The influence of data regularity in the critical exponent for a class of semilinear evolutions equations (1910.01823v1)

Abstract: In this paper we find the critical exponent for the global existence (in time) of small data solutions to the Cauchy problem for the semilinear dissipative evolution equations % [ u_{tt}+(-\Delta)^\delta u_{tt}+(-\Delta)^\alpha u+(-\Delta)^\theta u_t=|u_t|^p, \quad t\geq 0,\,\, x\in\R^n,] % with $p>1$, $2\theta \in [0, \alpha]$ and $\delta \in (\theta,\alpha]$. We show that, under additional regularity $\left(H^{\alpha+\delta}(\R^n)\cap L^{m}(\R^n) \right)\times \left(H^{2\delta}(\R^n)\cap L^{m}(\R^n)\right) $ for initial data, with $m\in (1,2]$, the critical exponent is given by $p_c=1+\frac{2m\theta}{n}$. The nonexistence of global solutions in the subcritical cases is proved, in the case of integers parameters $\alpha, \delta, \theta$, by using the test function method (under suitable sign assumptions on the initial data).