Nonexistence of weak solutions for some degenerate and singular hyperbolic problems on \(\mathbb{R}_+^{n+1}\) (Q2783080)

The model problem considered in this paper is the following NEWLINE\[NEWLINE\begin{cases} u_{tt}-|x|^\sigma\Delta u\geq |u|^q=0 &\text{in} {\mathbb R}^n\times (0,\infty),\\ u(x,0)=u_0(x) &\text{in} {\mathbb R}^n\\ u_t(x,0)=u_1(x) &\text{in} {\mathbb R}^n, \end{cases} \tag{1}NEWLINE\]NEWLINE where \(n\geq 1\), \(q>1\) and \(\sigma\leq 2\). By using a functional method introduced in the context of the elliptic equations in [Proc. Steklov Inst. Math. 227, 186-216 (1999); translation from Tr. Mat. Inst. Steklova 227, 192-222 (1999; Zbl 1056.35507) and Dokl. Math. 57, No. 2, 250-253 (1998); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 359, No. 4, 456-460 (1998; Zbl 0976.35100)] it is proved that, if \(\sigma<\min\{2, n\}\), \(q(2(n-1)-\sigma)<2(n+1)-3\sigma\) and \(u_1\) satisfies a positivity condition, then the problem (1) does not have any nontrivial (weak) solution. If \(\sigma =2\) and \(1<q\leq 3\) when \(n=1\) or \(n>2\) and if \(\sigma =2\) and \(1<q\) when \(n=2\), then the same conclusion follows. Some other results related to more general hyperbolic operators are given.NEWLINENEWLINEFor the entire collection see [Zbl 0981.00017].