Critical Non-linearity for some Evolution Equations with Fujita-type Critical Exponent

We consider the Cauchy problem for a class of non-linear evolution equations in the form (Formula presented.) here, L(∂t,∂x) is a linear partial differential operator with constant coefficients, of order m≥1 with respect to the time variable t, and ℓ is a natural number satisfying 0≤ℓ≤m-1. For several different choices of L, many authors have investigated the existence of global (in time) solutions to this problem when F(s)=|s|p is a power non-linearity, looking for a critical exponentpc>1 such that global small data solutions exist in the supercritical case p>pc, whereas no global weak solutions exist, under suitable sign assumptions on the data, in the subcritical case 1