Abstract :
Consider the quasilinear Cauchy problemut=Δu−a(x) up uq, x Rd, t>0u(x, 0)=φ(x) 0, x Rd,where a>0, p and q satisfyp 0 and q 1 or p>1 and q=0,and0 φ L1(Rd)∩C3, αb(Rd).This paper proves that the above equation possesses a unique positive classical solution and then investigates whether or not γ≡limt→∞ ∫Rd u(x, t) dx=0. In particular, it is shown that if a is on the order xm for large x, then γ=0 if dp+(d+1) q d+2+m. Under the assumption that for compactly supported φ, where u φ denotes the solution to the above equation with initial condition φ, it is shown that γ>0 if dp+2βq>d+2+max(m, −d). For a certain range of the parameters d, p, q, m, it is proved that (*) holds with β=(d+1)/2, and for many other parameter values it is proved that (*) holds with β=d/2. Note that if β=(d+1)/2, then the above condition for γ>0 becomes dp+(d+1) q>d+2+max(m, −d), which in light of the parameter restrictions is equivalent to dp+(d+1) q>d+2+m.
