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

We consider the parabolic, initial value problem $$v_t={\mathrm{\Delta }}_p(v)+\lambda g(x,v){\varphi }_p(v),\text{in }\mathrm{\Omega }\times (0,\mathrm{\infty }),$$ $$\begin{array}{cccc}& v=0,\text{in }\mathrm{\partial }\mathrm{\Omega }\times (0,\mathrm{\infty }),v=v_0\ge 0,\text{in }\mathrm{\Omega }\times \{0\},& & \text{(IVP)}\end{array}$$ where $\mathrm{\Omega }$ is a bounded domain in ${\mathbb R}^N$,forsomeinteger$N\ge1$,withsmoothboundary$\partial\Omega$,$\phi_p(s):=|s|^{p-1} {\rm sgn}s$,$s\in{\mathbb R}$,$\Delta_p$denotesthe$p$-Laplacian,with$p>\max{2,N}$,$v_0\in C^0(\overline{\Omega})$,and$\lambda>0$\mathrm{.}Thefunction$g:\overline{\Omega } \times [0,\infty)\to(0,\infty)$is$C^0$and,foreach$x\in\overline{\Omega }$,thefunction$g(x,\cdot):[0,\infty)\to(0,\infty)$ is Lipschitz continuous and strictly decreasing. Clearly, (IVP) has the trivial solution $v\equiv 0$, for all $\lambda >0$. In addition, there exists $0<{\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}(g)<{\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}(g)$ (${\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}(g)$ may be $\mathrm{\infty }$) such that: $(a)$ if $\lambda \in ̸({\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}(g),{\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}(g))$ then (IVP) has no non-trivial, positive equilibrium; $(b)$ if $\lambda \in ({\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}(g),{\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}(g))$ 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 $0<\lambda <{\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}(g)$ then the trivial solution is globally asymptotically stable; $(b)$ if ${\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}(g)<\lambda <{\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}(g)$ then $e_{\lambda }$ is globally asymptotically stable; $(c)$ if ${\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}(g)<\lambda$ then any non-trivial solution blows up in finite time.