Short-time asymptotic expansions of semilinear evolution equations (Q2799613)

Short-time asymptotic expansions are constructed for solutions of abstract semilinear evolution equations of the form NEWLINE\[NEWLINE\begin{aligned} \partial_tu(t) &=A u(t)+F(t,u(t), T_1u(t),\dots, T_nu(t)), \\ u(0)&=f,\end{aligned}NEWLINE\]NEWLINENEWLINENEWLINEin a suitable Banach algebra \(X\). Here, the linear operator \(A\) is the generator of an analytic semigroup \(G(t)\) in \(X\), and \(T_1,\dots,T_n\) are linear operators, essentially representing differentiation.NEWLINENEWLINEAssuming that \(G(t)\) possesses an asymptotic expansion as \(t\rightarrow 0^+\) of the form \( G(t)\sim G_0(t)+G_1(t)t+G_2(t)t^2+\cdots, \) where \(G_j\) are bounded linear operators, and the nonlinearity \(F\) is analytic in its arguments, asymptotic expansions \(u(t)\sim u_0(t)+u_1(t)t+u_2(t)t^2+\cdots\) of the solution \(u\) are derived such that also \(T_iu(t)\sim T_iu_0(t)+T_iu_1(t)t+T_iu_2(t)t^2+\cdots\) holds as \(t\rightarrow 0^+\). The proof is based on the symbolic calculus of pseudo-differential operators and heat kernel expansions, and the functions \(u_i\) are explicit and recursively determined.NEWLINENEWLINEAs an application short time asymptotic expansions for the solution of a semilinear parabolic equation on a bounded domain with nonlinear forcing term are constructed, and for stochastic processes that are solutions of backward stochastic differential equations.