Positive self-similar solutions of a parabolic equation with spatially weighted absorption in a half-space

Abstract

We study positive solutions of $u_t-Δu+x_1^{2α}|u|^{q-1}u=0$ in the half-space $\{x_1>0\}$, with $α>-1$ and $q>1$, vanishing on $\{x_1=0\}$ and at $t=0$ away from the origin. Let $q_c=(N+3+2α)/(N+1)$. For $1<q<q_c$ we prove that there is exactly one positive self-similar solution continuous away from the origin, and that its profile satisfies Gaussian bounds that are sharp up to multiplicative constants. For $q\ge q_c$ there is none. We also show that, after removing the Gaussian factor, the profile has an angular limit shape $Φ$ in every mode, give an integral formula for its Kummer coefficients, prove that $Φ$ is real-analytic and is the uniform limit of the rescaled profiles, that $Φ/(η_1/|η|)$ is monotone along meridians, and that $Φ$ is proportional to $η_1/|η|$ if and only if $2α+q=1$. Every self-similar solution of finite energy, or continuous away from the origin, is dominated in absolute value by the positive one, and for $(N+4+2α)/(N+2)\le q<q_c$ the only nontrivial ones are $\pm u_s$. For $α=0$ existence, uniqueness and the upper bound are due to Marcus and Véron; the lower bound with the sharp exponent and the description of $Φ$ are new also in that case. The existence and uniqueness proofs use a Brezis--Oswald type comparison principle and explicit barriers, given by explicit formulas; the analysis of $Φ$ uses Kummer functions.