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An inhomogeneous porous medium equation with non-integrable data: asymptotics

Matteo Muratori, Troy Petitt, Fernando Quirós

Abstract

We investigate the asymptotic behavior as $t\to+\infty$ of solutions to a weighted porous medium equation in $ \mathbb{R}^N $, whose weight $ρ(x)$ behaves at spatial infinity like $ |x|^{-γ} $ with subcritical power, namely $ γ\in [0,2) $. Inspired by some results by Alikakos-Rostamian and Kamin-Ughi from the 1980s on the unweighted problem, we focus on solutions whose initial data $u_0(x)$ are not globally integrable with respect to the weight and behave at infinity like $ |x|^{-α} $, for $α\in(0,N-γ)$. In the special case $ ρ(x)=|x|^{-γ} $ and $ u_0(x)=|x|^{-α} $ we show that self-similar solutions of Barenblatt type, i.e. reminiscent of the usual source-type solutions, still exist, although they are no longer compactly supported. Moreover, they exhibit a transition phenomenon which is new even for the unweighted equation. We prove that such self-similar solutions are attractors for the original problem, and convergence takes place globally in suitable weighted $ L^p $ spaces for $p\in[1,\infty)$ and even globally in $L^\infty$ under some mild additional regularity assumptions on the weight. Among the fundamental tools that we exploit, it is worth mentioning a global smoothing effect for non-integrable data.

An inhomogeneous porous medium equation with non-integrable data: asymptotics

Abstract

We investigate the asymptotic behavior as of solutions to a weighted porous medium equation in , whose weight behaves at spatial infinity like with subcritical power, namely . Inspired by some results by Alikakos-Rostamian and Kamin-Ughi from the 1980s on the unweighted problem, we focus on solutions whose initial data are not globally integrable with respect to the weight and behave at infinity like , for . In the special case and we show that self-similar solutions of Barenblatt type, i.e. reminiscent of the usual source-type solutions, still exist, although they are no longer compactly supported. Moreover, they exhibit a transition phenomenon which is new even for the unweighted equation. We prove that such self-similar solutions are attractors for the original problem, and convergence takes place globally in suitable weighted spaces for and even globally in under some mild additional regularity assumptions on the weight. Among the fundamental tools that we exploit, it is worth mentioning a global smoothing effect for non-integrable data.
Paper Structure (18 sections, 18 theorems, 289 equations)

This paper contains 18 sections, 18 theorems, 289 equations.

Table of Contents

  1. Introduction
  2. Preliminary material and statements of the main results
  3. Conditions on the weight
  4. A key functional space
  5. Assumptions on the initial data
  6. The non-integrable self-similar solution
  7. Well-posedness and setup for the asymptotic analysis
  8. Main results and paper organization
  9. A priori estimates
  10. Smoothing and stability estimates
  11. Two nonstandard Cauchy estimates
  12. Asymptotic convergence
  13. Detailed ODE analysis and self-similarity
  14. Reduction to an unweighted ODE and main result
  15. Existence of local solutions
  16. ...and 3 more sections

Key Result

Theorem 2.2

Let $N\geq3$, $m>1$, $\gamma \in [0,2)$, $\alpha \in (0,N-\gamma)$ and $b,c>0$. Let $g_\alpha \in C^2((0,+\infty)) \cap C([0,+\infty))$ be the solution of ode1. Then the unique solution of problem singularWPME is which satisfies and the two-sided estimate for some constants $c_1,c_2>0$ depending only on $N,m,\gamma,\alpha,c$. Moreover, the following dichotomy holds:

Theorems & Definitions (44)

  • Remark 2.1
  • Theorem 2.2: Non-integrable self-similar solutions
  • Remark 2.3: A noteworthy transition effect
  • Definition 2.4
  • Theorem 2.5: Global weighted $L^p$ convergence
  • Corollary 2.6
  • Remark 2.7
  • Theorem 2.8: Global uniform convergence
  • Proposition 2.9
  • Remark 2.10: On low dimensions
  • ...and 34 more