The Critical Exponent for Evolution Models with Power Non-linearity

In this note we derive Lr − Lq estimates for the solutions to the Cauchy problem $$\displaystyle u_{tt} +(-\varDelta )^{\sigma } u = 0\,, \qquad t\geq 0, \ x\in {\mathbb {R}}^n, \qquad u(0,x)=0, \;\; u_t(0,x)=g(x), $$ with σ > 1. Moreover, we derived the critical index pc(n) for the existence of global in time small data solutions to the associated semilinear Cauchy problem with power nonlinearity |u|p, p > 1.