Extinction for a Singular Diffusion Equation with Strong Gradient Absorption Revisited

When <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>N</m:mi> </m:mrow> <m:mo>/</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>N</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo><</m:mo> <m:mi>p</m:mi> <m:mo><</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> {2N/(N+1)<p<2} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>0</m:mn> <m:mo><</m:mo> <m:mi>q</m:mi> <m:mo><</m:mo> <m:mrow> <m:mi>p</m:mi> <m:mo>/</m:mo> <m:mn>2</m:mn> </m:mrow> </m:mrow> </m:math> {0<q<p/2} , non-negative solutions to the singular diffusion equation with gradient absorption <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:mrow> <m:msub> <m:mo>∂</m:mo> <m:mi>t</m:mi> </m:msub> <m:mo>⁡</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>-</m:mo> <m:mrow> <m:msub> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> </m:mrow> <m:mo>+</m:mo> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mrow> <m:mo>∇</m:mo> <m:mo>⁡</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mi>q</m:mi> </m:msup> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo mathvariant="italic" separator="true"> </m:mo> <m:mrow> <m:mrow> <m:mtext>in </m:mtext> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi mathvariant="normal">∞</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>×</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> </m:mrow> </m:mrow> </m:mrow> </m:math> ∂ₜu-Δₚu+|∇ u|q=0 (0,∞)×% RN vanish after a finite time. This phenomenon is usually referred to as finite-time extinction and takes place provided the initial condition <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>u</m:mi> <m:mn>0</m:mn> </m:msub> </m:math> {u₀} decays sufficiently rapidly as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mo>→</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:math> {|x|→∞} . On the one hand, the optimal decay of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>u</m:mi> <m:mn>0</m:mn> </m:msub> </m:math> {u₀} at infinity guaranteeing the occurrence of finite-time extinction is identified. On the other hand, assuming further that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo><</m:mo> <m:mi>q</m:mi> <m:mo><</m:mo> <m:mrow> <m:mi>p</m:mi> <m:mo>/</m:mo> <m:mn>2</m:mn> </m:mrow> </m:mrow> </m:math> {p-1<q<p/2}</jats:t