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Boundedness of solutions of nonautonomous degenerate logistic equations

José M. Arrieta, Marcos Molina-Rodríguez, Lucas A. Santos

Abstract

In this work we analyze the boundedness properties of the solutions of a nonautonomous parabolic degenerate logistic equation in a bounded domain. The equation is degenerate in the sense that the logistic nonlinearity vanishes in a moving region, $K(t)$, inside the domain. The boundedness character of the solutions depends not only on, roughly speaking, the first eigenvalue of the Laplace operator in $K(t)$ but also on the way this moving set evolves inside the domain and in particular on the speed at which it moves.

Boundedness of solutions of nonautonomous degenerate logistic equations

Abstract

In this work we analyze the boundedness properties of the solutions of a nonautonomous parabolic degenerate logistic equation in a bounded domain. The equation is degenerate in the sense that the logistic nonlinearity vanishes in a moving region, , inside the domain. The boundedness character of the solutions depends not only on, roughly speaking, the first eigenvalue of the Laplace operator in but also on the way this moving set evolves inside the domain and in particular on the speed at which it moves.
Paper Structure (11 sections, 21 theorems, 157 equations, 13 figures)

This paper contains 11 sections, 21 theorems, 157 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Local existence
  4. Positivity, Comparison and Global Existence of Solutions
  5. Initial Data Independence
  6. A general result
  7. Boundedness
  8. Estimates of the solution locally in space and time
  9. Boundedness for $\lambda < \lambda_0(K(t)), \forall t$
  10. Boundedness for a jumping $K(t)$
  11. Unboundedness

Key Result

Lemma 2.1

Let $f_i(t,x,u)=\lambda u - n_i(t,x) |u|^{\rho-1} u$ with $n_i\in L^\infty(\mathbb{R}\times \Omega)$, $n_1(t,x)\geq n_2(t,x)\geq 0$ a.e. $t\in (t_0,+\infty)$, a.e. $x\in \bar{\Omega}$. Then, for as long as solutions exist, we have that: Moreover, if $\tilde{\Omega}\subset \Omega$ then for any initial condition $u_0\geq 0$ defined in $\Omega$ if we denote by $\tilde{u}_0= {u_0}_{|\tilde{\Omega}}$

Figures (13)

  • Figure 1: A representation of the cases shown in \ref{['introeq2_0']}.
  • Figure 2: Domain $\Omega$ and a continuously moving $K(t)$.
  • Figure 3: Impact of the time-dependency of $K(t)$ on the variety of scenarios.
  • Figure 4: Domain $\Omega$ with a rotating sector.
  • Figure 5: Domain $\Omega$ with "jumping" subdomains $K_0$ and $K_1$.
  • ...and 8 more figures

Theorems & Definitions (32)

  • Lemma 2.1
  • Corollary 2.2: Global Existence
  • Remark 2.3
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Proposition 3.4
  • Lemma 4.1
  • Remark 4.2
  • Proposition 4.3
  • ...and 22 more