Dynamics of complex singularities in 1D nonlinear parabolic PDE's (Q2759173)

The author considers a local-in-time smoothing of a special class of nonlinear parabolic PDE. In order to indicate a finite-time-blow-up at a time \(T\) the loss of analyticity at \(T\) caused by complex singularities is studied. If \(S_\tau= \{z\in \mathbb{C}:|Im z|<\tau\}\) is a strip in the complex plane it is considered the scale of weighted Bergman spaces \(B^p_\alpha (S_\pi)\), where this space is defined by all in \(S_\tau\) analytic functions \(f\) with NEWLINE\[NEWLINE\int^1_0 (1-r)^\alpha M^p_p(r,f) dr<\infty \text{ and }M_p(r,f)= \left( {1\over 2\pi} \int^{2\pi}_0 \bigl(f(re^{i\varphi}) \bigr)^pd \varphi\right)^{1/ p}.NEWLINE\]NEWLINE For the nonlinear parabolic PDE with a quadratic nonlinearity \(u_t+ u_{xx xx} =u^2\) the following main result can be formulated: Theorem 0.1 Let \(u_0 \in B^2_\alpha (S_{\tau_0})\) for some \(\tau_0>0\), \(\alpha\leq 3\), then there exists a real number \(T>0\) such that \(u(t)\in B^2_\alpha (S_{\tau_0 +t})\) for all \(t\in [0,T]\).