Global asymptotic stability of bifurcating, positive equilibria of p-Laplacian boundary value problems with p-concave nonlinearities

Abstract

We consider the parabolic, initial value problem $$v_t={\Delta }_p(v)+\lambda g(x,v){\varphi }_p(v),\text{in Ω×(0,∞),}$$ \[ v =0, \text{in $\partial \Omega \times (0,\infty ),$}\tag{IVP} v =v_0\ge0, \text{in $\Omega \times \{0\},$} \] where $\Omega$ is a bounded domain in ${\mathbb{R}}^N$, for some integer $N\ge 1$, with smooth boundary $\partial \Omega$, ${\varphi }_p(s):=|s{|}^{p-1}\mathrm{sgn}s$, $s\in \mathbb{R}$, ${\Delta }_p$ denotes the $p$-Laplacian, with $p>\max \{2,N\}$, $v_0\in C^0(\overline{\Omega })$, and $\lambda >0$. The function $g:\overline{\Omega }\times [0,\infty )\to (0,\infty )$ is $C^0$ and, for each $x\in \overline{\Omega }$, the function $g(x,\cdot ):[0,\infty )\to (0,\infty )$ is Lipschitz continuous and strictly decreasing. Clearly, (IVP) has the trivial solution $v\equiv 0$, for all $\lambda >0$. In addition, there exists $0<{\lambda }_{\min }(g)<{\lambda }_{\max }(g)$ (${\lambda }_{\max }(g)$ may be $\infty$) such that: $(a)$ if $\lambda \notin ({\lambda }_{\min }(g),{\lambda }_{\max }(g))$ then (IVP) has no non-trivial, positive equilibrium; $(b)$ if $\lambda \in ({\lambda }_{\min }(g),{\lambda }_{\max }(g))$ then (IVP) has a unique, non-trivial, positive equilibrium $e_{\lambda }\in W_0^{1,p}(\Omega )$. We prove the following results on the positive solutions of (IVP): $(a)$ if $0<\lambda <{\lambda }_{\min }(g)$ then the trivial solution is globally asymptotically stable; $(b)$ if ${\lambda }_{\min }(g)<\lambda <{\lambda }_{\max }(g)$ then $e_{\lambda }$ is globally asymptotically stable; $(c)$ if ${\lambda }_{\max }(g)<\lambda$ then any non-trivial solution blows up in finite time.