Modulus of continuity in semilinear classical damped wave equations - Michael Reissig (Technische Universitat Bergakademie Freiberg)

Abstract:

In this lecture we show by examples how a modulus of continuity appears in several problems from the theory of PDE's to describe the threshold in qualitative properties of solutions. Among other things we consider the Cauchy problem for the semilinear damped wave equation

\[

u_{tt} -  \Delta u + u_t= h(u), \qquad u(0,x)=\phi(x), \qquad u_t(0,x)= \psi(x),

\]

where $h(s)=|s|^{1+ \frac2{n}}\mu(|s|)$. Here $n$ is the space dimension, $1+\frac{2}{n}$ is the Fujita exponent, and $\mu$ is a modulus of continuity. Our goal is to obtain sharp conditions on $\mu$ to obtain a threshold between global (in time) existence of small data solutions (stability of the zero solution) and blow-up behavior even of small data solutions.

Reference:

M. Ebert Rempel, G. Girardi, M. Reissig, "Critical regularity of nonlinearities in semilinear classical damped wave equations", submitted (2019).