The Strauss exponent for some k-evolution equation in the class of Boussinesq equations

Abstract

In this paper, we prove the existence of global small data solutions to the evolution equation$\{\begin{array}{cc}{v}_{tt}+A{v}_{tt}+Av+A^{2}v=Af\left(v\right),& t\ge 0,x\in R^n,\\ v(0,x)=v_0\left(x\right),v_t(0,x)=v_1\left(x\right),\end{array}$ where $A=F^{-1}\left(a{\left(\xi \right)}^2\right)$ with $a\left(\xi \right)$ homogeneous of order k, and $f\left(v\right)=|v{|}^{\alpha }$ or it is a more general power nonlinearity. We prove our result for $\alpha >\gamma \left(r\right)$, where γ is the Strauss exponent for nonlinear equations, and r is the rank of the Hessian of $a\left(\xi \right)$. We also consider the damped case, obtained adding $+A{v}_t$ to the left-hand side of the equation. We show that the effect of the dissipation is very weak, compared to the dispersion, however, it is sufficient to lower the existence exponent to some smaller, modified, Strauss exponent.