Global small data solutions for semilinear waves with two dissipative terms

In this work, we prove the existence of global (in time) small data solutions for wave equations with two dissipative terms and with power nonlinearity $$|u|^p$$ | u | p or nonlinearity of derivative type $$|u_t|^p$$ | u t | p , in any space dimension $$n\geqslant 1$$ n ⩾ 1 , for supercritical powers $$p>{\bar{p}}$$ p > p ¯ . The presence of two dissipative terms strongly influences the nature of the problem, allowing us to derive $$L^r-L^q$$ L r - L q long time decay estimates for the solution in the full range $$1\leqslant r\leqslant q\leqslant \infty $$ 1 ⩽ r ⩽ q ⩽ ∞ . The optimality of the critical exponents is guaranteed by a nonexistence result for subcritical powers $$p<{\bar{p}}$$ p < p ¯ .