Blow-up solutions of quasilinear hyperbolic equations with critical Sobolev exponent (Q2896309)

In this article the Cauchy problem for the following \(p\)-wave equation is considered NEWLINE\[NEWLINE \begin{aligned} &u_{tt}-\Delta_pu=|u|^{p_*-2}u\quad {\text{ in}}\quad \mathbb R^N\times \mathbb R,\\ &u(x, 0)=u_0(x)\in {\overset{\cdot}{W}}^{1, p}(\mathbb R^N)\cap L^2(\mathbb R^N),\quad u_t(x, 0)=u_1(x)\in L^p(\mathbb R^N), \end{aligned} NEWLINE\]NEWLINE where \(N\geq 3\), \(2<p<N\), \(\Delta_pu={\text{ div}}\Bigl(|\nabla u|^{p-2}\nabla u\Bigr)\) is the \(p\)-Laplacian and \(p_*={{pN}\over {N-p}}\) is the critical Sobolev exponent.NEWLINENEWLINEThe authors prove that there exists a solution \(u\in {\mathcal C}^0(I, \overset{\cdot}{W}^{1, p}(\mathbb R^N))\bigcap {\mathcal C}^1(I, L^p(\mathbb R^N))\) of the considered problem which is defined on the maximal finite time interval \(I\) and satisfying \(E((u_0, u_1))< E((w_p, 0))\), \(K(u_0)<0\). Here \(w_p(x)\) is the ground state defined as NEWLINE\[NEWLINE\begin{multlined} w_p(x)=\Bigl(1+{{p-1}\over {(N-p)N^{{1\over {p-1}}}}}|x|^{{p\over {p-1}}}\Bigr)^{{p-N}\over p}, \\ E(u) ={1\over 2} \int_{\mathbb R^N}|u_1|^2dx+{1\over p}\int_{\mathbb R^N}|\nabla u_0|^pdx-{1\over {p_*}}\int_{\mathbb R^N}|u_0|^{p_*}dx, K(u)=\int_{\mathbb R^N}|\nabla u|^pdx -\int_{\mathbb R^N}|u|^{p_*}dx.\end{multlined}NEWLINE\]