Existence of solution for the \(n\)-dimension second order semilinear hyperbolic equations (Q541269)

In this article is considered the Cauchy problem \[ u_{tt}-|t|^m\Delta u=f(t, x, u),\qquad t<0,\quad x\in\mathbb R^n, \] \[ u(0, x)=\varphi_1(x),\quad u_t(0, x)=\varphi_2(x),\quad x\in\mathbb R^n, \] where \(m<0\), \(f(t, x, u)\) is a smooth function of its arguments with compact support with respect to \(x\), and \(|\partial_u^j f(t, x, u)|\leq C(-t)^{\alpha}|u|^{k-j}\) for \(0\leq j\leq k\), \(|\partial_u^j f(t, x, u)|\leq C(-t)^{\alpha}\) for \(j\geq k\), \(\alpha=0\) for small \(|t|\). For large \(|t|\) and \(m\leq -4\) the parameter \(\alpha\) satisfies the inequality \(4(1+\alpha)-(k+1)m\leq 0\). For large \(|t|\) and \(m\in (-4, 0)\backslash\{-2\}\) the parameter \(\alpha\) satisfies the inequality \(4(1+\alpha+k)-m\leq 0\), \(k\in \mathbb N^+\). Moreover, \(\|\varphi_1\|_{H^{s-\frac{m}{2(m+2)}}}+ \|\varphi_2\|_{H^{s-\frac{m+4}{2(m+2)}}}\leq \varepsilon_0\) for small \(\varepsilon_0>0\) and \(s>\frac n2\). The author proves that the considered Cauchy problem has a unique solution \(u\in {\mathcal C}((-\infty, 0], H^s(\mathbb R^n))\cap {\mathcal C}^1((-\infty, 0), H^{s-1}(\mathbb R^n))\) for \(m\in (-\infty, 0)\backslash\{-2\}\).