Several negative results are presented concerning the solvability in Sobolev classes of the Cauchy problem for the inhomogeneous second-order uniformly parabolic equations without lower order terms in one space dimension. The main coefficient is assumed to be a bounded measurable function of $(t,x)$ bounded away from zero. We also discuss upper and lower estimates of certain kind on the fundamental solutions of such equations.
Reaction diffusion equations with super-linear absorption: universal bounds, uniqueness for the Cauchy problem, boundedness of stationary solutions
Uniqueness/nonuniqueness for nonnegative solutions of the Cauchy problem for $u_t=Δu-u^p$ in a punctured space
Well-posedness of the Cauchy problem for the fractional power dissipative equations
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