First and second critical exponents for an inhomogeneous damped wave equation with mixed nonlinearities

Abstract

We investigate the Cauchy problem for the nonlinear damped wave equation $u_{tt}-\Delta u +u_t = |u|^p+|\nabla u|^q +w(x)$, where $N\geq 1$, $p, q>1$, $w\in L^1_{loc}(\mathbb{R}^N)$, $w\geq 0$ and $w\not\equiv 0$. Namely, we first obtain the Fujita critical exponent for the considered problem. Next, we determine its second critical exponent in the sense of Lee and Ni. In particular, we show that the nonlinear gradient term $|\nabla u|^q$ induces a phenomenon of discontinuity of the Fujita critical exponent.