Critical Global Asymptotics in Higher-order Semilinear Parabolic Equations

We consider a higher-order semilinear parabolic equationut =−(−∆)mu−g(x,u) in RN×R+, m>1. The nonlinear term is homogeneous: g(x,su)≡ |s|P−1sg(x,u) and g(sx,u) ≡ |s|Qg(x,u) for any s ∈ R, with exponents P > 1, and Q > −2m. We also assume that g satisfies necessary coercivity and monotonicity conditions for global existence of solutions with sufficiently small initial data. The equation is invariant under a group of scaling transformations. We show that there exists a critical exponent P = 1+(2m+Q)/N such that the asymptotic behavior as t →∞ of a class of global small solutions is not group-invariant and is given by a logarithmic perturbation of the fundamental solution b(x,t)= t−N/2mf(xt−1/2m) of the parabolic operator ∂/∂t+(−∆)m, so that for t 1,u(x,t)=C0(lnt)−N/(2m+Q)[b(x,t) +o(1)], where C0 is a constant depending on m, N, and Q only.