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Nonlocal spatially inhomogeneous Hamilton-Jacobi equation with unusual free boundary
Long time behavior for the inhomogeneous PME in a medium with rapidly decaying density
| 1. | School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel |
| 2. | Departamento de Matemática Aplicada, E.T.S.I. de Caminos, Canales y Puertos, Universidad Politécnica de Madrid. 28040 Madrid |
| 3. | Departamento de Matemáticas and ICMAT. Universidad Autónoma de Madrid, Cantoblanco. 28049 Madrid, Spain |
$ \rho(x)\, \partial_t u= \Delta u^m $ in $Q$:$=\mathbb{R}^N\times\mathbb{R}_+$
$u(x, 0)=u_0 $ in $\mathbb{R}^N$
in dimensions $ N\ge 3$. We assume that $ m> 1 $ and $
\rho(x) $ is positive and bounded with $ \rho(x)\le
C|x|^{-\gamma} $ as $ |x|\to\infty$ with $\gamma>2$. The
initial data $u_0$ are nonnegative and have finite energy, i.e.,
$ \int \rho(x)u_0 dx< \infty$.
We show that in this case nontrivial solutions to the problem have
a long-time universal behavior in separate variables of the form
$u(x,t)$~$ t^{-1/(m-1)}W(x),$
where $V=W^m$ is the unique bounded, positive solution of the
sublinear elliptic equation $-\Delta V=c\,\rho(x)V^{1/m}$, in
$\mathbb{R}^N$ vanishing as $|x|\to\infty$; $c=1/(m-1)$. Such a
behavior of $u$ is typical of Dirichlet problems on bounded
domains with zero boundary data. It strongly departs from the
behavior in the case of slowly decaying densities, $\rho(x)$~$
|x|^{-\gamma}$ as $|x|\to\infty$ with 0 ≤ $ \gamma<2$, previously
studied by the authors.
If $\rho(x)$ has an intermediate decay, $\rho$~$ |x|^{-\gamma}$
as $|x|\to\infty$ with $2<\gamma<\gamma_2$:$=N-(N-2)/m$, solutions
still enjoy the finite propagation property (as in the case of
lower $\gamma$). In this range a more precise description may be
given at the diffusive scale in terms of source-type solutions
$U(x,t)$ of the related singular equation $|x|^{-\gamma}u_t=
\Delta u^m$. Thus in this range we have two different
space-time scales in which the behavior of solutions is
non-trivial. The corresponding results complement each other and
agree in the intermediate region where both apply, thus providing
an example of matched asymptotics.
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