Adaptivity and blow-up detection for nonlinear evolution PDEs

I will review some recent work on the problem of reliable automatic detection of blow-up behaviour for nonlinear parabolic PDEs. The adaptive algorithms developed are based on rigorous conditional a posteriori error bounds. The use of space-time adaptivity is crucial in making the problem computationally tractable. The results presented are applicable to quite general spatial operators, rendering the approach potentially useful in informing respective PDE theory. The new adaptive algorithm is shown to accurately estimate the blow-up time of a number of problems, including ones exhibiting regional blow-up.