Blow-up for a porous medium equation with a localized source

摘要

In this paper we investigate the blow-up properties of the positive solutions to a localized porous medium equation vτ=Δvm+avp1vq1(x0,τ) subject to homogeneous Dirichlet condition and positive initial datum v0(x). Under appropriate hypotheses, we establish the local existence and obtain that in the case of p1+q1m, the solution of the above problem blows up for large initial datum while it admits a global solution for small initial datum. Moreover, for the special case p1=0, q1>m and a is large, under an additional hypothesis on the initial datum, we can also obtain the asymptotic behavior of the blow-up solution.