Infinite Fujita exponents for nonlinear diffusion equations with localized sources

This paper deals with Cauchy problem to nonlinear diffusion ut=Δum+λ1up1(x,t)+λ2up2(x∗(t),t) with m≥1m≥1, pi,λi≥0pi,λi≥0 (i=1,2i=1,2) and x∗(t)x∗(t) Hölder continuous. A new phenomenon is observed that the critical Fujita exponent pc=+∞pc=+∞ whenever λ2>0λ2>0. More precisely, the solution blows up under any nontrivial and nonnegative initial data for all p=max{p1,p2}∈(1,+∞)p=max{p1,p2}∈(1,+∞). This result is then extended to a coupled system with localized sources as well as the cases with other nonlinearities.