The Cauchy problem is considered for certain equations with a boundary condition and an initial condition. A solution of the equations exists if and only if O <p <n+2/n. This paper deals with the question of existence (and uniqueness) when the initial data is a measure, for example a Dirac mass. Physically this corresponds to the important case when the initial temperature (or initial density etc. .) is extremely high near one point. The main novelty of this paper is to show that a solution exists only under some severe restrictions on the parameter P (or m); namely P must be less than n+2/n (m>n+2/n). For example, one striking conclusion reached is the fact that an equation possesses no solution in any dimension n> or = 1 and on any time interval (O, T). This result pinpoints the sharp contrast between linear and nonlinear equations from the point of view of existence.