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
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
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

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