  We consider non-local in time semilinear subdiffusion equations on a bounded
domain, where the kernel in the integro-differential operator belongs to a
large class, which covers many relevant cases from physics applications, in
particular the important case of fractional dynamics. The elliptic operator in
the equation is given in divergence form with bounded measurable coefficients.
We prove a well-posedness result in the setting of bounded weak solutions and
study the stability and instability of the zero function in the special case
where the nonlinearity vanishes at 0. We also establish a blowup result for
positive convex and superlinear nonlinearities.
