Concavity of solutions of the porous medium equation

Abstract:

We consider the problem \[ \left ( {\text {P}} \right )\quad \quad \left \{ {\begin {array}{*{20}{c}} {{u_t} = {{({u^m})}_{xx}},} \\ {u(x,0) = {u_0}(x)} \\ \end {array} } \right .\quad \begin {array}{*{20}{c}} {{\text {with}} x \in {\mathbf {R}}, t > 0} \\ {{\text {for}} x \in {\mathbf {R}}} \\ \end {array} \] where $m > 1$ and ${u_0}$ is a continuous, nonnegative function that vanishes outside an interval $(a, b)$ and such that $(u_0^{m - 1})'' \leq - C \leq 0$ in $(a, b)$. Using a Trotter-Kato formula we show that the solution conserves the concavity in time: for every $t > 0, u(x,t)$ vanishes outside an interval $\Omega (t) = ({}_{\zeta 1}(t), {}_{\zeta 2}(t))$ and \[ {({u^{m - 1}})_{xx}} \leq - \frac {C} {{1 + C(m(m + 1)/(m - 1))t}}\] in $\Omega (t)$. Consequently the interfaces $x{ = _{\zeta i}}(t)$, $i = 1, 2$, are concave curves. These results also give precise information about the large time behavior of solutions and interfaces.