Long-time dynamics of an integro-differential equation describing the evolution of a spherical flame (Q1431097)

The author investigate a flame ball model derived in \textit{G. Joulin} [Combust. Sci. Tech. 43, 99--113 (1985)]: \[ R\partial_{1/2}R=R\,\text{Log}\;R+Eq(t)-\lambda R^3,\quad R(0)=0,\quad \partial_{1/2}R=\frac{1}{\sqrt{\pi}}\int_0^t\frac{\dot R(s)}{\sqrt{t-s}}.\eqno(1) \] It describes the evolution of a spherical flame, initiated by a point source energy input \(Eq(t)\), and at which heat losses are applied, represented here by the parameter \(\lambda\), in the burnt gases. The intensity of this energy input is measured by the positive constant \(E\), and its time evolution is described by the function \(q\). \(q\) is a smooth, nonnegative function, with connected support and unit total mass, and its initial values satisfy the following assumption: \[ q(t)\sim q_0t^{\beta}\quad \text{with }0\leq \beta< 1/2. \] Also, \(q\) tends to \(0\) as \(t\to +\infty\). In \textit{J. Audounet} \textit{V. Giovangigli} and \textit{J.-M. Roquejoffre} [Physica D 121, No. 3--4, 295--316 (1998; Zbl 0938.80003)], the case with no heat losses (\(\lambda=0\)) has been studied, and the threshold phenomenon has been proved: there exists a critical energy \(E_{cr}(q)>0\) such that, if \(E<E_{cr}(q)\), the flame quenches; if \(E>E_{cr}(q)\), it propagates; if \(E=E_{cr}(q)\), the flame stabilized to \(1\). The goal of the present article is to find a similar result when \(\lambda >0\), and to study the asymptotic expressions of \(R(t)\) as well. Consider the equation \[ \text{Log }R=\lambda R^2.\eqno(2) \] If \(\lambda=\lambda_{cr}=1/2e\), (2) has a unique solution \(R_{cr}=\sqrt{e}\); if \(\lambda>\lambda_{cr}=1/2e\), (2) has no solution; if \(\lambda<\lambda_{cr}=1/2e\), (2) has two solution: \(R_1<R_{cr}<R_2\). There are three main theorems in this article, which gives one of the conclusions: {(i)} \(\lim\limits_{t\to \infty}R(t)=0\) or \(R_i\;(i=1,2)\); {(ii)} \(\lim\limits_{t\to \infty}R(t)=0\) or \(\lim\limits_{t\to t_{max}}R(t)=0\); {(iii)} \(\lim\limits_{t\to \infty}R(t)=0\) or \(\lim\limits_{t\to t_{\max}}R(t)=0\) or \(\lim\limits_{t\to \infty}R(t)=R_{cr}\); corresponding to the cases: \(\lambda<\lambda_{cr}\), \(\lambda>\lambda_{cr}\) and \(\lambda=\lambda_{cr}\), respectively. There are two other theorems about the asymptotic expressions of \(R(t)\). The above work has been extended to the equation \[ R\partial_{1/2}R=f(R)+Eq(t) \] as well, where \(f\) is polynomial, negative at \(x=0\) and has several zeros.