We present a one-dimensional semilinear parabolic equation for which the spatial derivative of solutions becomes unbounded in finite time while the solutions themselves remain bounded. In our example the derivative blows up in the interior of the space interval rather than at the boundary, as in earlier examples. In the case of monotone solutions we show that gradient blow-up occurs at a single point, and we study the shape of the singularity. Our argument for gradient blow-up also provides a pair of "naive viscosity sub- and super-solutions" which violate the comparison principle.
Sigurd B. Angenent. Marek Fila. "Interior gradient blow-up in a semilinear parabolic equation." Differential Integral Equations 9 (5) 865 - 877, 1996. https://doi.org/10.57262/die/1367871520