaction-diffusion systems on evolving domains is characterised by Lyapunov exponents of the evolution family associated with the linearised system (obtained by linearising the original system along a spatially independent solution). This framework allows for the inclusion of the analysis of the long-time behavior of the solutions of reaction-diffusion systems. Applications to two special types of evolving domains are considered: (i) time-dependent domains which evolve to a final limiting fixed domain and (ii) time-dependent domains which are eventually time periodic. Reaction-diffusion systems have been widely proposed as plausible mechanisms for pattern formation in morphogenesis

Photoinduced Excited State Electron Transfer at Liquid/Liquid Interfaces

Several aspects of the photoinduced electron transfer (ET) reaction betweencoumarin 314 (C314) and N,N-dimethylaniline (DMA) at the water/DMA interface areinvestigated by molecular dynamics simulations. New DMA and water/DMA potentialenergy surfaces are developed and used to characterize the neat water/DMA interface.The adsorption free energy, the rotational dynamics and the solvation dynamics of C314at the liquid/liquid interface are investigated and are generally in reasonable agreementwith available experimental data. The solvent free energy curves for the ET reactionbetween excited C314 and DMA molecules are calculated and compared with thosecalculated for a simple point charge model of the solute. It is found that thereorganization free energy is very small when the full molecular description of the soluteis taken into account. An estimate of the ET rate constant is in reasonable agreement withexperiment. Our calculations suggest that the polarity of the surface “reported” by thesolute, as reflected by solvation dynamics and the reorganization free energy, is strongly solute-dependent

Vector-valued Lizorkin-Triebel spaces and sharp trace theory for functions in Sobolev spaces with mixed Lp-norm for parabolic problems

The trace problem on the hypersurface yn = 0 is investigated for a function u = u(y,t) is an element of L-q(0,T;W-p(m)(R-+(n))) with atu partial derivative(t)u is an element of L-q(0,T;L-p(R-+(n))), that is, Sobolev spaces with mixed Lebesgue norm Lp,q (R-+(n) x (0, T)) = Lq (0,T; L-p(R-+(n))) are considered; here p = (p(1),...,p(n)) is a vector and R-+(n) = Rn-1 x (0, infinity). Such function spaces are useful in the context of parabolic equations. They allow, in particular, different exponents of summability in space and time. It is shown that the sharp regularity of the trace in the time variable is characterized by the Lizorkin-Triebel space F-q,pn(1-1)/((pnma))(0,T; L (p) over tilde $ (Rn-1)), p = ((p) over tilde ,p(n)). A similar result is established for first order spatial derivatives of u. These results allow one to determine the exact spaces for the data in the inhomogeneous Dirichlet and Neumann problems for parabolic equations of the second order if the solution is in the space L-q(0,T; W-p(2)(Omega)) boolean AND W-q(1)(0,T; L-p(Omega)) with p <= q