The influence of data regularity in the critical exponent for a class of semilinear evolution equations

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 utt+(-Δ)δutt+(-Δ)αu+(-Δ)θut=|ut|p,t≥0,x∈Rn,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} u_{tt}+(-\Delta )^\delta u_{tt}+(-\Delta )^\alpha u+(-\Delta )^\theta u_t=|u_t|^p, \quad t\ge 0,\,\, x\in {\mathbb {R}}^n, \end{aligned}$$\end{document}with p>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>1$$\end{document}, 2θ∈[0,α]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\theta \in [0, \alpha ]$$\end{document} and δ∈(θ,α]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta \in (\theta ,\alpha ]$$\end{document}. We show that, under additional regularity Hα+δ(Rn)∩Lm(Rn)×H2δ(Rn)∩Lm(Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( H^{\alpha +\delta }({\mathbb {R}}^n)\cap L^{m}({\mathbb {R}}^n) \right) \times \left( H^{2\delta }({\mathbb {R}}^n)\cap L^{m}({\mathbb {R}}^n)\right) $$\end{document} for initial data, with m∈(1,2]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\in (1,2]$$\end{document}, the critical exponent is given by pc=1+2mθn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_c=1+\frac{2m\theta }{n}$$\end{document}. The nonexistence of global solutions in the subcritical cases is proved, in the case of integers parameters α,δ,θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha , \delta , \theta $$\end{document}, by using the test function method (under suitable sign assumptions on the initial data).