Critical exponent for the semilinear wave equations with a damping increasing in the far field

We consider the Cauchy problem of the semilinear wave equation with a damping term {utt - Delta u + c(t, x)u(t) = vertical bar u(p), (t, x) is an element of (0, x) x R-N, u(0, x) = epsilon u(0)(x), u(t)(0, x) = epsilon u(1) (x), x is an element of R-N, where p > 1 and the coefficient of the damping term has the form c(t, x) = a(0)(1 + vertical bar x vertical bar(2))(-alpha/2) (1 + t)(-beta) with some a(0) > 0, alpha < 0, beta is an element of (- 1, 1]. In particular, we mainly consider the cases alpha < 0, beta = 0 or 0, beta = 1, which imply alpha + beta < 1, namely, the damping is spatially increasing and effective. Our aim is to prove that the critical exponent is given by p = 1+ 2/N - alpha This shows that the critical exponent is the same as that of the corresponding parabolic equation c(t, x)nu(t) - Delta(nu) = vertical bar nu vertical bar(p). The global existence part is proved by a weighted energy estimates with an exponential-type weight function and a special case of the Caffarelli-Kohn-Nirenberg inequality. The blow-up part is proved by a test-function method introduced by Ikeda and Sobajima [15]. We also give an upper estimate of the lifespan.