Oscillatory blow-up and gradient estimates for semilinear heat equations

For reaction-diffusion with blow-up nonlinearities, we consider the question whether the sup norm of any positive blow-up solution must be eventually monotone nondecreasing in time. While some sufficient conditions are known, especially for radial solutions, this natural and basic question for the blow-up theory does not seem to have been addressed so far in full generality. We construct surprising (nonradial) counter-examples of blow-up solutions with oscillatory $L^\infty$ norm, for any Sobolev supercritical power nonlinearity, which show that this property may fail. In addition, this provides examples of type II blow-up for any supercritical power, which considerably increases the known range of powers for which type II blow-up may occur. Moreover, whereas all the type II blow-up rates known so far were at most polynomial, the blow-up in our counter-examples can be arbitrarily singular. As a related question, we clarify the gradient estimates obtained and used in previous works. In particular we show that these estimates hold only at times when the $L^\infty$ norm is maximal with respect to the past.