HIGHER-ORDER QUASILINEAR PARABOLIC EQUATIONS WITH SINGULAR INITIAL DATA
Victor A. Galaktionov, Andrey Evgenievich Shishkov · Communications in Contemporary Mathematics · 2006
As a basic model, we study the 2mth-order quasilinear parabolic equation of diffusion-absorption type [Formula: see text] where Δm,pis the 2mth-order p-Laplacian [Formula: see text]. We consider the Cauchy problem in RN× R+with arbitrary singular initial data u0≠ 0 such that u0(x) = 0 for any x ≠ 0. We prove that, in the most delicate case p = q and [Formula: see text], this Cauchy problem admits the unique trivial solution u(·, t) = 0 for t > 0. For λ < λ0, such nontrivial very singular solutions are known to exist for some semilinear higher-order models. This extends the well-known result by Brezis and Friedman established in 1983 for the semilinear heat equation with p = q = m = 1.