The inhomogeneous PME in several space dimensions. Existence and uniqueness of finite energy solutions
Guillermo Reyes Juan-Luis Vázquez
We study the questions of existence and uniqueness of non-negative solutions to the Cauchy problem

$\rho(x)\partial_t u= \Delta u^m\qquad$ in $Q$:$=\mathbb R^n\times\mathbb R_+$

$u(x, 0)=u_0$

in dimensions $n\ge 3$. We deal with a class of solutions having finite energy

$E(t)=\int_{\mathbb R^n} \rho(x)u(x,t) dx$

for all $t\ge 0$. We assume that $m> 1$ (slow diffusion) and the density $\rho(x)$ is positive, bounded and smooth. We prove existence of weak solutions starting from data $u_0\ge 0$ with finite energy. We show that uniqueness takes place if $\rho$ has a moderate decay as $|x|\to\infty$ that essentially amounts to the condition $\rho\notin L^1(\mathbb R^n)$. We also identify conditions on the density that guarantee finite speed of propagation and energy conservation, $E(t)=$const. Our results are based on a new a priori estimate of the solutions.

keywords: semigroup solution. Inhomogeneous porous medium flow classes of uniqueness Cauchy problem
The Cauchy problem for the inhomogeneous porous medium equation
Guillermo Reyes Juan-Luis Vázquez
We consider the initial value problem for the filtration equation in an inhomogeneous medium
$p(x)u_t = \Delta u^m, m>1$.

The equation is posed in the whole space $\mathbb R^n$ , $n \geq 2$, for $0 < t < \infty$; $p(x)$ is a positive and bounded function with a certain behaviour at infinity. We take initial data $u(x,0) = u_0(x) \geq 0$, and prove that this problem is well-posed in the class of solutions with finite "energy", that is, in the weighted space $L^1_p$, thus completing previous work of several authors on the issue. Indeed, it generates a contraction semigroup.
    We also study the asymptotic behaviour of solutions in two space dimensions when $p$ decays like a non-integrable power as $|x| \rightarrow \infty$ : $p(x)$ $|x|^\alpha$ ~ $1$ with $\alpha \epsilon (0,2)$ (infinite mass medium). We show that the intermediate asymptotics is given by the unique selfsimilar solution $U_2(x, t; E)$ of the singular problem
$ |x|^{- \alpha} u_t = \Delta u_m$ in $\mathbb R^2 \times \mathbb R_+ $
$ |x|^{- \alpha} u(x,0) = E\delta(x), E = ||u_0||_{L^1_p}$
keywords: Degenerate parabolic equations intermediate asymptotics. inhomogeneous media
Asymptotic behavior of the solutions of the inhomogeneous Porous Medium Equation with critical vanishing density
Sofía Nieto Guillermo Reyes
We study the long-time behavior of non-negative, finite-energy solutions to the initial value problem for the Porous Medium Equation with variable density, i.e. solutions of the problem \begin{eqnarray*} \rho (x) \partial_{t} u = \Delta u^{m}, \quad in \quad Q:= R^n \times R_+, \\ u(x,0)=u_{0}(x), \quad in\quad R^n, \end{eqnarray*} where $m>1$, $u_0\in L^1(R^n, \rho(x)dx)$ and $n\ge 3$. We assume that $\rho (x)\sim C|x|^{-2}$ as $|x|\to\infty$ in $R^n$. Such a decay rate turns out to be critical. We show that the limit behavior can be described in terms of a family of source-type solutions of the associated singular equation $|x|^{-2}u_t = \Delta u^{m}$. The latter have a self-similar structure and exhibit a logarithmic singularity at the origin.
keywords: Inhomogeneous porous media intermediate asymptotics scaling methods.
Long time behavior for the inhomogeneous PME in a medium with slowly decaying density
Guillermo Reyes Juan-Luis Vázquez
We study the long-time behavior of non-negative solutions to the Cauchy problem

(P) $\qquad \rho(x) \partial_t u= \Delta u^m\qquad$ in $Q$:$=\mathbb R^n\times\mathbb R_+$

$u(x, 0)=u_0$

in dimensions $n\ge 3$. We assume that $m> 1$ (slow diffusion) and $\rho(x)$ is positive, bounded and behaves like $\rho(x)$~$|x|^{-\gamma}$ as $|x|\to\infty$, with $0\le \gamma<2$. The data $u_0$ are assumed to be nonnegative and such that $\int \rho(x)u_0 dx< \infty$.
Our asymptotic analysis leads to the associated singular equation $|x|^{-\gamma}u_t= \Delta u^m,$ which admits a one-parameter family of selfsimilar solutions $ U_E(x,t)=t^{-\alpha}F_E(xt^{-\beta})$, $E>0$, which are source-type in the sense that $|x|^{-\gamma}u(x,0)=E\delta(x)$. We show that these solutions provide the first term in the asymptotic expansion of generic solutions to problem (P) for large times, both in the weighted $L^1$ sense

$u(t)=U_E(t)+o(1)\qquad$ in $L^1_\rho$

and in the uniform sense $u(t)=U_E(t)+o(t^{-\alpha})$ in $L^\infty $ as $t\to \infty$ for the explicit rate $\alpha=\alpha(m,n,\gamma)>0$ which is precisely the time-decay rate of $U_E$. For a given solution, the proper choice of the parameter is $E=\int \rho(x)u_0 dx$.

keywords: intermediate asymptotics scaling methods Inhomogeneous porous medium flow
Existence of multiple solutions for a semilinear problem and a counterexample by E. N. Dancer
Alfonso Castro Guillermo Reyes
As shown by Dancer's counterexample [12], one cannot expect to have (generically) more than five solutions to the semilinear boundary value problem
$\mathbf{(P)}\qquad \left\{\begin{array}{ ll}-\Delta u =f(u, x)\quad&\text{in}B:=\{x\in\mathbb{R}^n:\, |x|<1\}, \\u =0&\text{on}\partial B\end{array}\right.$
when $f(0, x)=0$ and $\partial f/\partial u$ crosses the first two eigenvalues of $-\Delta$ with Dirichlet boundary conditions on $\partial B$, as $|u|$ grows from $0$ to $\infty$. Despite the fact that the nonlinearity $ f$ in [12] can be taken "arbitrarily close" to autonomous, we prove that the dependence of $f$ on $x$ is indeed essential in the arguments. More precisely, we show that (P) with $f$ independent of $x$ and satisfying the assumptions in [12] has at least six, and generically seven solutions. Under more stringent conditions on the non-linearity, we prove that there are up to nine solutions. Importantly, we do not assume any symmetry on $f$ for any of our results.
keywords: Semilinear elliptic equation Morse index Leray-Schauder degree Lyapunov-Schmidt reduction
Nonuniqueness of solutions to semilinear parabolic equations with singular coefficients
Luisa Moschini Guillermo Reyes Alberto Tesei
We prove nonuniqueness of solutions of the Cauchy problem for a semilinear parabolic equation with inverse-square potential in certain Lebesgue spaces. The nonuniqueness results proved in [5] are the limiting case of the present ones as the strength of the potential vanishes. Similar results are obtained for a related semilinear parabolic equation with singular coefficients. The proofs rely on investigating by variational methods in suitable weighted Sobolev spaces the equation satisfied by the profile of a radial similarity solution.
keywords: singular coefficients inverse-square potential nonuniqueness of solutions weighted Sobolev spaces. Semilinear parabolic equations
The Cauchy problem for a nonhomogeneous heat equation with reaction
Arturo de Pablo Guillermo Reyes Ariel Sánchez
We study the behaviour of the solutions to the Cauchy problem $$ \left\{\begin{array}{ll} \rho(x)u_t=\Delta u+u^p,&\quad x\in\mathbb{R}^N ,\;t\in(0,T),\\ u(x,\, 0)=u_0(x),&\quad x\in\mathbb{R}^N , \end{array}\right. $$ with $p>0$ and a positive density $\rho$ satisfying $\rho(x)\sim|x|^{-\sigma}$ for large $|x|$, $0<\sigma<2< N$. We consider both the cases of a bounded density and the singular density $\rho(x)=|x|^{-\sigma}$. We are interested in describing sharp decay conditions on the data at infinity that guarantee local/global existence of solutions, which are unique in classes of functions with the same decay. We prove that larger data give rise to instantaneous complete blow-up. We also deal with the occurrence of finite-time blow-up. We prove that the global existence exponent is $p_0=1$, while the Fujita exponent depends on $\sigma$, namely $p_c=1+\frac2{N-\sigma}$.
    We show that instantaneous blow-up at space infinity takes place when $p\le1$.
    We also briefly discuss the case $2<\sigma< N$: we prove that the Fujita exponent in this case does not depend on $\sigma$, $\tilde{p}_c=1+\frac2{N-2}$, and for initial values not too small at infinity a phenomenon of instantaneous complete blow-up occurs in the range $1< p < \tilde{p}_c$
keywords: blow-up Reaction-diffusion equations well-posedness initial value problem critical exponents.
Long time behavior for the inhomogeneous PME in a medium with rapidly decaying density
Shoshana Kamin Guillermo Reyes Juan Luis Vázquez
We study the long-time behavior of nonnegative solutions to the Cauchy problem

$ \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.

keywords: matched asymptotics. scaling methods Inhomogeneous porous media

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