\(L^p-L^q\) estimates for the wave equation with a time-dependent potential (Q1568369)

The authors try to investigate the Cauchy problem \[ \begin{cases} (\square+ m)u(t,x)= 0,\\ u(0,x)= 0,\\ \partial_t u(0,x)= f(x)\end{cases} \] under possibly weak conditions on the potential \(m\). The assumptions on \(m(t,x)\) are that \(m(t,x)\geq 0\), the derivative \(\partial_t m(t,x)\) exists and there exist nonnegative functions \(\mu_-(t)\), \(\mu_+(t)\in L^1_{\text{loc}}(\mathbb{R}_+)\) such that the inequality \(-\mu_-(t) m(t,x)\leq \partial_t m(t,x)\leq \mu_+(t) m(t,x)\) holds. The authors show that for exponents \(p\), \(q\) with \((1/p,1/q)\) belonging to some trapezoid a unique weak solution exists and the inequality \[ \|u(t)\|_q\leq C_{pq}(t) t^{1- n(1/p- 1/q)}\|f\|_p \] holds, where \(C_{pq}(t)\) is a function of \(t\) and an explicite expression of its dependence on \(m\) and \(\mu_\pm\) is given.