Sign-changing solutions of a quasilinear heat equation with a source term
2013
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 $m = 1$.
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