arXiv subjects
Publications and source records attributed to Renato Manfrin.
We show here that the quasilinear Kirchhoff-Pokhozhaev equation $$u_{tt}-\big(a\int_{\mathbb{R}^n} |\nabla u |^2 dx + b \big)^{-2} \Delta u = 0,$$ with $a\neq0$, admits conservation laws of all orders.
We prove that the special Kirchhoff equation studied by Pokhozhaev admits a third-order conservation law. We further show that if the energy of the solution is sufficiently small, then the $L^2$-norms of the derivatives up to third order of the solution remain uniformly bounded with respect to time.
We prove a second order identity for the Kirchhoff equation which yields, in particular, a simple and direct proof of Pokhozhaev's second order conservation law when the nonlinearity has the special form $(C_1 s +C_2)^{-2}$. As applications, we give: an estimate of order $\varepsilon^{-4}$ for the lifespan $T_\varepsilon$ of the solution of the Cauchy problem with initial data of size $\varepsilon$ in Sobolev spaces when the nonlinearity is given by any $C^2$ function $m(s)>0$; a necessary and sufficient condition for boundedness of a second order energy of the solutions.