Optimal regularity and fine asymptotics for the porous medium equation in bounded domains

We prove the optimal global regularity of nonnegative solutions to the porous medium equation in smooth bounded domains with the zero Dirichlet boundary condition after certain waiting time <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>T</m:mi> <m:mo>*</m:mo> </m:msup> </m:math> {T*} . More precisely, we show that solutions are <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>C</m:mi> <m:mrow> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mi>α</m:mi> </m:mrow> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mover accent="true"> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>¯</m:mo> </m:mover> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {C2,α(Ω̄)} in space, with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>α</m:mi> <m:mo>=</m:mo> <m:mfrac> <m:mn>1</m:mn> <m:mi>m</m:mi> </m:mfrac> </m:mrow> </m:math> {α=1/m} , and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>C</m:mi> <m:mi mathvariant="normal">∞</m:mi> </m:msup> </m:math> {C∞} in time (uniformly in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:mover accent="true"> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>¯</m:mo> </m:mover> </m:mrow> </m:math> {x∈Ω̄} ), for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>t</m:mi> <m:mo>></m:mo> <m:msup> <m:mi>T</m:mi> <m:mo>*</m:mo> </m:msup> </m:mrow> </m:math> {t>T*} . Furthermore, this allows us to refine the asymptotics of solutions for large times, improving the best known results so far in two ways: we establish a faster rate of convergence <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>O</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi>t</m:mi> <m:mrow> <m:mrow> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>-</m:mo> <m:mi>γ</m:mi> </m:mrow> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {O(t-1-γ)} , and we prove that the convergence holds in the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>C</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>α</m:mi> </m:mrow> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mover accent="true"> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>¯</m:mo> </m:mover> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {C1,α(Ω̄)} topology.