Shadow limit using Renormalization Group method and Center Manifold method

International audienceMulti-scale analysis and biological applications are two subjects of focus in Willi's research over his professional life. Abstract We study a shadow limit (the infinite diffusion coefficient-limit) of a system of ODEs coupled with a semilinear heat equation in a bounded domain with Neumann boundary conditions. In the literature, it was established formally that in the limit, the original semilinear heat equation reduces to an ODE involving the space averages of the solution to the semilinear heat equation and of the nonlinearity. It is coupled with the original system of ODEs for every space point x .We present derivation of the limit using the renormalization group (RG) and the center manifold approaches. The RG approach provides also further approximating expansion terms. The error estimate in the terms of the inverse of the diffusion coefficient is obtained for the finite time intervals. For the infinite times, the center manifolds for the starting problem and for its shadow limit approximation are compared and it is proved that their distance is of the order of the inverse of the diffusion coefficient

Regularity of the density for the stochastic heat equation

We study the smoothness of the density of a semilinear heat equation with multiplicative spacetime white noise. Using Malliavin calculus, we reduce the problem to a question of negative moments of solutions of a linear heat equation with multiplicative white noise. Then we settle this question by proving that solutions to the linear equation have negative moments of all orders

Blow-up Time for a Semilinear Heat Equation with a Gradient Term

We consider the semilinear Heat equation with a gradient term, which takes the form....... The full paper can be donwload using the link below

Homogenization of a semilinear heat equation

We consider the homogenization of a semilinear heat equation with vanishing viscosity and with oscillating positive potential depending on $u/\varepsilon$. According to the rate between the frequency of oscillations in the potential and the vanishing factor in the viscosity, we obtain different regimes in the limit evolution and we discuss the locally uniform convergence of the solutions to the effective problem. The interesting feature of the model is that in the strong diffusion regime the effective operator is discontinuous in the gradient entry. We get a complete characterization of the limit solution in dimension $n=1$, whereas in dimension $n>1$ we discuss the main properties of the solutions to the effective problem selected at the limit and we prove uniqueness for some classes of initial data