The Life Span of Classical Solutions to Nonlinear Wave Equations with Weighted Terms in Three Space Dimensions

The paper considers the Cauchy problem with small initial values for semilinear wave equations with weighted nonlinear terms. Similar to Strauss exponent p₀(n) which is the positive root of the quadratic equation \(1+{1\over 2}(n+1)p-{1\over 2}(n-1)p^{2}=0\) $$1+\frac{1}{2}(n+1)p-\frac{1}{2}(n-1)p^2=0$$ , we get smaller critical exponents p_m(n),p _m ^* (n) and have global existence in time when p>p_m(n) or p>p _m ^* (n). In addition, for the blow-up case, the introduction of the spacial weight shows the optimality of new upper and lower bound.