Sign-changing solutions of a quasilinear heat equation with a source term

Abstract

The Cauchy problem of a heat equation with a source term $$ \psi_t=\Delta \left(|\psi|^{m-1}\psi\right)+|\psi|^{\gamma-1}\psi\ \ \mbox{in}\ \ (0, \infty)\times R^n $$ is considered, where $\gamma>m>1$. We are interested in global solutions with H$\ddot{o}$lder continuity which satisfy the equation in the distribution sense, and with a fixed number of sign changes at any given time $ t > 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| \rightarrow \infty$ with an algebraic rate determined uniquely by $ n, m$ and $\gamma$. Our results extend previous study on positive self-similar solutions. Moreover, they demonstrate vital difference from the well-studied semi-linear case of $m = 1$.