Blow-up and Decay of Global Solutions for a Free Boundary Problem with Competing Nonlocal Nonlinearity and Absorption

In this paper, we are concerned with the characterization of the blow-up and global solutions for free boundary parabolic equation with competing nonlocal nonlinearity and absorption

$$\begin{aligned} u_t(t,x) = u_{xx}(t,x) + u^{p}(t,x) \int _0^{s(t)}u^{q}(t,x)dx -\gamma u^\alpha (t,x), t>0,\ 0<x <s(t), \end{aligned}$$

(1)

where \(q, \alpha \ge 1\), \(p=0\) or \(p\ge 1\) and \(\gamma > 0\) are given constants. This study is motivated from the works [Abdelhedi and Zaag: J. Differential Equations 272, 1–45, (2021); Souplet: SIAM J. Math. Anal. 29, 1301–1334, (1998); Zhou and Lin: J. Funct. Anal. 262, 3409–3429, (2012)] arisen from the investigation of many physical and biological phenomena such as population dynamics, combustion theory, phase separation in binary mixtures, theory of nuclear reactor dynamics... We first prove the local existence, uniqueness and stability of solution thanks to the “extension trick" introduced in [Du et al.: Math. Ann. 386(3-4), 2061–2106, (2023); Wang and Du: Discrete Contin. Dyn. Syst. Ser. B 26(4), 2201–2238, (2021)]. Second, by improving the comparison principle used in [Souplet: SIAM J. Math. Anal. 29, 1301–1334, (1998); Zhou and Lin: J. Funct. Anal. 262, 3409–3429, (2012)], we find a sharp criterion characterizing the blow-up and global solutions of (1) in term of power coefficients and initial data. We further show that there exists a threshold for the initial data that determines whether blow-up, global fast, or global slow solutions occur and find an upper bound for the existence time of blow-up solutions in two different cases \(\alpha >1\) and \(\alpha =1\). Our proofs are mainly based on the comparison principle by improving several techniques in previous works, combined new idea to handle differential inequalities and unified local existence theory for nonlocal semilinear parabolic equations.