Asymptotic analysis of a non-linear non-local integro-differential equation arising from bosonic quantum field dynamics
Let \(d\in\mathbb{N}\setminus\{0\}\), \(h\) a strictly positive parameter, \(S(\mathbb{R}^d,C)\) the Schwartz space, \(f\) a function defined on \(S(\mathbb{R}^d,C)\), \(\vec\xi_0\in\mathbb{R}^d\setminus\{0\}\), \(X\subseteq R\), \(Y\subseteq\mathbb{R}^d\), \({\mathcal B}(X,Y)\) the space of bounded functions from \(X\) to \(Y\), \({\mathcal C}(X,Y)\) the space of continuous functions from \(X\) to \(Y\), \(\vec u\in{\mathcal B}(X,Y)\), \(X'\subseteq X\), \(\| u\|_{\infty,X'}= \sup_{x\in X'} p|\vec u(x)|\), \(\not\mkern-7mu\int^b_a\vec u(\sigma)\,d\sigma= (b-a)^{-1} \int^b_a\vec u(\sigma)\,d\sigma\) and the functions \({\mathcal F}^{(h)}\) and \(F^{(0)}\) from \({\mathcal B} (\mathbb{R}^+,\mathbb{R}^d)\) to \({\mathcal C}(\mathbb{R}^+,\mathbb{R}^d)\) defined by \[ {\mathcal F}^{(h)}(\vec u)(t)= -2{\mathcal R} \int_{\mathbb{R}^d} \int^{th^{-1}}_0 \exp\Biggl(-ir\Biggl(\vec\eta^2- 2\vec\eta \not\mkern-7mu\int^t_{t-hr}\vec u_\sigma d\sigma\Biggr)\Biggr)\vec\eta|f(\vec\eta)|^2\,drd\vec\eta, \] \[ F^{(0)}(\vec u)(t)= -2{\mathcal R} \int^{+\infty}_0 \int_{\mathbb{R}^d}\exp(-ir(\vec\eta^2- 2\vec\eta\vec u_t))\vec\eta|f(\vec\eta)|^2\, d\vec\eta dr. \] The author considers the nonlinear integro-differential equation \[ {d\over dt}\vec\xi^{(h)}_t={\mathcal F}^{(h)}(\vec\xi^{(h)}_t),\;h>0,\qquad \vec\xi^{(h)}_{t=0}= \vec\xi_0,\;h>0,\tag{1} \] and the limit equation as \(h\to 0\) \[ {d\over dt}\vec\xi^{(0)}_t= F^{(0)}(\vec\xi^{(0)}_t),\quad \vec\xi^{(0)}_{t=0}= \vec\xi_0.\tag{2} \] The author shows the existence and uniqueness of strong global solutions for these equations and a result of uniform convergence on every compact interval of the solution of the one parameter family towards the solution of the limit equation. (1) admits a unique solution \(\vec\xi^{(h)}\in C^1(\mathbb{R}^+_t, \mathbb{R}^d)\) and (2) admits a maximal solution \(\vec\xi^{(0)}\in C^1([0, T_{\max}),\mathbb{R}^d)\) with \(T_{\max}> 0\). The author proves that (1) has a unique solution on \(C^1(\mathbb{R}^+,\mathbb{R}^d)\) and if \(d\geq 3\), (2) has a unique maximal solution \(\vec\xi^{(0)}\) and \(\lim_{t\to+\infty} \vec\xi^{(0)}_t= 0\). Moreover, for \(d\geq 3\) and \(T\in (0,T_{\max})\) then \((\vec\xi^{(h)}_t)_{t\in [0,T]}\) converges uniformly to \((\vec\xi^{(0)}_t)_{t\in [0,T]}\) as \(h\to 0\).
scientific article; zbMATH DE number 6358112