Time--space white noise eliminates global solutions in reaction diffusion equations

We prove that perturbing the reaction--diffusion equation $u_t=u_{xx} + (u_+)^p$ ($p>1$), with time--space white noise produces that solutions explodes with probability one for every initial datum, opposite to the deterministic model where a positive stationary solution exists.Comment: New results included. To be published in Physica

Analysis for time discrete approximations of blow-up solutions of semilinear parabolic equations

We prove a posteriori error estimates for time discrete approximations, for semilinear parabolic equations with solutions that might blow up in finite time. In particular we consider the backward Euler and the Crank–Nicolson methods. The main tools that are used in the analysis are the reconstruction technique and energy methods combined with appropriate fixed point arguments. The final estimates we derive are conditional and lead to error control near the blow up time

Totally discrete explicit and semi-implicit Euler methods for a blow-up problem in several space dimensions

The equation u t =Δu+u p with homogeneous Dirichlet boundary conditions has solutions with blow-up if p>1. An adaptive time-step procedure is given to reproduce the asymptotic behavior of the solutions in the numerical approximations. We prove that the numerical methods reproduce the blow-up cases, the blow-up rate and the blow-up time. We also localize the numerical blow-up set.Fil: Groisman, Pablo Jose. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin

TOTALLY DISCRETE EXPLICIT AND SEMI-IMPLICIT EULER METHODS FOR A BLOW-UP PROBLEM IN SEVERAL SPACE DIMENSIONS.

Abstract. The equation ut = ∆u + u p with homegeneous Dirichlet boundary conditions has solutions with blow-up if p> 1. An adaptive time-step procedure is given to reproduce the asymptotic behvior of the solutions in the numerical approximations. We prove that the numerical method reproduces the blow-up cases, the blow-up rate and the blow-up time. We also localize the numerical blow-up set. 1. Introduction. We study the behavior of an adaptive time step procedure for the following parabolic proble