Decay of mass for the equation ut = Δu-a(x) up |delu|q

Consider the quasilinear Cauchy problem u(t) = Delta(u) - u(x) u(P) \del u\(q), x is an element of R-d, t > 0 u(x, 0) = phi(x) greater than or equal to 0, x is an element of R-d, where a > 0, p and q satisfy p greater than or equal to 0 and q greater than or equal to 1 or p > 1 and q = 0, and 0 less than or equal to phi is an element of L-1(R-d) boolean AND C-b(3,alpha)(R-d). This paper proves that the above equation possesses a unique positive classical solution and then investigates whether or not gamma = lim(t --> infinity) integral R-d u(x, t) dx = 0. In particular, it is shown that if a is on the order \x\(m) for large \x\, then gamma = 0 if dp + (d + 1) q less than or equal to d + 2 + m. Under the assumption that for compactly supported phi. \\del u(epsilon phi)(., t)\\(infinity) less than or equal to epsilon c phi/(t + 1)(beta), for some beta greater than or equal to 0, and for epsilon is an element of(0,1], (*) where u(epsilon phi) denotes the solution to the above equation with initial condition epsilon phi, it is shown that gamma > 0 if dp + 2 beta q > d + 2 + max(m, -d). For a certain range of the parameters d, p, q, m, it is proved that (*) holds with beta = (d + 1)/2, and for many other parameter values it is proved that (*) holds with beta = d/2. Note that if beta =(d + 1)/2, then the above condition for gamma > 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. (C) 2000 Academic Press.