554 research outputs found
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 CrankāNicolson 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
Renormalizing Partial Differential Equations
In this review paper, we explain how to apply Renormalization Group ideas to
the analysis of the long-time asymptotics of solutions of partial differential
equations. We illustrate the method on several examples of nonlinear parabolic
equations. We discuss many applications, including the stability of profiles
and fronts in the Ginzburg-Landau equation, anomalous scaling laws in
reaction-diffusion equations, and the shape of a solution near a blow-up point.Comment: 34 pages, Latex; [email protected]; [email protected]
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