[Paper Review] Global existence and blow-up of solutions to the double nonlinear porous medium equation

Bolys Sabitbek, Berikbol T. Torebek|arXiv (Cornell University)|Apr 14, 2021

Advanced Mathematical Physics Problems28 references4 citations

TL;DR

This paper establishes global existence and finite-time blow-up of weak solutions for a double nonlinear porous medium equation with a novel nonlinearity condition. By introducing potential wells and analyzing initial energy levels, it proves that solutions blow up in finite time for negative or critical initial energy, while global existence and asymptotic decay are established for subcritical and critical cases under specific initial energy and positivity conditions on the initial data's energy functional.

ABSTRACT

In this study, we examine a double nonlinear porous medium equation subject to a novel nonlinearity condition within a bounded domain. First, we introduce the blow-up solution for the problem under consideration for the negative initial energy. By introducing a set of potential wells, we construct invariant sets of solutions for the double nonlinear porous medium equation. For subcritical and critical initial energy scenarios, we derive the global existence and asymptotic behavior of weak solutions, as well as blow-up phenomena occurring within a finite time for the positive solution to the double nonlinear porous medium equation.

Motivation & Objective

• To analyze global existence and blow-up behavior of weak solutions for a double nonlinear porous medium equation with a new nonlinearity condition.
• To extend the potential well method to the double nonlinear setting involving $p$-Laplacian and porous medium terms.
• To classify solution behavior based on initial energy levels: negative, subcritical, and critical.
• To establish asymptotic decay estimates for global solutions under critical energy conditions.
• To identify open problems, including supercritical energy and fast diffusion cases ($0 < m < 1$).

Proposed method

• Introduce a novel nonlinearity condition $(H)$ involving $f(u)$, $F(u)$, and parameters $\alpha$, $\gamma$, $\sigma$, $\beta$, and the first eigenvalue $\lambda_{1,p}$ of the $p$-Laplacian.
• Define the energy functional $J(u)$ and the Nehari functional $I(u)$ to classify initial data into different energy regimes.
• Construct potential wells and invariant sets to control solution evolution and prevent blow-up under certain initial energy and positivity conditions.
• Use the concavity method and energy estimates to prove finite-time blow-up for negative initial energy.
• Apply Sobolev-type inequalities and $L^{m+1}$-norm decay estimates to derive asymptotic behavior for global solutions.
• Analyze critical energy case $J(u_0) = d$ by distinguishing between $I(u_0) \geq 0$ and $I(u_0) < 0$ to determine global existence or blow-up.

Experimental results

Research questions

• RQ1 Under what conditions on the initial energy $J(u_0)$ does the solution to the double nonlinear porous medium equation exist globally or blow up in finite time?
• RQ2 How does the novel nonlinearity condition $(H)$, involving $\alpha$, $\gamma$, $\sigma$, and $\beta$, affect the existence and blow-up behavior of solutions?
• RQ3 Can the potential well method be extended to the double nonlinear setting involving $p$-Laplacian and porous medium terms to classify solution behavior?
• RQ4 What is the asymptotic decay rate of global weak solutions when $J(u_0) = d$ and $I(u_0) \geq 0$?
• RQ5 What are the open problems in the supercritical energy regime ($J(u_0) > d$) and for the fast diffusion case ($0 < m < 1$)?

Key findings

• Solutions blow up in finite time when $J(u_0) < 0$, regardless of the initial data's positivity, due to the concavity method and energy estimates.
• For $J(u_0) < d$ and $I(u_0) \geq 0$, global weak solutions exist and satisfy $||u(t)||_{L^{m+1}(\Omega)} \leq (||u_0||_{L^{m+1}}^{m+1-pm} + \lambda t)^{-1/(pm - m - 1)}$ for $t \geq t_1 > 0$ when $pm > m + 1$, with $\lambda > 0$.
• When $pm = m + 1$, the global solution decays exponentially: $||u(t)||_{L^{m+1}} \leq e^{-(m+1)(1 - \delta_1)Ct} ||u_0||_{L^{m+1}}$ for $t \geq t_1 > 0$, with $C > 0$.
• For $J(u_0) = d$ and $I(u_0) \geq 0$, global existence and asymptotic decay are established, with decay rates depending on the relation between $pm$ and $m+1$.
• When $J(u_0) = d$ and $I(u_0) < 0$, the solution blows up in finite time, with $\lim_{t \to T} \int_0^t \int_\Omega u^{m+1} \, dx d\tau = +\infty$ for some $T > 0$.
• The supercritical energy case ($J(u_0) > d$) and the fast diffusion case ($0 < m < 1$) remain open for future research.