Sign-Changing Solutions Of A Quasilinear Heat Equation With A Source Term

The Cauchy problem of a heat equation with a source term ψt = Δ(|ψ|m-1 ψ) + |ψ|γ-1 ψ in (0,∞) × Rn is considered, where γ \u3e m \u3e 1. We are interested in global solutions with Hölder continuity which satisfy the equation in the distribution sense, and with a fixed number of sign changes at any given time t \u3e 0. Through detailed analysis of the self-similarity problem, we prove the existence of two type of such solutions, one with compact support and the other decays to zero as |x| → ∞ with an algebraic rate determined uniquely by n,m and γ. Our results extend previous study on positive self-similar solutions. Moreover, they demonstrate vital difference from the well-studied semi-linear case of m = 1