= Similarity reduction for a cubic absorption equation with dilating drift
{title2=$u_t=u_{xx}-\frac{x}{2t}u_x-\frac12u^3$}
The transformation $\tau=\log t$, $\xi=x/\sqrt t$, $v=\sqrt t\,u$ gives $v_\tau=v_{\xi\xi}+(v-v^3)/2$. Thus the stationary half-line profile $v=\tanh(\xi/2)$ yields $u=t^{-1/2}\tanh[x/(2\sqrt t)]$, with zero value at the origin and far-field value $t^{-1/2}$. For two bounded <classical solutions>, their difference has reaction coefficient $[1-(v_1^2+v_1v_2+v_2^2)]/2\leq1/2$. Multiplying it by $e^{-\tau/2}$ therefore produces a linear parabolic equation with nonnegative absorption. A positive growing quadratic spatial barrier and the <maximum principle> give uniqueness on the unbounded half-line. This proof identifies the finite-time boundedness or growth hypothesis needed at <spatial infinity> rather than assuming a supremum is automatically attained there.
Back to article page
