Free boundary problems with nonlocal and local diffusions II: Spreading-vanishing and long-time behavior

TL;DR

This paper investigates the long-term behavior of solutions to free boundary problems involving both nonlocal and local diffusions, establishing conditions for spreading or vanishing and analyzing the solution's asymptotic behavior.

Contribution

It introduces criteria for spreading and vanishing in free boundary problems with mixed diffusions and characterizes the long-time dynamics when spreading occurs.

Findings

Established a spreading-vanishing dichotomy.

Derived criteria for spreading and vanishing.

Analyzed the asymptotic behavior of solutions during spreading.

Abstract

This is part II of our study on the free boundary problems with nonlocal and local diffusions. In part I, we obtained the existence, uniqueness, regularity and estimates of global solution. In part II here, we show a spreading-vanishing dichotomy, and provide the criteria of spreading and vanishing, as well as the long time behavior of solution when spreading happens.